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Properties

Label 32-150e16-1.1-c3e16-0-0
Degree 3232
Conductor 6.568×10346.568\times 10^{34}
Sign 11
Analytic cond. 1.41682×10151.41682\times 10^{15}
Root an. cond. 2.974942.97494
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 12·3-s + 24·4-s + 15·5-s + 96·6-s + 46·7-s − 32·8-s + 54·9-s − 120·10-s − 83·11-s − 288·12-s + 22·13-s − 368·14-s − 180·15-s + 16·16-s − 79·17-s − 432·18-s − 15·19-s + 360·20-s − 552·21-s + 664·22-s − 143·23-s + 384·24-s + 10·25-s − 176·26-s − 108·27-s + 1.10e3·28-s + ⋯
L(s)  = 1  − 2.82·2-s − 2.30·3-s + 3·4-s + 1.34·5-s + 6.53·6-s + 2.48·7-s − 1.41·8-s + 2·9-s − 3.79·10-s − 2.27·11-s − 6.92·12-s + 0.469·13-s − 7.02·14-s − 3.09·15-s + 1/4·16-s − 1.12·17-s − 5.65·18-s − 0.181·19-s + 4.02·20-s − 5.73·21-s + 6.43·22-s − 1.29·23-s + 3.26·24-s + 2/25·25-s − 1.32·26-s − 0.769·27-s + 7.45·28-s + ⋯

Functional equation

Λ(s)=((216⋅316⋅532)s/2 ΓC(s)16 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((216⋅316⋅532)s/2 ΓC(s+3/2)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{32}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 216⋅316⋅5322^{16} \cdot 3^{16} \cdot 5^{32}
Sign: 11
Analytic conductor: 1.41682×10151.41682\times 10^{15}
Root analytic conductor: 2.974942.97494
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 216⋅316⋅532, ( :[3/2]16), 1)(32,\ 2^{16} \cdot 3^{16} \cdot 5^{32} ,\ ( \ : [3/2]^{16} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.0045852731640.004585273164
L(12)L(\frac12) ≈\approx 0.0045852731640.004585273164
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 (1+pT+p2T2+p3T3+p4T4)4 ( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{4}
3 (1+pT+p2T2+p3T3+p4T4)4 ( 1 + p T + p^{2} T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{4}
5 1−3pT+43pT2−36p3T3+11516pT4−25629p2T5+78496p3T6−191994p4T7+358861p5T8−191994p7T9+78496p9T10−25629p11T11+11516p13T12−36p18T13+43p19T14−3p22T15+p24T16 1 - 3 p T + 43 p T^{2} - 36 p^{3} T^{3} + 11516 p T^{4} - 25629 p^{2} T^{5} + 78496 p^{3} T^{6} - 191994 p^{4} T^{7} + 358861 p^{5} T^{8} - 191994 p^{7} T^{9} + 78496 p^{9} T^{10} - 25629 p^{11} T^{11} + 11516 p^{13} T^{12} - 36 p^{18} T^{13} + 43 p^{19} T^{14} - 3 p^{22} T^{15} + p^{24} T^{16}
good7 (1−23T+181pT2−346p2T3+90348pT4−6303478T5+39104630pT6−3056032827T7+112722035404T8−3056032827p3T9+39104630p7T10−6303478p9T11+90348p13T12−346p17T13+181p19T14−23p21T15+p24T16)2 ( 1 - 23 T + 181 p T^{2} - 346 p^{2} T^{3} + 90348 p T^{4} - 6303478 T^{5} + 39104630 p T^{6} - 3056032827 T^{7} + 112722035404 T^{8} - 3056032827 p^{3} T^{9} + 39104630 p^{7} T^{10} - 6303478 p^{9} T^{11} + 90348 p^{13} T^{12} - 346 p^{17} T^{13} + 181 p^{19} T^{14} - 23 p^{21} T^{15} + p^{24} T^{16} )^{2}
11 1+83T+2131T2+46325T3+4980270T4+219524939T5+3816268172T6+100157669234T7+7409673674820T8+25268743328145pT9+4888950571730110T10+48722406629213165T11+7133107727485187410T12+ 1 + 83 T + 2131 T^{2} + 46325 T^{3} + 4980270 T^{4} + 219524939 T^{5} + 3816268172 T^{6} + 100157669234 T^{7} + 7409673674820 T^{8} + 25268743328145 p T^{9} + 4888950571730110 T^{10} + 48722406629213165 T^{11} + 7133107727485187410 T^{12} + 33 ⁣⋯ ⁣7033\!\cdots\!70T13+ T^{13} + 48 ⁣⋯ ⁣7048\!\cdots\!70T14+ T^{14} + 27 ⁣⋯ ⁣7527\!\cdots\!75T15+ T^{15} + 20 ⁣⋯ ⁣6020\!\cdots\!60T16+ T^{16} + 27 ⁣⋯ ⁣7527\!\cdots\!75p3T17+ p^{3} T^{17} + 48 ⁣⋯ ⁣7048\!\cdots\!70p6T18+ p^{6} T^{18} + 33 ⁣⋯ ⁣7033\!\cdots\!70p9T19+7133107727485187410p12T20+48722406629213165p15T21+4888950571730110p18T22+25268743328145p22T23+7409673674820p24T24+100157669234p27T25+3816268172p30T26+219524939p33T27+4980270p36T28+46325p39T29+2131p42T30+83p45T31+p48T32 p^{9} T^{19} + 7133107727485187410 p^{12} T^{20} + 48722406629213165 p^{15} T^{21} + 4888950571730110 p^{18} T^{22} + 25268743328145 p^{22} T^{23} + 7409673674820 p^{24} T^{24} + 100157669234 p^{27} T^{25} + 3816268172 p^{30} T^{26} + 219524939 p^{33} T^{27} + 4980270 p^{36} T^{28} + 46325 p^{39} T^{29} + 2131 p^{42} T^{30} + 83 p^{45} T^{31} + p^{48} T^{32}
