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Properties

Label 32-630e16-1.1-c1e16-0-8
Degree 3232
Conductor 6.158×10446.158\times 10^{44}
Sign 11
Analytic cond. 1.68221×10111.68221\times 10^{11}
Root an. cond. 2.242892.24289
Motivic weight 11
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  − 16·2-s + 2·3-s + 136·4-s + 8·5-s − 32·6-s + 4·7-s − 816·8-s − 9-s − 128·10-s + 11-s + 272·12-s + 2·13-s − 64·14-s + 16·15-s + 3.87e3·16-s + 11·17-s + 16·18-s − 2·19-s + 1.08e3·20-s + 8·21-s − 16·22-s + 11·23-s − 1.63e3·24-s + 28·25-s − 32·26-s − 7·27-s + 544·28-s + ⋯
L(s)  = 1  − 11.3·2-s + 1.15·3-s + 68·4-s + 3.57·5-s − 13.0·6-s + 1.51·7-s − 288.·8-s − 1/3·9-s − 40.4·10-s + 0.301·11-s + 78.5·12-s + 0.554·13-s − 17.1·14-s + 4.13·15-s + 969·16-s + 2.66·17-s + 3.77·18-s − 0.458·19-s + 243.·20-s + 1.74·21-s − 3.41·22-s + 2.29·23-s − 333.·24-s + 28/5·25-s − 6.27·26-s − 1.34·27-s + 102.·28-s + ⋯

Functional equation

Λ(s)=((216⋅332⋅516⋅716)s/2 ΓC(s)16 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅332⋅516⋅716)s/2 ΓC(s+1/2)16 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 3232
Conductor: 216⋅332⋅516⋅7162^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16}
Sign: 11
Analytic conductor: 1.68221×10111.68221\times 10^{11}
Root analytic conductor: 2.242892.24289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (32, 216⋅332⋅516⋅716, ( :[1/2]16), 1)(32,\ 2^{16} \cdot 3^{32} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 1.2753998491.275399849
L(12)L(\frac12) ≈\approx 1.2753998491.275399849
L(32)L(\frac{3}{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+T)16 ( 1 + T )^{16}
3 1−2T+5T2−5T3+8T4−8T5−20T6+17pT7−61pT8+17p2T9−20p2T10−8p3T11+8p4T12−5p5T13+5p6T14−2p7T15+p8T16 1 - 2 T + 5 T^{2} - 5 T^{3} + 8 T^{4} - 8 T^{5} - 20 T^{6} + 17 p T^{7} - 61 p T^{8} + 17 p^{2} T^{9} - 20 p^{2} T^{10} - 8 p^{3} T^{11} + 8 p^{4} T^{12} - 5 p^{5} T^{13} + 5 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16}
5 (1−T+T2)8 ( 1 - T + T^{2} )^{8}
7 1−4T+15T2−46T3+85T4−234T5+295T6+50pT7−39T8+50p2T9+295p2T10−234p3T11+85p4T12−46p5T13+15p6T14−4p7T15+p8T16 1 - 4 T + 15 T^{2} - 46 T^{3} + 85 T^{4} - 234 T^{5} + 295 T^{6} + 50 p T^{7} - 39 T^{8} + 50 p^{2} T^{9} + 295 p^{2} T^{10} - 234 p^{3} T^{11} + 85 p^{4} T^{12} - 46 p^{5} T^{13} + 15 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16}
