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Properties

Label 24-338e12-1.1-c3e12-0-0
Degree 2424
Conductor 2.223×10302.223\times 10^{30}
Sign 11
Analytic cond. 3.95724×10153.95724\times 10^{15}
Root an. cond. 4.465714.46571
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  + 12·2-s − 9·3-s + 60·4-s − 36·5-s − 108·6-s + 25·7-s + 112·8-s + 65·9-s − 432·10-s + 37·11-s − 540·12-s + 300·14-s + 324·15-s − 336·16-s − 99·17-s + 780·18-s − 81·19-s − 2.16e3·20-s − 225·21-s + 444·22-s − 267·23-s − 1.00e3·24-s + 266·25-s − 58·27-s + 1.50e3·28-s + 119·29-s + 3.88e3·30-s + ⋯
L(s)  = 1  + 4.24·2-s − 1.73·3-s + 15/2·4-s − 3.21·5-s − 7.34·6-s + 1.34·7-s + 4.94·8-s + 2.40·9-s − 13.6·10-s + 1.01·11-s − 12.9·12-s + 5.72·14-s + 5.57·15-s − 5.25·16-s − 1.41·17-s + 10.2·18-s − 0.978·19-s − 24.1·20-s − 2.33·21-s + 4.30·22-s − 2.42·23-s − 8.57·24-s + 2.12·25-s − 0.413·27-s + 10.1·28-s + 0.761·29-s + 23.6·30-s + ⋯

Functional equation

Λ(s)=((212⋅1324)s/2 ΓC(s)12 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((212⋅1324)s/2 ΓC(s+3/2)12 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 2424
Conductor: 212⋅13242^{12} \cdot 13^{24}
Sign: 11
Analytic conductor: 3.95724×10153.95724\times 10^{15}
Root analytic conductor: 4.465714.46571
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (24, 212⋅1324, ( :[3/2]12), 1)(24,\ 2^{12} \cdot 13^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )

Particular Values

L(2)L(2) ≈\approx 0.85832918420.8583291842
L(12)L(\frac12) ≈\approx 0.85832918420.8583291842
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)6 ( 1 - p T + p^{2} T^{2} )^{6}
13 1 1
good3 1+p2T+16T2−383T3−2903T4−1330pT5+9848p2T6+594275T7+114521pT8−15120890T9−65767805T10+130939027T11+2123613139T12+130939027p3T13−65767805p6T14−15120890p9T15+114521p13T16+594275p15T17+9848p20T18−1330p22T19−2903p24T20−383p27T21+16p30T22+p35T23+p36T24 1 + p^{2} T + 16 T^{2} - 383 T^{3} - 2903 T^{4} - 1330 p T^{5} + 9848 p^{2} T^{6} + 594275 T^{7} + 114521 p T^{8} - 15120890 T^{9} - 65767805 T^{10} + 130939027 T^{11} + 2123613139 T^{12} + 130939027 p^{3} T^{13} - 65767805 p^{6} T^{14} - 15120890 p^{9} T^{15} + 114521 p^{13} T^{16} + 594275 p^{15} T^{17} + 9848 p^{20} T^{18} - 1330 p^{22} T^{19} - 2903 p^{24} T^{20} - 383 p^{27} T^{21} + 16 p^{30} T^{22} + p^{35} T^{23} + p^{36} T^{24}
5 (1+18T+353T2+1063pT3+15993pT4+1003083T5+12801634T6+1003083p3T7+15993p7T8+1063p10T9+353p12T10+18p15T11+p18T12)2 ( 1 + 18 T + 353 T^{2} + 1063 p T^{3} + 15993 p T^{4} + 1003083 T^{5} + 12801634 T^{6} + 1003083 p^{3} T^{7} + 15993 p^{7} T^{8} + 1063 p^{10} T^{9} + 353 p^{12} T^{10} + 18 p^{15} T^{11} + p^{18} T^{12} )^{2}