13 1−22T−8233T2+135730T3+32280125T4−931341006T5−64913082488T6+5039072939438T7+38938197973430T8−18402018257260710T9+296409225739147903T10+46568068796243862704T11− 1 - 22 T - 8233 T^{2} + 135730 T^{3} + 32280125 T^{4} - 931341006 T^{5} - 64913082488 T^{6} + 5039072939438 T^{7} + 38938197973430 T^{8} - 18402018257260710 T^{9} + 296409225739147903 T^{10} + 46568068796243862704 T^{11} - 16 ⁣⋯ ⁣1416\!\cdots\!14T12− T^{12} - 83 ⁣⋯ ⁣6083\!\cdots\!60T13+ T^{13} + 54 ⁣⋯ ⁣9054\!\cdots\!90T14+ T^{14} + 73 ⁣⋯ ⁣2873\!\cdots\!28T15− T^{15} - 13 ⁣⋯ ⁣1113\!\cdots\!11T16+ T^{16} + 73 ⁣⋯ ⁣2873\!\cdots\!28p3T17+ p^{3} T^{17} + 54 ⁣⋯ ⁣9054\!\cdots\!90p6T18− p^{6} T^{18} - 83 ⁣⋯ ⁣6083\!\cdots\!60p9T19− p^{9} T^{19} - 16 ⁣⋯ ⁣1416\!\cdots\!14p12T20+46568068796243862704p15T21+296409225739147903p18T22−18402018257260710p21T23+38938197973430p24T24+5039072939438p27T25−64913082488p30T26−931341006p33T27+32280125p36T28+135730p39T29−8233p42T30−22p45T31+p48T32 p^{12} T^{20} + 46568068796243862704 p^{15} T^{21} + 296409225739147903 p^{18} T^{22} - 18402018257260710 p^{21} T^{23} + 38938197973430 p^{24} T^{24} + 5039072939438 p^{27} T^{25} - 64913082488 p^{30} T^{26} - 931341006 p^{33} T^{27} + 32280125 p^{36} T^{28} + 135730 p^{39} T^{29} - 8233 p^{42} T^{30} - 22 p^{45} T^{31} + p^{48} T^{32}
17 1+79T+13253T2+1431530T3+132103900T4+13646894087T5+1283825852483T6+6299111717903pT7+9723144385868115T8+829392753779236860T9+63950188092072426467T10+ 1 + 79 T + 13253 T^{2} + 1431530 T^{3} + 132103900 T^{4} + 13646894087 T^{5} + 1283825852483 T^{6} + 6299111717903 p T^{7} + 9723144385868115 T^{8} + 829392753779236860 T^{9} + 63950188092072426467 T^{10} + 52 ⁣⋯ ⁣6352\!\cdots\!63T11+ T^{11} + 40 ⁣⋯ ⁣5640\!\cdots\!56T12+ T^{12} + 29 ⁣⋯ ⁣1029\!\cdots\!10T13+ T^{13} + 23 ⁣⋯ ⁣3023\!\cdots\!30T14+ T^{14} + 16 ⁣⋯ ⁣6616\!\cdots\!66T15+ T^{15} + 11 ⁣⋯ ⁣7911\!\cdots\!79T16+ T^{16} + 16 ⁣⋯ ⁣6616\!\cdots\!66p3T17+ p^{3} T^{17} + 23 ⁣⋯ ⁣3023\!\cdots\!30p6T18+ p^{6} T^{18} + 29 ⁣⋯ ⁣1029\!\cdots\!10p9T19+ p^{9} T^{19} + 40 ⁣⋯ ⁣5640\!\cdots\!56p12T20+ p^{12} T^{20} + 52 ⁣⋯ ⁣6352\!\cdots\!63p15T21+63950188092072426467p18T22+829392753779236860p21T23+9723144385868115p24T24+6299111717903p28T25+1283825852483p30T26+13646894087p33T27+132103900p36T28+1431530p39T29+13253p42T30+79p45T31+p48T32 p^{15} T^{21} + 63950188092072426467 p^{18} T^{22} + 829392753779236860 p^{21} T^{23} + 9723144385868115 p^{24} T^{24} + 6299111717903 p^{28} T^{25} + 1283825852483 p^{30} T^{26} + 13646894087 p^{33} T^{27} + 132103900 p^{36} T^{28} + 1431530 p^{39} T^{29} + 13253 p^{42} T^{30} + 79 p^{45} T^{31} + p^{48} T^{32}
19 1+15T−26911T2+134795T3+334581480T4−6246163335T5−2441902724295T6+83122703225735T7+15092172749204005T8−718284879818480945T9− 1 + 15 T - 26911 T^{2} + 134795 T^{3} + 334581480 T^{4} - 6246163335 T^{5} - 2441902724295 T^{6} + 83122703225735 T^{7} + 15092172749204005 T^{8} - 718284879818480945 T^{9} - 11 ⁣⋯ ⁣2311\!\cdots\!23T10+ T^{10} + 49 ⁣⋯ ⁣0549\!\cdots\!05T11+ T^{11} + 43 ⁣⋯ ⁣5243\!\cdots\!52pT12− p T^{12} - 24 ⁣⋯ ⁣4024\!\cdots\!40T13− T^{13} - 46 ⁣⋯ ⁣1046\!\cdots\!10T14+ T^{14} + 56 ⁣⋯ ⁣7056\!\cdots\!70T15+ T^{15} + 26 ⁣⋯ ⁣6526\!\cdots\!65T16+ T^{16} + 56 ⁣⋯ ⁣7056\!\cdots\!70p3T17− p^{3} T^{17} - 46 ⁣⋯ ⁣1046\!\cdots\!10p6T18− p^{6} T^{18} - 24 ⁣⋯ ⁣4024\!\cdots\!40p9T19+ p^{9} T^{19} + 43 ⁣⋯ ⁣5243\!\cdots\!52p13T20+ p^{13} T^{20} + 49 ⁣⋯ ⁣0549\!\cdots\!05p15T21− p^{15} T^{21} - 11 ⁣⋯ ⁣2311\!\cdots\!23p18T22−718284879818480945p21T23+15092172749204005p24T24+83122703225735p27T25−2441902724295p30T26−6246163335p33T27+334581480p36T28+134795p39T29−26911p42T30+15p45T31+p48T32 p^{18} T^{22} - 718284879818480945 p^{21} T^{23} + 15092172749204005 p^{24} T^{24} + 83122703225735 p^{27} T^{25} - 2441902724295 p^{30} T^{26} - 6246163335 p^{33} T^{27} + 334581480 p^{36} T^{28} + 134795 p^{39} T^{29} - 26911 p^{42} T^{30} + 15 p^{45} T^{31} + p^{48} T^{32}