good11 1−T−35T2+182T3+577T4−5289T5+8314T6+80986T7−380701T8+165840T9+6417590T10−21138520T11−17857183T12+372829025T13−755353711T14−1808389533T15+15175368627T16−1808389533pT17−755353711p2T18+372829025p3T19−17857183p4T20−21138520p5T21+6417590p6T22+165840p7T23−380701p8T24+80986p9T25+8314p10T26−5289p11T27+577p12T28+182p13T29−35p14T30−p15T31+p16T32 1 - T - 35 T^{2} + 182 T^{3} + 577 T^{4} - 5289 T^{5} + 8314 T^{6} + 80986 T^{7} - 380701 T^{8} + 165840 T^{9} + 6417590 T^{10} - 21138520 T^{11} - 17857183 T^{12} + 372829025 T^{13} - 755353711 T^{14} - 1808389533 T^{15} + 15175368627 T^{16} - 1808389533 p T^{17} - 755353711 p^{2} T^{18} + 372829025 p^{3} T^{19} - 17857183 p^{4} T^{20} - 21138520 p^{5} T^{21} + 6417590 p^{6} T^{22} + 165840 p^{7} T^{23} - 380701 p^{8} T^{24} + 80986 p^{9} T^{25} + 8314 p^{10} T^{26} - 5289 p^{11} T^{27} + 577 p^{12} T^{28} + 182 p^{13} T^{29} - 35 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32}
13 1−2T−50T2+148T3+1054T4−3995T5−11532T6+43443T7+117342T8+20138T9−2880550T10−3340678T11+58331516T12−10068559T13−672470458T14+306068259T15+6775158987T16+306068259pT17−672470458p2T18−10068559p3T19+58331516p4T20−3340678p5T21−2880550p6T22+20138p7T23+117342p8T24+43443p9T25−11532p10T26−3995p11T27+1054p12T28+148p13T29−50p14T30−2p15T31+p16T32 1 - 2 T - 50 T^{2} + 148 T^{3} + 1054 T^{4} - 3995 T^{5} - 11532 T^{6} + 43443 T^{7} + 117342 T^{8} + 20138 T^{9} - 2880550 T^{10} - 3340678 T^{11} + 58331516 T^{12} - 10068559 T^{13} - 672470458 T^{14} + 306068259 T^{15} + 6775158987 T^{16} + 306068259 p T^{17} - 672470458 p^{2} T^{18} - 10068559 p^{3} T^{19} + 58331516 p^{4} T^{20} - 3340678 p^{5} T^{21} - 2880550 p^{6} T^{22} + 20138 p^{7} T^{23} + 117342 p^{8} T^{24} + 43443 p^{9} T^{25} - 11532 p^{10} T^{26} - 3995 p^{11} T^{27} + 1054 p^{12} T^{28} + 148 p^{13} T^{29} - 50 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32}
17 1−11T+36T2+123T3−1898T4+9669T5−17977T6−83870T7+834166T8−3577974T9+6674756T10+20040864T11−226666002T12+969820747T13−1237622341T14−9875829222T15+67383134907T16−9875829222pT17−1237622341p2T18+969820747p3T19−226666002p4T20+20040864p5T21+6674756p6T22−3577974p7T23+834166p8T24−83870p9T25−17977p10T26+9669p11T27−1898p12T28+123p13T29+36p14T30−11p15T31+p16T32 1 - 11 T + 36 T^{2} + 123 T^{3} - 1898 T^{4} + 9669 T^{5} - 17977 T^{6} - 83870 T^{7} + 834166 T^{8} - 3577974 T^{9} + 6674756 T^{10} + 20040864 T^{11} - 226666002 T^{12} + 969820747 T^{13} - 1237622341 T^{14} - 9875829222 T^{15} + 67383134907 T^{16} - 9875829222 p T^{17} - 1237622341 p^{2} T^{18} + 969820747 p^{3} T^{19} - 226666002 p^{4} T^{20} + 20040864 p^{5} T^{21} + 6674756 p^{6} T^{22} - 3577974 p^{7} T^{23} + 834166 p^{8} T^{24} - 83870 p^{9} T^{25} - 17977 p^{10} T^{26} + 9669 p^{11} T^{27} - 1898 p^{12} T^{28} + 123 p^{13} T^{29} + 36 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32}