7 1−25T−267T2+1544pT3−41021T4−1296635T5+23995119T6+9910501pT7−1232685738pT8−31346785682pT9+154528034457p2T10+1657960494149p2T11−89824498888557p2T12+1657960494149p5T13+154528034457p8T14−31346785682p10T15−1232685738p13T16+9910501p16T17+23995119p18T18−1296635p21T19−41021p24T20+1544p28T21−267p30T22−25p33T23+p36T24 1 - 25 T - 267 T^{2} + 1544 p T^{3} - 41021 T^{4} - 1296635 T^{5} + 23995119 T^{6} + 9910501 p T^{7} - 1232685738 p T^{8} - 31346785682 p T^{9} + 154528034457 p^{2} T^{10} + 1657960494149 p^{2} T^{11} - 89824498888557 p^{2} T^{12} + 1657960494149 p^{5} T^{13} + 154528034457 p^{8} T^{14} - 31346785682 p^{10} T^{15} - 1232685738 p^{13} T^{16} + 9910501 p^{16} T^{17} + 23995119 p^{18} T^{18} - 1296635 p^{21} T^{19} - 41021 p^{24} T^{20} + 1544 p^{28} T^{21} - 267 p^{30} T^{22} - 25 p^{33} T^{23} + p^{36} T^{24}
11 1−37T−3374T2+90165T3+6920025T4−72282084T5−9465779646T6−122761204227T7+992732213841pT8+2107273243932p2T9−5376348087443175T10−186152687993983949T11+1687873828970897175T12−186152687993983949p3T13−5376348087443175p6T14+2107273243932p11T15+992732213841p13T16−122761204227p15T17−9465779646p18T18−72282084p21T19+6920025p24T20+90165p27T21−3374p30T22−37p33T23+p36T24 1 - 37 T - 3374 T^{2} + 90165 T^{3} + 6920025 T^{4} - 72282084 T^{5} - 9465779646 T^{6} - 122761204227 T^{7} + 992732213841 p T^{8} + 2107273243932 p^{2} T^{9} - 5376348087443175 T^{10} - 186152687993983949 T^{11} + 1687873828970897175 T^{12} - 186152687993983949 p^{3} T^{13} - 5376348087443175 p^{6} T^{14} + 2107273243932 p^{11} T^{15} + 992732213841 p^{13} T^{16} - 122761204227 p^{15} T^{17} - 9465779646 p^{18} T^{18} - 72282084 p^{21} T^{19} + 6920025 p^{24} T^{20} + 90165 p^{27} T^{21} - 3374 p^{30} T^{22} - 37 p^{33} T^{23} + p^{36} T^{24}
17 1+99T−5434T2−1327755T3−64109215T4+2302353130T5+284397483536T6+14376654383487T7+1730888463481815T8+160829292478556350T9+2491616674911046785T10− 1 + 99 T - 5434 T^{2} - 1327755 T^{3} - 64109215 T^{4} + 2302353130 T^{5} + 284397483536 T^{6} + 14376654383487 T^{7} + 1730888463481815 T^{8} + 160829292478556350 T^{9} + 2491616674911046785 T^{10} - 92 ⁣⋯ ⁣3392\!\cdots\!33T11− T^{11} - 97 ⁣⋯ ⁣9197\!\cdots\!91T12− T^{12} - 92 ⁣⋯ ⁣3392\!\cdots\!33p3T13+2491616674911046785p6T14+160829292478556350p9T15+1730888463481815p12T16+14376654383487p15T17+284397483536p18T18+2302353130p21T19−64109215p24T20−1327755p27T21−5434p30T22+99p33T23+p36T24 p^{3} T^{13} + 2491616674911046785 p^{6} T^{14} + 160829292478556350 p^{9} T^{15} + 1730888463481815 p^{12} T^{16} + 14376654383487 p^{15} T^{17} + 284397483536 p^{18} T^{18} + 2302353130 p^{21} T^{19} - 64109215 p^{24} T^{20} - 1327755 p^{27} T^{21} - 5434 p^{30} T^{22} + 99 p^{33} T^{23} + p^{36} T^{24}