23 1+143T−39148T2−4449205T3+970599945T4+76383676149T5−15610293398058T6−29434057471659pT7+224832517917945450T8+5977086678069106955T9− 1 + 143 T - 39148 T^{2} - 4449205 T^{3} + 970599945 T^{4} + 76383676149 T^{5} - 15610293398058 T^{6} - 29434057471659 p T^{7} + 224832517917945450 T^{8} + 5977086678069106955 T^{9} - 30 ⁣⋯ ⁣0230\!\cdots\!02T10− T^{10} - 81 ⁣⋯ ⁣8181\!\cdots\!81T11+ T^{11} + 44 ⁣⋯ ⁣7644\!\cdots\!76T12+ T^{12} + 14 ⁣⋯ ⁣4014\!\cdots\!40T13− T^{13} - 58 ⁣⋯ ⁣8058\!\cdots\!80T14− T^{14} - 73 ⁣⋯ ⁣4273\!\cdots\!42T15+ T^{15} + 75 ⁣⋯ ⁣1975\!\cdots\!19T16− T^{16} - 73 ⁣⋯ ⁣4273\!\cdots\!42p3T17− p^{3} T^{17} - 58 ⁣⋯ ⁣8058\!\cdots\!80p6T18+ p^{6} T^{18} + 14 ⁣⋯ ⁣4014\!\cdots\!40p9T19+ p^{9} T^{19} + 44 ⁣⋯ ⁣7644\!\cdots\!76p12T20− p^{12} T^{20} - 81 ⁣⋯ ⁣8181\!\cdots\!81p15T21− p^{15} T^{21} - 30 ⁣⋯ ⁣0230\!\cdots\!02p18T22+5977086678069106955p21T23+224832517917945450p24T24−29434057471659p28T25−15610293398058p30T26+76383676149p33T27+970599945p36T28−4449205p39T29−39148p42T30+143p45T31+p48T32 p^{18} T^{22} + 5977086678069106955 p^{21} T^{23} + 224832517917945450 p^{24} T^{24} - 29434057471659 p^{28} T^{25} - 15610293398058 p^{30} T^{26} + 76383676149 p^{33} T^{27} + 970599945 p^{36} T^{28} - 4449205 p^{39} T^{29} - 39148 p^{42} T^{30} + 143 p^{45} T^{31} + p^{48} T^{32}
29 1−255T+12174T2+4692930T3−197902415T4+77636219010T5−67519657328140T6+9799994553632640T7+775272839415067910T8− 1 - 255 T + 12174 T^{2} + 4692930 T^{3} - 197902415 T^{4} + 77636219010 T^{5} - 67519657328140 T^{6} + 9799994553632640 T^{7} + 775272839415067910 T^{8} - 23 ⁣⋯ ⁣8023\!\cdots\!80T9+ T^{9} + 23 ⁣⋯ ⁣8223\!\cdots\!82T10− T^{10} - 38 ⁣⋯ ⁣3538\!\cdots\!35T11+ T^{11} + 17 ⁣⋯ ⁣8817\!\cdots\!88T12− T^{12} - 80 ⁣⋯ ⁣1080\!\cdots\!10T13− T^{13} - 37 ⁣⋯ ⁣7037\!\cdots\!70T14+ T^{14} + 40 ⁣⋯ ⁣3040\!\cdots\!30T15+ T^{15} + 33 ⁣⋯ ⁣0533\!\cdots\!05T16+ T^{16} + 40 ⁣⋯ ⁣3040\!\cdots\!30p3T17− p^{3} T^{17} - 37 ⁣⋯ ⁣7037\!\cdots\!70p6T18− p^{6} T^{18} - 80 ⁣⋯ ⁣1080\!\cdots\!10p9T19+ p^{9} T^{19} + 17 ⁣⋯ ⁣8817\!\cdots\!88p12T20− p^{12} T^{20} - 38 ⁣⋯ ⁣3538\!\cdots\!35p15T21+ p^{15} T^{21} + 23 ⁣⋯ ⁣8223\!\cdots\!82p18T22− p^{18} T^{22} - 23 ⁣⋯ ⁣8023\!\cdots\!80p21T23+775272839415067910p24T24+9799994553632640p27T25−67519657328140p30T26+77636219010p33T27−197902415p36T28+4692930p39T29+12174p42T30−255p45T31+p48T32 p^{21} T^{23} + 775272839415067910 p^{24} T^{24} + 9799994553632640 p^{27} T^{25} - 67519657328140 p^{30} T^{26} + 77636219010 p^{33} T^{27} - 197902415 p^{36} T^{28} + 4692930 p^{39} T^{29} + 12174 p^{42} T^{30} - 255 p^{45} T^{31} + p^{48} T^{32}
31 1+378T+114781T2+24745335T3+6454636985T4+43573938429pT5+280096152458152T6+51120991195516749T7+10803844060837049370T8+ 1 + 378 T + 114781 T^{2} + 24745335 T^{3} + 6454636985 T^{4} + 43573938429 p T^{5} + 280096152458152 T^{6} + 51120991195516749 T^{7} + 10803844060837049370 T^{8} + 20 ⁣⋯ ⁣7020\!\cdots\!70T9+ T^{9} + 39 ⁣⋯ ⁣3539\!\cdots\!35T10+ T^{10} + 23 ⁣⋯ ⁣4023\!\cdots\!40pT11+ p T^{11} + 14 ⁣⋯ ⁣3514\!\cdots\!35T12+ T^{12} + 25 ⁣⋯ ⁣9525\!\cdots\!95T13+ T^{13} + 46 ⁣⋯ ⁣4546\!\cdots\!45T14+ T^{14} + 80 ⁣⋯ ⁣5080\!\cdots\!50T15+ T^{15} + 14 ⁣⋯ ⁣6014\!\cdots\!60T16+ T^{16} + 80 ⁣⋯ ⁣5080\!\cdots\!50p3T17+ p^{3} T^{17} + 46 ⁣⋯ ⁣4546\!\cdots\!45p6T18+ p^{6} T^{18} + 25 ⁣⋯ ⁣9525\!\cdots\!95p9T19+ p^{9} T^{19} + 14 ⁣⋯ ⁣3514\!\cdots\!35p12T20+ p^{12} T^{20} + 23 ⁣⋯ ⁣4023\!\cdots\!40p16T21+ p^{16} T^{21} + 39 ⁣⋯ ⁣3539\!\cdots\!35p18T22+ p^{18} T^{22} + 20 ⁣⋯ ⁣7020\!\cdots\!70p21T23+10803844060837049370p24T24+51120991195516749p27T25+280096152458152p30T26+43573938429p34T27+6454636985p36T28+24745335p39T29+114781p42T30+378p45T31+p48T32 p^{21} T^{23} + 10803844060837049370 p^{24} T^{24} + 51120991195516749 p^{27} T^{25} + 280096152458152 p^{30} T^{26} + 43573938429 p^{34} T^{27} + 6454636985 p^{36} T^{28} + 24745335 p^{39} T^{29} + 114781 p^{42} T^{30} + 378 p^{45} T^{31} + p^{48} T^{32}