19 1+2T−54T2+52T3+1522T4−5531T5−18356T6+142945T7−156428T8−2136646T9+5666688T10+23411554T11−135081014T12−508657467T13+5117825840T14+6906391627T15−121849980107T16+6906391627pT17+5117825840p2T18−508657467p3T19−135081014p4T20+23411554p5T21+5666688p6T22−2136646p7T23−156428p8T24+142945p9T25−18356p10T26−5531p11T27+1522p12T28+52p13T29−54p14T30+2p15T31+p16T32 1 + 2 T - 54 T^{2} + 52 T^{3} + 1522 T^{4} - 5531 T^{5} - 18356 T^{6} + 142945 T^{7} - 156428 T^{8} - 2136646 T^{9} + 5666688 T^{10} + 23411554 T^{11} - 135081014 T^{12} - 508657467 T^{13} + 5117825840 T^{14} + 6906391627 T^{15} - 121849980107 T^{16} + 6906391627 p T^{17} + 5117825840 p^{2} T^{18} - 508657467 p^{3} T^{19} - 135081014 p^{4} T^{20} + 23411554 p^{5} T^{21} + 5666688 p^{6} T^{22} - 2136646 p^{7} T^{23} - 156428 p^{8} T^{24} + 142945 p^{9} T^{25} - 18356 p^{10} T^{26} - 5531 p^{11} T^{27} + 1522 p^{12} T^{28} + 52 p^{13} T^{29} - 54 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32}
23 1−11T−8T2+511T3−1175T4−318pT5+31375T6−146176T7+863162T8+3529440T9−34166005T10+8815708T11+134850764T12−1234120523T13+19738400159T14+2588217123T15−586357209291T16+2588217123pT17+19738400159p2T18−1234120523p3T19+134850764p4T20+8815708p5T21−34166005p6T22+3529440p7T23+863162p8T24−146176p9T25+31375p10T26−318p12T27−1175p12T28+511p13T29−8p14T30−11p15T31+p16T32 1 - 11 T - 8 T^{2} + 511 T^{3} - 1175 T^{4} - 318 p T^{5} + 31375 T^{6} - 146176 T^{7} + 863162 T^{8} + 3529440 T^{9} - 34166005 T^{10} + 8815708 T^{11} + 134850764 T^{12} - 1234120523 T^{13} + 19738400159 T^{14} + 2588217123 T^{15} - 586357209291 T^{16} + 2588217123 p T^{17} + 19738400159 p^{2} T^{18} - 1234120523 p^{3} T^{19} + 134850764 p^{4} T^{20} + 8815708 p^{5} T^{21} - 34166005 p^{6} T^{22} + 3529440 p^{7} T^{23} + 863162 p^{8} T^{24} - 146176 p^{9} T^{25} + 31375 p^{10} T^{26} - 318 p^{12} T^{27} - 1175 p^{12} T^{28} + 511 p^{13} T^{29} - 8 p^{14} T^{30} - 11 p^{15} T^{31} + p^{16} T^{32}
29 1−17T+46T2+223T3+3250T4−18084T5−204302T6+750182T7+3378518T8+10195995T9−165509581T10+1120124pT11+4544588759T12−14362280831T13−142821670546T14+116125876596T15+6133655884938T16+116125876596pT17−142821670546p2T18−14362280831p3T19+4544588759p4T20+1120124p6T21−165509581p6T22+10195995p7T23+3378518p8T24+750182p9T25−204302p10T26−18084p11T27+3250p12T28+223p13T29+46p14T30−17p15T31+p16T32 1 - 17 T + 46 T^{2} + 223 T^{3} + 3250 T^{4} - 18084 T^{5} - 204302 T^{6} + 750182 T^{7} + 3378518 T^{8} + 10195995 T^{9} - 165509581 T^{10} + 1120124 p T^{11} + 4544588759 T^{12} - 14362280831 T^{13} - 142821670546 T^{14} + 116125876596 T^{15} + 6133655884938 T^{16} + 116125876596 p T^{17} - 142821670546 p^{2} T^{18} - 14362280831 p^{3} T^{19} + 4544588759 p^{4} T^{20} + 1120124 p^{6} T^{21} - 165509581 p^{6} T^{22} + 10195995 p^{7} T^{23} + 3378518 p^{8} T^{24} + 750182 p^{9} T^{25} - 204302 p^{10} T^{26} - 18084 p^{11} T^{27} + 3250 p^{12} T^{28} + 223 p^{13} T^{29} + 46 p^{14} T^{30} - 17 p^{15} T^{31} + p^{16} T^{32}