19 1+81T−14726T2−186209T3+166884549T4−6779420944T5−866610386870T6+89433153834823T7+664547746074283T8−502153099717971364T9+11987237177864360773T10+56646495765690815391pT11− 1 + 81 T - 14726 T^{2} - 186209 T^{3} + 166884549 T^{4} - 6779420944 T^{5} - 866610386870 T^{6} + 89433153834823 T^{7} + 664547746074283 T^{8} - 502153099717971364 T^{9} + 11987237177864360773 T^{10} + 56646495765690815391 p T^{11} - 88 ⁣⋯ ⁣5388\!\cdots\!53T12+56646495765690815391p4T13+11987237177864360773p6T14−502153099717971364p9T15+664547746074283p12T16+89433153834823p15T17−866610386870p18T18−6779420944p21T19+166884549p24T20−186209p27T21−14726p30T22+81p33T23+p36T24 T^{12} + 56646495765690815391 p^{4} T^{13} + 11987237177864360773 p^{6} T^{14} - 502153099717971364 p^{9} T^{15} + 664547746074283 p^{12} T^{16} + 89433153834823 p^{15} T^{17} - 866610386870 p^{18} T^{18} - 6779420944 p^{21} T^{19} + 166884549 p^{24} T^{20} - 186209 p^{27} T^{21} - 14726 p^{30} T^{22} + 81 p^{33} T^{23} + p^{36} T^{24}
23 1+267T+5996T2−5270773T3−672443396T4−31932625905T5−1227012468944T6+664024075258251T7+222364660120074076T8+18529838508574461255T9− 1 + 267 T + 5996 T^{2} - 5270773 T^{3} - 672443396 T^{4} - 31932625905 T^{5} - 1227012468944 T^{6} + 664024075258251 T^{7} + 222364660120074076 T^{8} + 18529838508574461255 T^{9} - 82 ⁣⋯ ⁣3282\!\cdots\!32T10− T^{10} - 95 ⁣⋯ ⁣3595\!\cdots\!35pT11− p T^{11} - 20 ⁣⋯ ⁣1820\!\cdots\!18T12− T^{12} - 95 ⁣⋯ ⁣3595\!\cdots\!35p4T13− p^{4} T^{13} - 82 ⁣⋯ ⁣3282\!\cdots\!32p6T14+18529838508574461255p9T15+222364660120074076p12T16+664024075258251p15T17−1227012468944p18T18−31932625905p21T19−672443396p24T20−5270773p27T21+5996p30T22+267p33T23+p36T24 p^{6} T^{14} + 18529838508574461255 p^{9} T^{15} + 222364660120074076 p^{12} T^{16} + 664024075258251 p^{15} T^{17} - 1227012468944 p^{18} T^{18} - 31932625905 p^{21} T^{19} - 672443396 p^{24} T^{20} - 5270773 p^{27} T^{21} + 5996 p^{30} T^{22} + 267 p^{33} T^{23} + p^{36} T^{24}
29 1−119T−110183T2+8876420T3+7293704953T4−379237642413T5−340028891349473T6+10660672822726647T7+12332118877300770900T8− 1 - 119 T - 110183 T^{2} + 8876420 T^{3} + 7293704953 T^{4} - 379237642413 T^{5} - 340028891349473 T^{6} + 10660672822726647 T^{7} + 12332118877300770900 T^{8} - 19 ⁣⋯ ⁣1419\!\cdots\!14T9− T^{9} - 37 ⁣⋯ ⁣7537\!\cdots\!75T10+ T^{10} + 63 ⁣⋯ ⁣6963\!\cdots\!69pT11+ p T^{11} + 11 ⁣⋯ ⁣7511\!\cdots\!75p2T12+ p^{2} T^{12} + 63 ⁣⋯ ⁣6963\!\cdots\!69p4T13− p^{4} T^{13} - 37 ⁣⋯ ⁣7537\!\cdots\!75p6T14− p^{6} T^{14} - 19 ⁣⋯ ⁣1419\!