37 1+514T+34978T2−40887955T3−9850861885T4−826553105378T5−303817369039032T6−85180199573646609T7+28253290189207526495T8+ 1 + 514 T + 34978 T^{2} - 40887955 T^{3} - 9850861885 T^{4} - 826553105378 T^{5} - 303817369039032 T^{6} - 85180199573646609 T^{7} + 28253290189207526495 T^{8} + 12 ⁣⋯ ⁣3512\!\cdots\!35T9+ T^{9} + 14 ⁣⋯ ⁣0714\!\cdots\!07T10+ T^{10} + 55 ⁣⋯ ⁣2855\!\cdots\!28T11+ T^{11} + 55 ⁣⋯ ⁣9655\!\cdots\!96T12+ T^{12} + 75 ⁣⋯ ⁣2075\!\cdots\!20T13− T^{13} - 45 ⁣⋯ ⁣5045\!\cdots\!50T14− T^{14} - 97 ⁣⋯ ⁣5997\!\cdots\!59T15− T^{15} - 76 ⁣⋯ ⁣5676\!\cdots\!56T16− T^{16} - 97 ⁣⋯ ⁣5997\!\cdots\!59p3T17− p^{3} T^{17} - 45 ⁣⋯ ⁣5045\!\cdots\!50p6T18+ p^{6} T^{18} + 75 ⁣⋯ ⁣2075\!\cdots\!20p9T19+ p^{9} T^{19} + 55 ⁣⋯ ⁣9655\!\cdots\!96p12T20+ p^{12} T^{20} + 55 ⁣⋯ ⁣2855\!\cdots\!28p15T21+ p^{15} T^{21} + 14 ⁣⋯ ⁣0714\!\cdots\!07p18T22+ p^{18} T^{22} + 12 ⁣⋯ ⁣3512\!\cdots\!35p21T23+28253290189207526495p24T24−85180199573646609p27T25−303817369039032p30T26−826553105378p33T27−9850861885p36T28−40887955p39T29+34978p42T30+514p45T31+p48T32 p^{21} T^{23} + 28253290189207526495 p^{24} T^{24} - 85180199573646609 p^{27} T^{25} - 303817369039032 p^{30} T^{26} - 826553105378 p^{33} T^{27} - 9850861885 p^{36} T^{28} - 40887955 p^{39} T^{29} + 34978 p^{42} T^{30} + 514 p^{45} T^{31} + p^{48} T^{32}
41 1+918T+332221T2+31400740T3−19545308845T4−9610122317136T5−1869108009716788T6+135769977213045144T7+ 1 + 918 T + 332221 T^{2} + 31400740 T^{3} - 19545308845 T^{4} - 9610122317136 T^{5} - 1869108009716788 T^{6} + 135769977213045144 T^{7} + 24 ⁣⋯ ⁣2024\!\cdots\!20T8+ T^{8} + 91 ⁣⋯ ⁣7091\!\cdots\!70T9+ T^{9} + 15 ⁣⋯ ⁣8515\!\cdots\!85T10− T^{10} - 21 ⁣⋯ ⁣1021\!\cdots\!10T11− T^{11} - 18 ⁣⋯ ⁣9018\!\cdots\!90T12− T^{12} - 44 ⁣⋯ ⁣3044\!\cdots\!30T13− T^{13} - 88 ⁣⋯ ⁣3088\!\cdots\!30T14+ T^{14} + 27 ⁣⋯ ⁣0027\!\cdots\!00T15+ T^{15} + 10 ⁣⋯ ⁣8510\!\cdots\!85T16+ T^{16} + 27 ⁣⋯ ⁣0027\!\cdots\!00p3T17− p^{3} T^{17} - 88 ⁣⋯ ⁣3088\!\cdots\!30p6T18− p^{6} T^{18} - 44 ⁣⋯ ⁣3044\!\cdots\!30p9T19− p^{9} T^{19} - 18 ⁣⋯ ⁣9018\!\cdots\!90p12T20− p^{12} T^{20} - 21 ⁣⋯ ⁣1021\!\cdots\!10p15T21+ p^{15} T^{21} + 15 ⁣⋯ ⁣8515\!\cdots\!85p18T22+ p^{18} T^{22} + 91 ⁣⋯ ⁣7091\!\cdots\!70p21T23+ p^{21} T^{23} + 24 ⁣⋯ ⁣2024\!\cdots\!20p24T24+135769977213045144p27T25−1869108009716788p30T26−9610122317136p33T27−19545308845p36T28+31400740p39T29+332221p42T30+918p45T31+p48T32 p^{24} T^{24} + 135769977213045144 p^{27} T^{25} - 1869108009716788 p^{30} T^{26} - 9610122317136 p^{33} T^{27} - 19545308845 p^{36} T^{28} + 31400740 p^{39} T^{29} + 332221 p^{42} T^{30} + 918 p^{45} T^{31} + p^{48} T^{32}
43 (1−506T+529623T2−217946122T3+131977207041T4−44544422354176T5+19671375227058705T6−5479302172478432416T7+ ( 1 - 506 T + 529623 T^{2} - 217946122 T^{3} + 131977207041 T^{4} - 44544422354176 T^{5} + 19671375227058705 T^{6} - 5479302172478432416 T^{7} + 19 ⁣⋯ ⁣0419\!\cdots\!04T8−5479302172478432416p3T9+19671375227058705p6T10−44544422354176p9T11+131977207041p12T12−217946122p15T13+529623p18T14−506p21T15+p24T16)2 T^{8} - 5479302172478432416 p^{3} T^{9} + 19671375227058705 p^{6} T^{10} - 44544422354176 p^{9} T^{11} + 131977207041 p^{12} T^{12} - 217946122 p^{15} T^{13} + 529623 p^{18} T^{14} - 506 p^{21} T^{15} + p^{24} T^{16} )^{2}
47 1+209T−475992T2−73076000T3+111624408835T4+6572303077612T5−18637926631261012T6+1150860426442005456T7+ 1 + 209 T - 475992 T^{2} - 73076000 T^{3} + 111624408835 T^{4} + 6572303077612 T^{5} - 18637926631261012 T^{6} + 1150860426442005456 T^{7} + 25 ⁣⋯ ⁣4025\!\cdots\!40T8− T^{8} - 41 ⁣⋯ ⁣7041\!\cdots\!70T9− T^{9} - 27 ⁣⋯ ⁣8827\!\cdots\!88T10+ T^{10} + 62 ⁣⋯ ⁣8362\!\cdots\!83T11+ T^{11} + 21 ⁣⋯ ⁣1621\!\cdots\!16T12− T^{12} - 56 ⁣⋯ ⁣3056\!\cdots\!30T13− T^{13} - 10 ⁣⋯ ⁣3010\!\cdots\!30T14+ T^{14} + 23 ⁣⋯ ⁣7623\!\cdots\!76T15+ T^{15} + 56 ⁣⋯ ⁣4956\!\cdots\!49T16+ T^{16} + 23 ⁣⋯ ⁣7623\!\cdots\!76p3T17− p^{3} T^{17} - 10 ⁣⋯ ⁣3010\!