31 (1−15T+294T2−2963T3+33518T4−256035T5+2103711T6−12738397T7+81705846T8−12738397pT9+2103711p2T10−256035p3T11+33518p4T12−2963p5T13+294p6T14−15p7T15+p8T16)2 ( 1 - 15 T + 294 T^{2} - 2963 T^{3} + 33518 T^{4} - 256035 T^{5} + 2103711 T^{6} - 12738397 T^{7} + 81705846 T^{8} - 12738397 p T^{9} + 2103711 p^{2} T^{10} - 256035 p^{3} T^{11} + 33518 p^{4} T^{12} - 2963 p^{5} T^{13} + 294 p^{6} T^{14} - 15 p^{7} T^{15} + p^{8} T^{16} )^{2}
37 1+2T−109T2−494T3+4396T4+37709T5+5949T6−1192233T7−6568281T8+428352pT9+180379971T10−663666882T11−2633817279T12+62063414607T13+300032298102T14−1464225969303T15−17940373399443T16−1464225969303pT17+300032298102p2T18+62063414607p3T19−2633817279p4T20−663666882p5T21+180379971p6T22+428352p8T23−6568281p8T24−1192233p9T25+5949p10T26+37709p11T27+4396p12T28−494p13T29−109p14T30+2p15T31+p16T32 1 + 2 T - 109 T^{2} - 494 T^{3} + 4396 T^{4} + 37709 T^{5} + 5949 T^{6} - 1192233 T^{7} - 6568281 T^{8} + 428352 p T^{9} + 180379971 T^{10} - 663666882 T^{11} - 2633817279 T^{12} + 62063414607 T^{13} + 300032298102 T^{14} - 1464225969303 T^{15} - 17940373399443 T^{16} - 1464225969303 p T^{17} + 300032298102 p^{2} T^{18} + 62063414607 p^{3} T^{19} - 2633817279 p^{4} T^{20} - 663666882 p^{5} T^{21} + 180379971 p^{6} T^{22} + 428352 p^{8} T^{23} - 6568281 p^{8} T^{24} - 1192233 p^{9} T^{25} + 5949 p^{10} T^{26} + 37709 p^{11} T^{27} + 4396 p^{12} T^{28} - 494 p^{13} T^{29} - 109 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32}
41 1−7T−58T2+821T3−5081T4+3487T5+202690T6−2313484T7+22320617T8−84574319T9−449228446T10+8174132915T11−61865466443T12+229869969509T13+311827818587T14−13738704403974T15+126160448679906T16−13738704403974pT17+311827818587p2T18+229869969509p3T19−61865466443p4T20+8174132915p5T21−449228446p6T22−84574319p7T23+22320617p8T24−2313484p9T25+202690p10T26+3487p11T27−5081p12T28+821p13T29−58p14T30−7p15T31+p16T32 1 - 7 T - 58 T^{2} + 821 T^{3} - 5081 T^{4} + 3487 T^{5} + 202690 T^{6} - 2313484 T^{7} + 22320617 T^{8} - 84574319 T^{9} - 449228446 T^{10} + 8174132915 T^{11} - 61865466443 T^{12} + 229869969509 T^{13} + 311827818587 T^{14} - 13738704403974 T^{15} + 126160448679906 T^{16} - 13738704403974 p T^{17} + 311827818587 p^{2} T^{18} + 229869969509 p^{3} T^{19} - 61865466443 p^{4} T^{20} + 8174132915 p^{5} T^{21} - 449228446 p^{6} T^{22} - 84574319 p^{7} T^{23} + 22320617 p^{8} T^{24} - 2313484 p^{9} T^{25} + 202690 p^{10} T^{26} + 3487 p^{11} T^{27} - 5081 p^{12} T^{28} + 821 p^{13} T^{29} - 58 p^{14} T^{30} - 7 p^{15} T^{31} + p^{16} T^{32}