\cdots\!14p9T15+12332118877300770900p12T16+10660672822726647p15T17−340028891349473p18T18−379237642413p21T19+7293704953p24T20+8876420p27T21−110183p30T22−119p33T23+p36T24 p^{9} T^{15} + 12332118877300770900 p^{12} T^{16} + 10660672822726647 p^{15} T^{17} - 340028891349473 p^{18} T^{18} - 379237642413 p^{21} T^{19} + 7293704953 p^{24} T^{20} + 8876420 p^{27} T^{21} - 110183 p^{30} T^{22} - 119 p^{33} T^{23} + p^{36} T^{24}
31 (1−625T+316140T2−104356466T3+29721828071T4−6530570693913T5+1262869136265616T6−6530570693913p3T7+29721828071p6T8−104356466p9T9+316140p12T10−625p15T11+p18T12)2 ( 1 - 625 T + 316140 T^{2} - 104356466 T^{3} + 29721828071 T^{4} - 6530570693913 T^{5} + 1262869136265616 T^{6} - 6530570693913 p^{3} T^{7} + 29721828071 p^{6} T^{8} - 104356466 p^{9} T^{9} + 316140 p^{12} T^{10} - 625 p^{15} T^{11} + p^{18} T^{12} )^{2}
37 1−274T−172179T2+31298256T3+20598937694T4−1795401545956T5−1915135012258020T6+108121656871938888T7+ 1 - 274 T - 172179 T^{2} + 31298256 T^{3} + 20598937694 T^{4} - 1795401545956 T^{5} - 1915135012258020 T^{6} + 108121656871938888 T^{7} + 13 ⁣⋯ ⁣1313\!\cdots\!13T8− T^{8} - 43 ⁣⋯ ⁣4043\!\cdots\!40T9− T^{9} - 83 ⁣⋯ ⁣9383\!\cdots\!93T10+ T^{10} + 63 ⁣⋯ ⁣3063\!\cdots\!30T11+ T^{11} + 46 ⁣⋯ ⁣6146\!\cdots\!61T12+ T^{12} + 63 ⁣⋯ ⁣3063\!\cdots\!30p3T13− p^{3} T^{13} - 83 ⁣⋯ ⁣9383\!\cdots\!93p6T14− p^{6} T^{14} - 43 ⁣⋯ ⁣4043\!\cdots\!40p9T15+ p^{9} T^{15} + 13 ⁣⋯ ⁣1313\!\cdots\!13p12T16+108121656871938888p15T17−1915135012258020p18T18−1795401545956p21T19+20598937694p24T20+31298256p27T21−172179p30T22−274p33T23+p36T24 p^{12} T^{16} + 108121656871938888 p^{15} T^{17} - 1915135012258020 p^{18} T^{18} - 1795401545956 p^{21} T^{19} + 20598937694 p^{24} T^{20} + 31298256 p^{27} T^{21} - 172179 p^{30} T^{22} - 274 p^{33} T^{23} + p^{36} T^{24}
41 1−1140T+431902T2−53291206T3+18188754925T4−16860355235744T5+5182958888100297T6−1001584729781787693T7+9902716607030270666pT8− 1 - 1140 T + 431902 T^{2} - 53291206 T^{3} + 18188754925 T^{4} - 16860355235744 T^{5} + 5182958888100297 T^{6} - 1001584729781787693 T^{7} + 9902716607030270666 p T^{8} - 12 ⁣⋯ ⁣6212\!\cdots\!62T9+ T^{9} + 19 ⁣⋯ ⁣9919\!\cdots\!99T10− T^{10} - 81 ⁣⋯ ⁣1581\!\cdots\!15T11+ T^{11} + 33 ⁣⋯ ⁣1633\!\cdots\!16T12− T^{12} - 81 ⁣⋯ ⁣1581\!\cdots\!15p3T13+ p^{3} T^{13} + 19 ⁣⋯ ⁣9919\!\cdots\!99p6T14− p^{6} T^{14} - 12 ⁣⋯ ⁣6212\!\cdots\!62p9T15+9902716607030270666p13T16−1001584729781787693p15T17+5182958888100297p18T18−16860355235744p21T19+18188754925p24T20−53291206p27T21+431902p30T22−1140p33T23+p36T24 p^{9} T^{15} + 9902716607030270666 p^{13} T^{16} - 1001584729781787693 p^{15} T^{17} + 5182958888100297 p^{18} T^{18} - 16860355235744 p^{21} T^{19} + 18188754925 p^{24} T^{20} - 53291206 p^{27} T^{21} + 431902 p^{30} T^{22} - 1140 p^{33} T^{23} + p^{36} T^{24}