\cdots\!30p6T18− p^{6} T^{18} - 56 ⁣⋯ ⁣3056\!\cdots\!30p9T19+ p^{9} T^{19} + 21 ⁣⋯ ⁣1621\!\cdots\!16p12T20+ p^{12} T^{20} + 62 ⁣⋯ ⁣8362\!\cdots\!83p15T21− p^{15} T^{21} - 27 ⁣⋯ ⁣8827\!\cdots\!88p18T22− p^{18} T^{22} - 41 ⁣⋯ ⁣7041\!\cdots\!70p21T23+ p^{21} T^{23} + 25 ⁣⋯ ⁣4025\!\cdots\!40p24T24+1150860426442005456p27T25−18637926631261012p30T26+6572303077612p33T27+111624408835p36T28−73076000p39T29−475992p42T30+209p45T31+p48T32 p^{24} T^{24} + 1150860426442005456 p^{27} T^{25} - 18637926631261012 p^{30} T^{26} + 6572303077612 p^{33} T^{27} + 111624408835 p^{36} T^{28} - 73076000 p^{39} T^{29} - 475992 p^{42} T^{30} + 209 p^{45} T^{31} + p^{48} T^{32}
53 1+1738T+1211032T2+263538985T3−187516548715T4−167565205042176T5−49773191933381108T6+1868145018506317003T7+ 1 + 1738 T + 1211032 T^{2} + 263538985 T^{3} - 187516548715 T^{4} - 167565205042176 T^{5} - 49773191933381108 T^{6} + 1868145018506317003 T^{7} + 60 ⁣⋯ ⁣2560\!\cdots\!25T8+ T^{8} + 19 ⁣⋯ ⁣6519\!\cdots\!65T9+ T^{9} + 45 ⁣⋯ ⁣9345\!\cdots\!93T10+ T^{10} + 23 ⁣⋯ ⁣7423\!\cdots\!74T11+ T^{11} + 10 ⁣⋯ ⁣6610\!\cdots\!66T12+ T^{12} + 86 ⁣⋯ ⁣7086\!\cdots\!70T13− T^{13} - 12 ⁣⋯ ⁣9012\!\cdots\!90T14− T^{14} - 72 ⁣⋯ ⁣4772\!\cdots\!47T15− T^{15} - 27 ⁣⋯ ⁣3627\!\cdots\!36T16− T^{16} - 72 ⁣⋯ ⁣4772\!\cdots\!47p3T17− p^{3} T^{17} - 12 ⁣⋯ ⁣9012\!\cdots\!90p6T18+ p^{6} T^{18} + 86 ⁣⋯ ⁣7086\!\cdots\!70p9T19+ p^{9} T^{19} + 10 ⁣⋯ ⁣6610\!\cdots\!66p12T20+ p^{12} T^{20} + 23 ⁣⋯ ⁣7423\!\cdots\!74p15T21+ p^{15} T^{21} + 45 ⁣⋯ ⁣9345\!\cdots\!93p18T22+ p^{18} T^{22} + 19 ⁣⋯ ⁣6519\!\cdots\!65p21T23+ p^{21} T^{23} + 60 ⁣⋯ ⁣2560\!\cdots\!25p24T24+1868145018506317003p27T25−49773191933381108p30T26−167565205042176p33T27−187516548715p36T28+263538985p39T29+1211032p42T30+1738p45T31+p48T32 p^{24} T^{24} + 1868145018506317003 p^{27} T^{25} - 49773191933381108 p^{30} T^{26} - 167565205042176 p^{33} T^{27} - 187516548715 p^{36} T^{28} + 263538985 p^{39} T^{29} + 1211032 p^{42} T^{30} + 1738 p^{45} T^{31} + p^{48} T^{32}
59 1−695T+650109T2−649504360T3+323116464260T4−187937135311545T5+115051021927005445T6−29063741037236700480T7+ 1 - 695 T + 650109 T^{2} - 649504360 T^{3} + 323116464260 T^{4} - 187937135311545 T^{5} + 115051021927005445 T^{6} - 29063741037236700480 T^{7} + 10 ⁣⋯ ⁣8510\!\cdots\!85T8− T^{8} - 44 ⁣⋯ ⁣1544\!\cdots\!15T9− T^{9} - 21 ⁣⋯ ⁣4821\!\cdots\!48T10+ T^{10} + 63 ⁣⋯ ⁣8563\!\cdots\!85T11− T^{11} - 44 ⁣⋯ ⁣2244\!\cdots\!22T12+ T^{12} + 28 ⁣⋯ ⁣4528\!\cdots\!45T13+ T^{13} + 42 ⁣⋯ ⁣5542\!\cdots\!55T14− T^{14} - 47 ⁣⋯ ⁣3547\!\cdots\!35T15− T^{15} - 64 ⁣⋯ ⁣9064\!\cdots\!90T16− T^{16} - 47 ⁣⋯ ⁣3547\!\cdots\!35p3T17+ p^{3} T^{17} + 42 ⁣⋯ ⁣5542\!\cdots\!55p6T18+ p^{6} T^{18} + 28 ⁣⋯ ⁣4528\!\cdots\!45p9T19− p^{9} T^{19} - 44 ⁣⋯ ⁣2244\!\cdots\!22p12T20+ p^{12} T^{20} + 63 ⁣⋯ ⁣8563\!\cdots\!85p15T21− p^{15} T^{21} - 21 ⁣⋯ ⁣4821\!\cdots\!48p18T22− p^{18} T^{22} - 44 ⁣⋯ ⁣1544\!\cdots\!15p21T23+ p^{21} T^{23} + 10 ⁣⋯ ⁣8510\!\cdots\!85p24T24−29063741037236700480p27T25+115051021927005445p30T26−187937135311545p33T27+323116464260p36T28−649504360p39T29+650109p42T30−695p45T31+p48T32 p^{24} T^{24} - 29063741037236700480 p^{27} T^{25} + 115051021927005445 p^{30} T^{26} - 187937135311545 p^{33} T^{27} + 323116464260 p^{36} T^{28} - 649504360 p^{39} T^{29} + 650109 p^{42} T^{30} - 695 p^{45} T^{31} + p^{48} T^{32}
61 1−352T−988859T2+383977960T3+370714761035T4−163175493974896T5−46548262854333468T6+31613097768425914484T7− 1 - 352 T - 988859 T^{2} + 383977960 T^{3} + 370714761035 T^{4} - 163175493974896 T^{5} - 46548262854333468 T^{6} + 31613097768425914484 T^{7} - 89 ⁣⋯ ⁣8089\!\cdots\!80T8− T^{8} - 23 ⁣⋯ ⁣3023\!\cdots\!30T9+ T^{9} + 38 ⁣⋯ ⁣8538\!\cdots\!85T10+ T^{10} + 32 ⁣⋯ ⁣9032\!\cdots\!90T11− T^{11} - 40 ⁣⋯ ⁣9040\!\cdots\!90T12− T^{12} - 21 ⁣⋯ ⁣3021\!\cdots\!30T13− T^{13} - 41 ⁣⋯ ⁣3041\!\cdots\!30T14+ T^{14} + 31 ⁣⋯ ⁣0031\!\cdots\!00T15+ T^{15} + 20 ⁣⋯ ⁣8520\!\cdots\!85T16+ T^{16} + 31 ⁣⋯ ⁣0031\!