43 1+13T−110T2−1567T3+10823T4+109505T5−883584T6−5592870T7+52864479T8+206005533T9−2766579456T10−7286801151T11+131529658809T12+225678713547T13−6233727820281T14−3123476792046T15+288754313221062T16−3123476792046pT17−6233727820281p2T18+225678713547p3T19+131529658809p4T20−7286801151p5T21−2766579456p6T22+206005533p7T23+52864479p8T24−5592870p9T25−883584p10T26+109505p11T27+10823p12T28−1567p13T29−110p14T30+13p15T31+p16T32 1 + 13 T - 110 T^{2} - 1567 T^{3} + 10823 T^{4} + 109505 T^{5} - 883584 T^{6} - 5592870 T^{7} + 52864479 T^{8} + 206005533 T^{9} - 2766579456 T^{10} - 7286801151 T^{11} + 131529658809 T^{12} + 225678713547 T^{13} - 6233727820281 T^{14} - 3123476792046 T^{15} + 288754313221062 T^{16} - 3123476792046 p T^{17} - 6233727820281 p^{2} T^{18} + 225678713547 p^{3} T^{19} + 131529658809 p^{4} T^{20} - 7286801151 p^{5} T^{21} - 2766579456 p^{6} T^{22} + 206005533 p^{7} T^{23} + 52864479 p^{8} T^{24} - 5592870 p^{9} T^{25} - 883584 p^{10} T^{26} + 109505 p^{11} T^{27} + 10823 p^{12} T^{28} - 1567 p^{13} T^{29} - 110 p^{14} T^{30} + 13 p^{15} T^{31} + p^{16} T^{32}
47 (1−5T+167T2−679T3+12865T4−55056T5+721843T6−3490742T7+35510129T8−3490742pT9+721843p2T10−55056p3T11+12865p4T12−679p5T13+167p6T14−5p7T15+p8T16)2 ( 1 - 5 T + 167 T^{2} - 679 T^{3} + 12865 T^{4} - 55056 T^{5} + 721843 T^{6} - 3490742 T^{7} + 35510129 T^{8} - 3490742 p T^{9} + 721843 p^{2} T^{10} - 55056 p^{3} T^{11} + 12865 p^{4} T^{12} - 679 p^{5} T^{13} + 167 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} )^{2}
53 1−18T−25T2+3094T3−20382T4−155341T5+2463947T6−2962263T7−127200023T8+760818902T9+3333886647T10−61667632700T11+106780502839T12+3773454519213T13−28393687643164T14−98984202945069T15+2205152767325745T16−98984202945069pT17−28393687643164p2T18+3773454519213p3T19+106780502839p4T20−61667632700p5T21+3333886647p6T22+760818902p7T23−127200023p8T24−2962263p9T25+2463947p10T26−155341p11T27−20382p12T28+3094p13T29−25p14T30−18p15T31+p16T32 1 - 18 T - 25 T^{2} + 3094 T^{3} - 20382 T^{4} - 155341 T^{5} + 2463947 T^{6} - 2962263 T^{7} - 127200023 T^{8} + 760818902 T^{9} + 3333886647 T^{10} - 61667632700 T^{11} + 106780502839 T^{12} + 3773454519213 T^{13} - 28393687643164 T^{14} - 98984202945069 T^{15} + 2205152767325745 T^{16} - 98984202945069 p T^{17} - 28393687643164 p^{2} T^{18} + 3773454519213 p^{3} T^{19} + 106780502839 p^{4} T^{20} - 61667632700 p^{5} T^{21} + 3333886647 p^{6} T^{22} + 760818902 p^{7} T^{23} - 127200023 p^{8} T^{24} - 2962263 p^{9} T^{25} + 2463947 p^{10} T^{26} - 155341 p^{11} T^{27} - 20382 p^{12} T^{28} + 3094 p^{13} T^{29} - 25 p^{14} T^{30} - 18 p^{15} T^{31} + p^{16} T^{32}
59 (1+T+317T2+659T3+48346T4+123360T5+4790140T6+11791354T7+334628690T8+11791354pT9+4790140p2T10+123360p3T11+48346p4T12+659p5T13+317p6T14+p7T15+p8T16)2 ( 1 + T + 317 T^{2} + 659 T^{3} + 48346 T^{4} + 123360 T^{5} + 4790140 T^{6} + 11791354 T^{7} + 334628690 T^{8} + 11791354 p T^{9} + 4790140 p^{2} T^{10} + 123360 p^{3} T^{11} + 48346 p^{4} T^{12} + 659 p^{5} T^{13} + 317 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} )^{2}