43 1+428T−16653T2−65902532T3−28639656928T4−5977021820724T5+249322836785029T6+859861367196461312T7+ 1 + 428 T - 16653 T^{2} - 65902532 T^{3} - 28639656928 T^{4} - 5977021820724 T^{5} + 249322836785029 T^{6} + 859861367196461312 T^{7} + 40 ⁣⋯ ⁣8740\!\cdots\!87T8+ T^{8} + 83 ⁣⋯ ⁣0083\!\cdots\!00T9− T^{9} - 29 ⁣⋯ ⁣4829\!\cdots\!48T10− T^{10} - 78 ⁣⋯ ⁣6478\!\cdots\!64T11− T^{11} - 30 ⁣⋯ ⁣7530\!\cdots\!75T12− T^{12} - 78 ⁣⋯ ⁣6478\!\cdots\!64p3T13− p^{3} T^{13} - 29 ⁣⋯ ⁣4829\!\cdots\!48p6T14+ p^{6} T^{14} + 83 ⁣⋯ ⁣0083\!\cdots\!00p9T15+ p^{9} T^{15} + 40 ⁣⋯ ⁣8740\!\cdots\!87p12T16+859861367196461312p15T17+249322836785029p18T18−5977021820724p21T19−28639656928p24T20−65902532p27T21−16653p30T22+428p33T23+p36T24 p^{12} T^{16} + 859861367196461312 p^{15} T^{17} + 249322836785029 p^{18} T^{18} - 5977021820724 p^{21} T^{19} - 28639656928 p^{24} T^{20} - 65902532 p^{27} T^{21} - 16653 p^{30} T^{22} + 428 p^{33} T^{23} + p^{36} T^{24}
47 (1−986T+712667T2−325045705T3+134208823841T4−44205691637095T5+15291628070110878T6−44205691637095p3T7+134208823841p6T8−325045705p9T9+712667p12T10−986p15T11+p18T12)2 ( 1 - 986 T + 712667 T^{2} - 325045705 T^{3} + 134208823841 T^{4} - 44205691637095 T^{5} + 15291628070110878 T^{6} - 44205691637095 p^{3} T^{7} + 134208823841 p^{6} T^{8} - 325045705 p^{9} T^{9} + 712667 p^{12} T^{10} - 986 p^{15} T^{11} + p^{18} T^{12} )^{2}
53 (1−89T+149636T2+19578912T3+11883866543T4−8498293720903T5+5004078025030288T6−8498293720903p3T7+11883866543p6T8+19578912p9T9+149636p12T10−89p15T11+p18T12)2 ( 1 - 89 T + 149636 T^{2} + 19578912 T^{3} + 11883866543 T^{4} - 8498293720903 T^{5} + 5004078025030288 T^{6} - 8498293720903 p^{3} T^{7} + 11883866543 p^{6} T^{8} + 19578912 p^{9} T^{9} + 149636 p^{12} T^{10} - 89 p^{15} T^{11} + p^{18} T^{12} )^{2}
59 1−1088T+328780T2+90009146T3−84336779359T4+13283513005596T5+21739034235208103T6−17322876409308491411T7+ 1 - 1088 T + 328780 T^{2} + 90009146 T^{3} - 84336779359 T^{4} + 13283513005596 T^{5} + 21739034235208103 T^{6} - 17322876409308491411 T^{7} + 17 ⁣⋯ ⁣0217\!\cdots\!02T8+ T^{8} + 27 ⁣⋯ ⁣3227\!\cdots\!32T9− T^{9} - 61 ⁣⋯ ⁣7161\!\cdots\!71T10− T^{10} - 47 ⁣⋯ ⁣9147\!\cdots\!91T11+ T^{11} + 35 ⁣⋯ ⁣9635\!\cdots\!96T12− T^{12} - 47 ⁣⋯ ⁣9147\!\cdots\!91p3T13− p^{3} T^{13} - 61 ⁣⋯ ⁣7161\!\cdots\!71p6T14+ p^{6} T^{14} + 27 ⁣⋯ ⁣3227\!\cdots\!32p9T15+ p^{9} T^{15} + 17 ⁣⋯ ⁣0217\!\cdots\!02p12T16−17322876409308491411p15T17+21739034235208103p18T18+13283513005596p21T19−84336779359p24T20+90009146p27T21+328780p30T22−1088p33T23+p36T24 p^{12} T^{16} - 17322876409308491411 p^{15} T^{17} + 21739034235208103 p^{18} T^{18} + 13283513005596 p^{21} T^{19} - 84336779359 p^{24} T^{20} + 90009146 p^{27} T^{21} + 328780 p^{30} T^{22} - 1088 p^{33} T^{23} + p^{36} T^{24}