\cdots\!00p3T17− p^{3} T^{17} - 41 ⁣⋯ ⁣3041\!\cdots\!30p6T18− p^{6} T^{18} - 21 ⁣⋯ ⁣3021\!\cdots\!30p9T19− p^{9} T^{19} - 40 ⁣⋯ ⁣9040\!\cdots\!90p12T20+ p^{12} T^{20} + 32 ⁣⋯ ⁣9032\!\cdots\!90p15T21+ p^{15} T^{21} + 38 ⁣⋯ ⁣8538\!\cdots\!85p18T22− p^{18} T^{22} - 23 ⁣⋯ ⁣3023\!\cdots\!30p21T23− p^{21} T^{23} - 89 ⁣⋯ ⁣8089\!\cdots\!80p24T24+31613097768425914484p27T25−46548262854333468p30T26−163175493974896p33T27+370714761035p36T28+383977960p39T29−988859p42T30−352p45T31+p48T32 p^{24} T^{24} + 31613097768425914484 p^{27} T^{25} - 46548262854333468 p^{30} T^{26} - 163175493974896 p^{33} T^{27} + 370714761035 p^{36} T^{28} + 383977960 p^{39} T^{29} - 988859 p^{42} T^{30} - 352 p^{45} T^{31} + p^{48} T^{32}
67 1+1554T+262833T2−850195060T3−493453069400T4+203098551275722T5+296141339334582468T6+26380457041566225046T7− 1 + 1554 T + 262833 T^{2} - 850195060 T^{3} - 493453069400 T^{4} + 203098551275722 T^{5} + 296141339334582468 T^{6} + 26380457041566225046 T^{7} - 10 ⁣⋯ ⁣8010\!\cdots\!80T8− T^{8} - 46 ⁣⋯ ⁣2046\!\cdots\!20T9+ T^{9} + 22 ⁣⋯ ⁣2722\!\cdots\!27T10+ T^{10} + 25 ⁣⋯ ⁣1825\!\cdots\!18T11+ T^{11} + 23 ⁣⋯ ⁣3123\!\cdots\!31T12− T^{12} - 72 ⁣⋯ ⁣2072\!\cdots\!20T13− T^{13} - 34 ⁣⋯ ⁣6034\!\cdots\!60T14+ T^{14} + 89 ⁣⋯ ⁣7689\!\cdots\!76T15+ T^{15} + 13 ⁣⋯ ⁣2413\!\cdots\!24T16+ T^{16} + 89 ⁣⋯ ⁣7689\!\cdots\!76p3T17− p^{3} T^{17} - 34 ⁣⋯ ⁣6034\!\cdots\!60p6T18− p^{6} T^{18} - 72 ⁣⋯ ⁣2072\!\cdots\!20p9T19+ p^{9} T^{19} + 23 ⁣⋯ ⁣3123\!\cdots\!31p12T20+ p^{12} T^{20} + 25 ⁣⋯ ⁣1825\!\cdots\!18p15T21+ p^{15} T^{21} + 22 ⁣⋯ ⁣2722\!\cdots\!27p18T22− p^{18} T^{22} - 46 ⁣⋯ ⁣2046\!\cdots\!20p21T23− p^{21} T^{23} - 10 ⁣⋯ ⁣8010\!\cdots\!80p24T24+26380457041566225046p27T25+296141339334582468p30T26+203098551275722p33T27−493453069400p36T28−850195060p39T29+262833p42T30+1554p45T31+p48T32 p^{24} T^{24} + 26380457041566225046 p^{27} T^{25} + 296141339334582468 p^{30} T^{26} + 203098551275722 p^{33} T^{27} - 493453069400 p^{36} T^{28} - 850195060 p^{39} T^{29} + 262833 p^{42} T^{30} + 1554 p^{45} T^{31} + p^{48} T^{32}
71 1+1543T−208329T2−1005016350T3+344257949820T4+698351279304729T5−62650049213568413T6− 1 + 1543 T - 208329 T^{2} - 1005016350 T^{3} + 344257949820 T^{4} + 698351279304729 T^{5} - 62650049213568413 T^{6} - 19 ⁣⋯ ⁣7119\!\cdots\!71T7+ T^{7} + 57 ⁣⋯ ⁣9557\!\cdots\!95T8+ T^{8} + 81 ⁣⋯ ⁣2081\!\cdots\!20T9+ T^{9} + 21 ⁣⋯ ⁣3521\!\cdots\!35T10− T^{10} - 47 ⁣⋯ ⁣3547\!\cdots\!35T11− T^{11} - 12 ⁣⋯ ⁣4012\!\cdots\!40T12+ T^{12} + 40 ⁣⋯ ⁣7040\!\cdots\!70T13+ T^{13} + 13 ⁣⋯ ⁣2013\!\cdots\!20T14+ T^{14} + 11 ⁣⋯ ⁣0011\!\cdots\!00T15− T^{15} - 41 ⁣⋯ ⁣1541\!\cdots\!15T16+ T^{16} + 11 ⁣⋯ ⁣0011\!\cdots\!00p3T17+ p^{3} T^{17} + 13 ⁣⋯ ⁣2013\!\cdots\!20p6T18+ p^{6} T^{18} + 40 ⁣⋯ ⁣7040\!\cdots\!70p9T19− p^{9} T^{19} - 12 ⁣⋯ ⁣4012\!\cdots\!40p12T20− p^{12} T^{20} - 47 ⁣⋯ ⁣3547\!\cdots\!35p15T21+ p^{15} T^{21} + 21 ⁣⋯ ⁣3521\!\cdots\!35p18T22+ p^{18} T^{22} + 81 ⁣⋯ ⁣2081\!\cdots\!20p21T23+ p^{21} T^{23} + 57 ⁣⋯ ⁣9557\!\cdots\!95p24T24− p^{24} T^{24} - 19 ⁣⋯ ⁣7119\!\cdots\!71p27T25−62650049213568413p30T26+698351279304729p33T27+344257949820p36T28−1005016350p39T29−208329p42T30+1543p45T31+p48T32 p^{27} T^{25} - 62650049213568413 p^{30} T^{26} + 698351279304729 p^{33} T^{27} + 344257949820 p^{36} T^{28} - 1005016350 p^{39} T^{29} - 208329 p^{42} T^{30} + 1543 p^{45} T^{31} + p^{48} T^{32}
73 1+778T−500693T2−273706210T3+585980941270T4+176624269337404T5−366545805128711908T6−53805977380721679492T7+ 1 + 778 T - 500693 T^{2} - 273706210 T^{3} + 585980941270 T^{4} + 176624269337404 T^{5} - 366545805128711908 T^{6} - 53805977380721679492 T^{7} + 19 ⁣⋯ ⁣7019\!\cdots\!70T8− T^{8} - 19 ⁣⋯ ⁣1019\!\cdots\!10T9− T^{9} - 88 ⁣⋯ ⁣6788\!\cdots\!67T10+ T^{10} + 63 ⁣⋯ ⁣3463\!\cdots\!34T11+ T^{11} + 35 ⁣⋯ ⁣5135\!\cdots\!51T12− T^{12} - 30 ⁣⋯ ⁣0030\!\cdots\!00pT13− p T^{13} - 11 ⁣⋯ ⁣2011\!\cdots\!20T14+ T^{14} + 10 ⁣⋯ ⁣2810\!\cdots\!28T15+ T^{15} + 50 ⁣⋯ ⁣0450\!\cdots\!04T16+ T^{16} + 10 ⁣⋯ ⁣2810\!