61 (1−27T+644T2−9523T3+127396T4−1294067T5+12537951T6−101639815T7+845706468T8−101639815pT9+12537951p2T10−1294067p3T11+127396p4T12−9523p5T13+644p6T14−27p7T15+p8T16)2 ( 1 - 27 T + 644 T^{2} - 9523 T^{3} + 127396 T^{4} - 1294067 T^{5} + 12537951 T^{6} - 101639815 T^{7} + 845706468 T^{8} - 101639815 p T^{9} + 12537951 p^{2} T^{10} - 1294067 p^{3} T^{11} + 127396 p^{4} T^{12} - 9523 p^{5} T^{13} + 644 p^{6} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} )^{2}
67 (1−10T+183T2−1911T3+28302T4−239130T5+2683324T6−21197185T7+215653584T8−21197185pT9+2683324p2T10−239130p3T11+28302p4T12−1911p5T13+183p6T14−10p7T15+p8T16)2 ( 1 - 10 T + 183 T^{2} - 1911 T^{3} + 28302 T^{4} - 239130 T^{5} + 2683324 T^{6} - 21197185 T^{7} + 215653584 T^{8} - 21197185 p T^{9} + 2683324 p^{2} T^{10} - 239130 p^{3} T^{11} + 28302 p^{4} T^{12} - 1911 p^{5} T^{13} + 183 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2}
71 (1+19T+400T2+3773T3+41473T4+130458T5+473407T6−19694168T7−122824232T8−19694168pT9+473407p2T10+130458p3T11+41473p4T12+3773p5T13+400p6T14+19p7T15+p8T16)2 ( 1 + 19 T + 400 T^{2} + 3773 T^{3} + 41473 T^{4} + 130458 T^{5} + 473407 T^{6} - 19694168 T^{7} - 122824232 T^{8} - 19694168 p T^{9} + 473407 p^{2} T^{10} + 130458 p^{3} T^{11} + 41473 p^{4} T^{12} + 3773 p^{5} T^{13} + 400 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} )^{2}
73 1+8T−390T2−908T3+103150T4−105353T5−16697378T6+75246967T7+1927206172T8−14189615254T9−139595267550T10+1686118517626T11+5218854241786T12−119151906708597T13+190749604993352T14+3821131738775017T15−31483648169073569T16+3821131738775017pT17+190749604993352p2T18−119151906708597p3T19+5218854241786p4T20+1686118517626p5T21−139595267550p6T22−14189615254p7T23+1927206172p8T24+75246967p9T25−16697378p10T26−105353p11T27+103150p12T28−908p13T29−390p14T30+8p15T31+p16T32 1 + 8 T - 390 T^{2} - 908 T^{3} + 103150 T^{4} - 105353 T^{5} - 16697378 T^{6} + 75246967 T^{7} + 1927206172 T^{8} - 14189615254 T^{9} - 139595267550 T^{10} + 1686118517626 T^{11} + 5218854241786 T^{12} - 119151906708597 T^{13} + 190749604993352 T^{14} + 3821131738775017 T^{15} - 31483648169073569 T^{16} + 3821131738775017 p T^{17} + 190749604993352 p^{2} T^{18} - 119151906708597 p^{3} T^{19} + 5218854241786 p^{4} T^{20} + 1686118517626 p^{5} T^{21} - 139595267550 p^{6} T^{22} - 14189615254 p^{7} T^{23} + 1927206172 p^{8} T^{24} + 75246967 p^{9} T^{25} - 16697378 p^{10} T^{26} - 105353 p^{11} T^{27} + 103150 p^{12} T^{28} - 908 p^{13} T^{29} - 390 p^{14} T^{30} + 8 p^{15} T^{31} + p^{16} T^{32}
79 (1−25T+807T2−13099T3+239467T4−2871909T5+37853854T6−357554245T7+46953672pT8−357554245pT9+37853854p2T10−2871909p3T11+239467p4T12−13099p5T13+807p6T14−25p7T15+p8T16)2 ( 1 - 25 T + 807 T^{2} - 13099 T^{3} + 239467 T^{4} - 2871909 T^{5} + 37853854 T^{6} - 357554245 T^{7} + 46953672 p T^{8} - 357554245 p T^{9} + 37853854 p^{2} T^{10} - 2871909 p^{3} T^{11} + 239467 p^{4} T^{12} - 13099 p^{5} T^{13} + 807 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} )^{2}