61 1+1704T+964913T2+32814990T3−190132450110T4−105404733375058T5−40545666342004564T6+2060628600521217352T7+ 1 + 1704 T + 964913 T^{2} + 32814990 T^{3} - 190132450110 T^{4} - 105404733375058 T^{5} - 40545666342004564 T^{6} + 2060628600521217352 T^{7} + 18 ⁣⋯ ⁣6918\!\cdots\!69T8+ T^{8} + 88 ⁣⋯ ⁣2488\!\cdots\!24T9− T^{9} - 33 ⁣⋯ ⁣0533\!\cdots\!05T10− T^{10} - 16 ⁣⋯ ⁣3016\!\cdots\!30T11− T^{11} - 87 ⁣⋯ ⁣6387\!\cdots\!63T12− T^{12} - 16 ⁣⋯ ⁣3016\!\cdots\!30p3T13− p^{3} T^{13} - 33 ⁣⋯ ⁣0533\!\cdots\!05p6T14+ p^{6} T^{14} + 88 ⁣⋯ ⁣2488\!\cdots\!24p9T15+ p^{9} T^{15} + 18 ⁣⋯ ⁣6918\!\cdots\!69p12T16+2060628600521217352p15T17−40545666342004564p18T18−105404733375058p21T19−190132450110p24T20+32814990p27T21+964913p30T22+1704p33T23+p36T24 p^{12} T^{16} + 2060628600521217352 p^{15} T^{17} - 40545666342004564 p^{18} T^{18} - 105404733375058 p^{21} T^{19} - 190132450110 p^{24} T^{20} + 32814990 p^{27} T^{21} + 964913 p^{30} T^{22} + 1704 p^{33} T^{23} + p^{36} T^{24}
67 1+1692T+127351T2−843881488T3+362460636146T4+759339717368988T5−94435239939881877T6− 1 + 1692 T + 127351 T^{2} - 843881488 T^{3} + 362460636146 T^{4} + 759339717368988 T^{5} - 94435239939881877 T^{6} - 20 ⁣⋯ ⁣6420\!\cdots\!64T7+ T^{7} + 15 ⁣⋯ ⁣8715\!\cdots\!87T8+ T^{8} + 89 ⁣⋯ ⁣6489\!\cdots\!64T9− T^{9} - 49 ⁣⋯ ⁣4849\!\cdots\!48T10− T^{10} - 19 ⁣⋯ ⁣8619\!\cdots\!86T11+ T^{11} + 25 ⁣⋯ ⁣7325\!\cdots\!73T12− T^{12} - 19 ⁣⋯ ⁣8619\!\cdots\!86p3T13− p^{3} T^{13} - 49 ⁣⋯ ⁣4849\!\cdots\!48p6T14+ p^{6} T^{14} + 89 ⁣⋯ ⁣6489\!\cdots\!64p9T15+ p^{9} T^{15} + 15 ⁣⋯ ⁣8715\!\cdots\!87p12T16− p^{12} T^{16} - 20 ⁣⋯ ⁣6420\!\cdots\!64p15T17−94435239939881877p18T18+759339717368988p21T19+362460636146p24T20−843881488p27T21+127351p30T22+1692p33T23+p36T24 p^{15} T^{17} - 94435239939881877 p^{18} T^{18} + 759339717368988 p^{21} T^{19} + 362460636146 p^{24} T^{20} - 843881488 p^{27} T^{21} + 127351 p^{30} T^{22} + 1692 p^{33} T^{23} + p^{36} T^{24}