\cdots\!28p3T17− p^{3} T^{17} - 11 ⁣⋯ ⁣2011\!\cdots\!20p6T18− p^{6} T^{18} - 30 ⁣⋯ ⁣0030\!\cdots\!00p10T19+ p^{10} T^{19} + 35 ⁣⋯ ⁣5135\!\cdots\!51p12T20+ p^{12} T^{20} + 63 ⁣⋯ ⁣3463\!\cdots\!34p15T21− p^{15} T^{21} - 88 ⁣⋯ ⁣6788\!\cdots\!67p18T22− p^{18} T^{22} - 19 ⁣⋯ ⁣1019\!\cdots\!10p21T23+ p^{21} T^{23} + 19 ⁣⋯ ⁣7019\!\cdots\!70p24T24−53805977380721679492p27T25−366545805128711908p30T26+176624269337404p33T27+585980941270p36T28−273706210p39T29−500693p42T30+778p45T31+p48T32 p^{24} T^{24} - 53805977380721679492 p^{27} T^{25} - 366545805128711908 p^{30} T^{26} + 176624269337404 p^{33} T^{27} + 585980941270 p^{36} T^{28} - 273706210 p^{39} T^{29} - 500693 p^{42} T^{30} + 778 p^{45} T^{31} + p^{48} T^{32}
79 1−920T+525809T2−278706170T3+308776177695T4+63942976170010T5−129966518173800480T6+12078879224764750140T7+ 1 - 920 T + 525809 T^{2} - 278706170 T^{3} + 308776177695 T^{4} + 63942976170010 T^{5} - 129966518173800480 T^{6} + 12078879224764750140 T^{7} + 57 ⁣⋯ ⁣7057\!\cdots\!70T8− T^{8} - 17 ⁣⋯ ⁣3017\!\cdots\!30T9+ T^{9} + 82 ⁣⋯ ⁣6782\!\cdots\!67T10− T^{10} - 83 ⁣⋯ ⁣4083\!\cdots\!40T11+ T^{11} + 38 ⁣⋯ ⁣1838\!\cdots\!18T12− T^{12} - 10 ⁣⋯ ⁣6010\!\cdots\!60T13+ T^{13} + 15 ⁣⋯ ⁣1015\!\cdots\!10T14+ T^{14} + 15 ⁣⋯ ⁣3015\!\cdots\!30T15− T^{15} - 68 ⁣⋯ ⁣6568\!\cdots\!65T16+ T^{16} + 15 ⁣⋯ ⁣3015\!\cdots\!30p3T17+ p^{3} T^{17} + 15 ⁣⋯ ⁣1015\!\cdots\!10p6T18− p^{6} T^{18} - 10 ⁣⋯ ⁣6010\!\cdots\!60p9T19+ p^{9} T^{19} + 38 ⁣⋯ ⁣1838\!\cdots\!18p12T20− p^{12} T^{20} - 83 ⁣⋯ ⁣4083\!\cdots\!40p15T21+ p^{15} T^{21} + 82 ⁣⋯ ⁣6782\!\cdots\!67p18T22− p^{18} T^{22} - 17 ⁣⋯ ⁣3017\!\cdots\!30p21T23+ p^{21} T^{23} + 57 ⁣⋯ ⁣7057\!\cdots\!70p24T24+12078879224764750140p27T25−129966518173800480p30T26+63942976170010p33T27+308776177695p36T28−278706170p39T29+525809p42T30−920p45T31+p48T32 p^{24} T^{24} + 12078879224764750140 p^{27} T^{25} - 129966518173800480 p^{30} T^{26} + 63942976170010 p^{33} T^{27} + 308776177695 p^{36} T^{28} - 278706170 p^{39} T^{29} + 525809 p^{42} T^{30} - 920 p^{45} T^{31} + p^{48} T^{32}
83 1−2967T+3285857T2−889502295T3−1458233474800T4+2342356175857219T5−2887246304185396218T6+ 1 - 2967 T + 3285857 T^{2} - 889502295 T^{3} - 1458233474800 T^{4} + 2342356175857219 T^{5} - 2887246304185396218 T^{6} + 27 ⁣⋯ ⁣4827\!\cdots\!48T7− T^{7} - 10 ⁣⋯ ⁣7010\!\cdots\!70T8− T^{8} - 95 ⁣⋯ ⁣3595\!\cdots\!35T9+ T^{9} + 16 ⁣⋯ ⁣2816\!\cdots\!28T10− T^{10} - 15 ⁣⋯ ⁣8115\!\cdots\!81T11+ T^{11} + 11 ⁣⋯ ⁣0611\!\cdots\!06T12− T^{12} - 42 ⁣⋯ ⁣6042\!\cdots\!60T13− T^{13} - 32 ⁣⋯ ⁣1032\!\cdots\!10T14+ T^{14} + 58 ⁣⋯ ⁣5358\!\cdots\!53T15− T^{15} - 50 ⁣⋯ ⁣9650\!\cdots\!96T16+ T^{16} + 58 ⁣⋯ ⁣5358\!\cdots\!53p3T17− p^{3} T^{17} - 32 ⁣⋯ ⁣1032\!\cdots\!10p6T18− p^{6} T^{18} - 42 ⁣⋯ ⁣6042\!\cdots\!60p9T19+ p^{9} T^{19} + 11 ⁣⋯ ⁣0611\!\cdots\!06p12T20− p^{12} T^{20} - 15 ⁣⋯ ⁣8115\!\cdots\!81p15T21+ p^{15} T^{21} + 16 ⁣⋯ ⁣2816\!\cdots\!28p18T22− p^{18} T^{22} - 95 ⁣⋯ ⁣3595\!\cdots\!35p21T23− p^{21} T^{23} - 10 ⁣⋯ ⁣7010\!\cdots\!70p24T24+ p^{24} T^{24} + 27 ⁣⋯ ⁣4827\!\cdots\!48p27T25−2887246304185396218p30T26+2342356175857219p33T27−1458233474800p36T28−889502295p39T29+3285857p42T30−2967p45T31+p48T32 p^{27} T^{25} - 2887246304185396218 p^{30} T^{26} + 2342356175857219 p^{33} T^{27} - 1458233474800 p^{36} T^{28} - 889502295 p^{39} T^{29} + 3285857 p^{42} T^{30} - 2967 p^{45} T^{31} + p^{48} T^{32}