83 1−2T−379T2+46T3+71929T4+111425T5−9247598T6−27250484T7+925626125T8+3684425840T9−76460286955T10−351032380757T11+5350114348114T12+23636419050310T13−345108599067805T14−759472537118196T15+25092808099894680T16−759472537118196pT17−345108599067805p2T18+23636419050310p3T19+5350114348114p4T20−351032380757p5T21−76460286955p6T22+3684425840p7T23+925626125p8T24−27250484p9T25−9247598p10T26+111425p11T27+71929p12T28+46p13T29−379p14T30−2p15T31+p16T32 1 - 2 T - 379 T^{2} + 46 T^{3} + 71929 T^{4} + 111425 T^{5} - 9247598 T^{6} - 27250484 T^{7} + 925626125 T^{8} + 3684425840 T^{9} - 76460286955 T^{10} - 351032380757 T^{11} + 5350114348114 T^{12} + 23636419050310 T^{13} - 345108599067805 T^{14} - 759472537118196 T^{15} + 25092808099894680 T^{16} - 759472537118196 p T^{17} - 345108599067805 p^{2} T^{18} + 23636419050310 p^{3} T^{19} + 5350114348114 p^{4} T^{20} - 351032380757 p^{5} T^{21} - 76460286955 p^{6} T^{22} + 3684425840 p^{7} T^{23} + 925626125 p^{8} T^{24} - 27250484 p^{9} T^{25} - 9247598 p^{10} T^{26} + 111425 p^{11} T^{27} + 71929 p^{12} T^{28} + 46 p^{13} T^{29} - 379 p^{14} T^{30} - 2 p^{15} T^{31} + p^{16} T^{32}
89 1+6T−361T2−3094T3+62940T4+748723T5−6298663T6−119355885T7+250393933T8+13841837530T9+33272999661T10−1196873353342T11−8138474562173T12+74723700233589T13+1028573157441158T14−2348330140909041T15−100635305983812723T16−2348330140909041pT17+1028573157441158p2T18+74723700233589p3T19−8138474562173p4T20−1196873353342p5T21+33272999661p6T22+13841837530p7T23+250393933p8T24−119355885p9T25−6298663p10T26+748723p11T27+62940p12T28−3094p13T29−361p14T30+6p15T31+p16T32 1 + 6 T - 361 T^{2} - 3094 T^{3} + 62940 T^{4} + 748723 T^{5} - 6298663 T^{6} - 119355885 T^{7} + 250393933 T^{8} + 13841837530 T^{9} + 33272999661 T^{10} - 1196873353342 T^{11} - 8138474562173 T^{12} + 74723700233589 T^{13} + 1028573157441158 T^{14} - 2348330140909041 T^{15} - 100635305983812723 T^{16} - 2348330140909041 p T^{17} + 1028573157441158 p^{2} T^{18} + 74723700233589 p^{3} T^{19} - 8138474562173 p^{4} T^{20} - 1196873353342 p^{5} T^{21} + 33272999661 p^{6} T^{22} + 13841837530 p^{7} T^{23} + 250393933 p^{8} T^{24} - 119355885 p^{9} T^{25} - 6298663 p^{10} T^{26} + 748723 p^{11} T^{27} + 62940 p^{12} T^{28} - 3094 p^{13} T^{29} - 361 p^{14} T^{30} + 6 p^{15} T^{31} + p^{16} T^{32}
97 1−26T−56T2+2492T3+90143T4−817018T5−11928114T6−57652506T7+2793757095T8+11900406246T9−106656506274T10−4250712480600T11+2907448664925T12+284134319779998T13+4080312688533132T14−249527458766268pT15−351229803717868731T16−249527458766268p2T17+4080312688533132p2T18+284134319779998p3T19+2907448664925p4T20−4250712480600p5T21−106656506274p6T22+11900406246p7T23+2793757095p8T24−57652506p9T25−11928114p10T26−817018p11T27+90143p12T28+2492p13T29−56p14T30−26p15T31+p16T32 1 - 26 T - 56 T^{2} + 2492 T^{3} + 90143 T^{4} - 817018 T^{5} - 11928114 T^{6} - 57652506 T^{7} + 2793757095 T^{8} + 11900406246 T^{9} - 106656506274 T^{10} - 4250712480600 T^{11} + 2907448664925 T^{12} + 284134319779998 T^{13} + 4080312688533132 T^{14} - 249527458766268 p T^{15} - 351229803717868731 T^{16} - 249527458766268 p^{2} T^{17} + 4080312688533132 p^{2} T^{18} + 284134319779998 p^{3} T^{19} + 2907448664925 p^{4} T^{20} - 4250712480600 p^{5} T^{21} - 106656506274 p^{6} T^{22} + 11900406246 p^{7} T^{23} + 2793757095 p^{8} T^{24} - 57652506 p^{9} T^{25} - 11928114 p^{10} T^{26} - 817018 p^{11} T^{27} + 90143 p^{12} T^{28} + 2492 p^{13} T^{29} - 56 p^{14} T^{30} - 26 p^{15} T^{31} + p^{16} 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

−2.62795830361231431315216521751, −2.53031063379358800228686114144, −2.43552873561276168320235241440, −2.38950179677767182420383218145, −2.22940446256406910489669197778, −2.21759646319098869761016061067, −2.19154741720087814963968006267, −2.15244198884246532912923334627, −1.95130698274305457789969861333, −1.84113802483322056501018123538, −1.81676894652539480540176291978, −1.76487442993915784144351759858, −1.74492879988632525760616594683, −1.49658977389916115675831477868, −1.35100470718697936698890405560, −1.34140063108091855632433071203, −1.11237675984058315613027233949, −1.08445874562826869742992800865, −1.04524203634278503185634118968, −0.859189649824870529058602089556, −0.819694694871868081607301204702, −0.77336241069871618124826443583, −0.73193651490986429127265168186, −0.48967435316127325420660519437, −0.46979706644094715782197522417, 0.46979706644094715782197522417, 0.48967435316127325420660519437, 0.73193651490986429127265168186, 0.77336241069871618124826443583, 0.819694694871868081607301204702, 0.859189649824870529058602089556, 1.04524203634278503185634118968, 1.08445874562826869742992800865, 1.11237675984058315613027233949, 1.34140063108091855632433071203, 1.35100470718697936698890405560, 1.49658977389916115675831477868, 1.74492879988632525760616594683, 1.76487442993915784144351759858, 1.81676894652539480540176291978, 1.84113802483322056501018123538, 1.95130698274305457789969861333, 2.15244198884246532912923334627, 2.19154741720087814963968006267, 2.21759646319098869761016061067, 2.22940446256406910489669197778, 2.38950179677767182420383218145, 2.43552873561276168320235241440, 2.53031063379358800228686114144, 2.62795830361231431315216521751

Graph of the ZZ-function along the critical line

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