89 1+2605T+2992849T2+2136344045T3+1262225805360T4+23016854835835pT5+3568950052505913070T6+ 1 + 2605 T + 2992849 T^{2} + 2136344045 T^{3} + 1262225805360 T^{4} + 23016854835835 p T^{5} + 3568950052505913070 T^{6} + 32 ⁣⋯ ⁣1032\!\cdots\!10T7+ T^{7} + 14 ⁣⋯ ⁣6014\!\cdots\!60T8+ T^{8} + 13 ⁣⋯ ⁣0513\!\cdots\!05T9+ T^{9} + 40 ⁣⋯ ⁣5240\!\cdots\!52T10+ T^{10} + 10 ⁣⋯ ⁣3510\!\cdots\!35T11+ T^{11} + 51 ⁣⋯ ⁣5851\!\cdots\!58T12− T^{12} - 50 ⁣⋯ ⁣9050\!\cdots\!90T13− T^{13} - 76 ⁣⋯ ⁣2076\!\cdots\!20T14− T^{14} - 30 ⁣⋯ ⁣0530\!\cdots\!05T15− T^{15} - 25 ⁣⋯ ⁣4025\!\cdots\!40T16− T^{16} - 30 ⁣⋯ ⁣0530\!\cdots\!05p3T17− p^{3} T^{17} - 76 ⁣⋯ ⁣2076\!\cdots\!20p6T18− p^{6} T^{18} - 50 ⁣⋯ ⁣9050\!\cdots\!90p9T19+ p^{9} T^{19} + 51 ⁣⋯ ⁣5851\!\cdots\!58p12T20+ p^{12} T^{20} + 10 ⁣⋯ ⁣3510\!\cdots\!35p15T21+ p^{15} T^{21} + 40 ⁣⋯ ⁣5240\!\cdots\!52p18T22+ p^{18} T^{22} + 13 ⁣⋯ ⁣0513\!\cdots\!05p21T23+ p^{21} T^{23} + 14 ⁣⋯ ⁣6014\!\cdots\!60p24T24+ p^{24} T^{24} + 32 ⁣⋯ ⁣1032\!\cdots\!10p27T25+3568950052505913070p30T26+23016854835835p34T27+1262225805360p36T28+2136344045p39T29+2992849p42T30+2605p45T31+p48T32 p^{27} T^{25} + 3568950052505913070 p^{30} T^{26} + 23016854835835 p^{34} T^{27} + 1262225805360 p^{36} T^{28} + 2136344045 p^{39} T^{29} + 2992849 p^{42} T^{30} + 2605 p^{45} T^{31} + p^{48} T^{32}
97 1−1251T−1544117T2+2105987265T3+1990526674490T4−3856701043597483T5+27092960650973008T6+ 1 - 1251 T - 1544117 T^{2} + 2105987265 T^{3} + 1990526674490 T^{4} - 3856701043597483 T^{5} + 27092960650973008 T^{6} + 37 ⁣⋯ ⁣9637\!\cdots\!96T7− T^{7} - 61 ⁣⋯ ⁣2061\!\cdots\!20T8− T^{8} - 42 ⁣⋯ ⁣2542\!\cdots\!25T9+ T^{9} + 23 ⁣⋯ ⁣0223\!\cdots\!02T10+ T^{10} + 19 ⁣⋯ ⁣8319\!\cdots\!83T11− T^{11} - 16 ⁣⋯ ⁣1416\!\cdots\!14T12− T^{12} - 16 ⁣⋯ ⁣4016\!\cdots\!40T13+ T^{13} + 25 ⁣⋯ ⁣7025\!\cdots\!70T14− T^{14} - 16 ⁣⋯ ⁣8916\!\cdots\!89T15− T^{15} - 13 ⁣⋯ ⁣5613\!\cdots\!56T16− T^{16} - 16 ⁣⋯ ⁣8916\!\cdots\!89p3T17+ p^{3} T^{17} + 25 ⁣⋯ ⁣7025\!\cdots\!70p6T18− p^{6} T^{18} - 16 ⁣⋯ ⁣4016\!\cdots\!40p9T19− p^{9} T^{19} - 16 ⁣⋯ ⁣1416\!\cdots\!14p12T20+ p^{12} T^{20} + 19 ⁣⋯ ⁣8319\!\cdots\!83p15T21+ p^{15} T^{21} + 23 ⁣⋯ ⁣0223\!\cdots\!02p18T22− p^{18} T^{22} - 42 ⁣⋯ ⁣2542\!\cdots\!25p21T23− p^{21} T^{23} - 61 ⁣⋯ ⁣2061\!\cdots\!20p24T24+ p^{24} T^{24} + 37 ⁣⋯ ⁣9637\!\cdots\!96p27T25+27092960650973008p30T26−3856701043597483p33T27+1990526674490p36T28+2105987265p39T29−1544117p42T30−1251p45T31+p48T32 p^{27} T^{25} + 27092960650973008 p^{30} T^{26} - 3856701043597483 p^{33} T^{27} + 1990526674490 p^{36} T^{28} + 2105987265 p^{39} T^{29} - 1544117 p^{42} T^{30} - 1251 p^{45} T^{31} + p^{48} T^{32}
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   L(s)=∏p ∏j=132(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.23026723542137874889234416451, −3.12413030831585182135256336205, −3.07066793471123357662495610534, −2.99174513803311828164114153020, −2.84376165920353446523209992180, −2.60387370843089757922783137795, −2.54251902550737608893983255800, −2.24993901962847763860694307091, −2.13201796070724937145546970367, −2.09099952085679018961472068231, −2.06093588460895550948822677029, −1.93220270524547281670977192624, −1.75031635394400088137280190157, −1.71414807594683907677617088271, −1.70114973653886076921567752101, −1.53913261105045436801308615954, −1.24171572037348567457698786732, −1.22486068123036822283335563093, −1.09528497788674332712177344649, −0.836364174382042418765522857434, −0.51775916447468740014259136599, −0.44988538317506098809196430437, −0.34008918723327171879180249980, −0.095975148005085959430155554357, −0.06677363693016618557041513944, 0.06677363693016618557041513944, 0.095975148005085959430155554357, 0.34008918723327171879180249980, 0.44988538317506098809196430437, 0.51775916447468740014259136599, 0.836364174382042418765522857434, 1.09528497788674332712177344649, 1.22486068123036822283335563093, 1.24171572037348567457698786732, 1.53913261105045436801308615954, 1.70114973653886076921567752101, 1.71414807594683907677617088271, 1.75031635394400088137280190157, 1.93220270524547281670977192624, 2.06093588460895550948822677029, 2.09099952085679018961472068231, 2.13201796070724937145546970367, 2.24993901962847763860694307091, 2.54251902550737608893983255800, 2.60387370843089757922783137795, 2.84376165920353446523209992180, 2.99174513803311828164114153020, 3.07066793471123357662495610534, 3.12413030831585182135256336205, 3.23026723542137874889234416451

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.