71 1+1221T−222229T2−229445802T3+548429440873T4+98641092965829T5−178758364328736769T6+98487688450745149485T7+ 1 + 1221 T - 222229 T^{2} - 229445802 T^{3} + 548429440873 T^{4} + 98641092965829 T^{5} - 178758364328736769 T^{6} + 98487688450745149485 T^{7} + 64 ⁣⋯ ⁣5664\!\cdots\!56T8− T^{8} - 32 ⁣⋯ ⁣6632\!\cdots\!66T9+ T^{9} + 14 ⁣⋯ ⁣1114\!\cdots\!11T10+ T^{10} + 13 ⁣⋯ ⁣1313\!\cdots\!13T11− T^{11} - 24 ⁣⋯ ⁣0524\!\cdots\!05T12+ T^{12} + 13 ⁣⋯ ⁣1313\!\cdots\!13p3T13+ p^{3} T^{13} + 14 ⁣⋯ ⁣1114\!\cdots\!11p6T14− p^{6} T^{14} - 32 ⁣⋯ ⁣6632\!\cdots\!66p9T15+ p^{9} T^{15} + 64 ⁣⋯ ⁣5664\!\cdots\!56p12T16+98487688450745149485p15T17−178758364328736769p18T18+98641092965829p21T19+548429440873p24T20−229445802p27T21−222229p30T22+1221p33T23+p36T24 p^{12} T^{16} + 98487688450745149485 p^{15} T^{17} - 178758364328736769 p^{18} T^{18} + 98641092965829 p^{21} T^{19} + 548429440873 p^{24} T^{20} - 229445802 p^{27} T^{21} - 222229 p^{30} T^{22} + 1221 p^{33} T^{23} + p^{36} T^{24}
73 (1−1554T+2675277T2−2821078736T3+2754872512643T4−2120598772053916T5+1451353422113036309T6−2120598772053916p3T7+2754872512643p6T8−2821078736p9T9+2675277p12T10−1554p15T11+p18T12)2 ( 1 - 1554 T + 2675277 T^{2} - 2821078736 T^{3} + 2754872512643 T^{4} - 2120598772053916 T^{5} + 1451353422113036309 T^{6} - 2120598772053916 p^{3} T^{7} + 2754872512643 p^{6} T^{8} - 2821078736 p^{9} T^{9} + 2675277 p^{12} T^{10} - 1554 p^{15} T^{11} + p^{18} T^{12} )^{2}
79 (1+875T+2505744T2+1612380868T3+2680533156433T4+1358001363523637T5+1672684167171147060T6+1358001363523637p3T7+2680533156433p6T8+1612380868p9T9+2505744p12T10+875p15T11+p18T12)2 ( 1 + 875 T + 2505744 T^{2} + 1612380868 T^{3} + 2680533156433 T^{4} + 1358001363523637 T^{5} + 1672684167171147060 T^{6} + 1358001363523637 p^{3} T^{7} + 2680533156433 p^{6} T^{8} + 1612380868 p^{9} T^{9} + 2505744 p^{12} T^{10} + 875 p^{15} T^{11} + p^{18} T^{12} )^{2}
83 (1−126T+1578473T2−79217558T3+1520212835457T4+29472962128638T5+980791484443862659T6+29472962128638p3T7+1520212835457p6T8−79217558p9T9+1578473p12T10−126p15T11+p18T12)2 ( 1 - 126 T + 1578473 T^{2} - 79217558 T^{3} + 1520212835457 T^{4} + 29472962128638 T^{5} + 980791484443862659 T^{6} + 29472962128638 p^{3} T^{7} + 1520212835457 p^{6} T^{8} - 79217558 p^{9} T^{9} + 1578473 p^{12} T^{10} - 126 p^{15} T^{11} + p^{18} T^{12} )^{2}
89 1+374T−2168759T2−167755306T3+2757626355714T4−328578956137010T5−1822411905315756507T6+ 1 + 374 T - 2168759 T^{2} - 167755306 T^{3} + 2757626355714 T^{4} - 328578956137010 T^{5} - 1822411905315756507 T^{6} + 86 ⁣⋯ ⁣8486\!\cdots\!84T7+ T^{7} + 43 ⁣⋯ ⁣3543\!\cdots\!35T8− T^{8} - 67 ⁣⋯ ⁣0667\!\cdots\!06T9+ T^{9} + 62 ⁣⋯ ⁣0862\!\cdots\!08T10+ T^{10} + 24 ⁣⋯ ⁣0624\!\cdots\!06T11− T^{11} - 70 ⁣⋯ ⁣1170\!\cdots\!11T12+ T^{12} + 24 ⁣⋯ ⁣0624\!\cdots\!06p3T13+ p^{3} T^{13} + 62 ⁣⋯ ⁣0862\!\cdots\!08p6T14− p^{6} T^{14} - 67 ⁣⋯ ⁣0667\!\cdots\!06p9T15+ p^{9} T^{15} + 43 ⁣⋯ ⁣3543\!\cdots\!35p12T16+ p^{12} T^{16} + 86 ⁣⋯ ⁣8486\!\cdots\!84p15T17−1822411905315756507p18T18−328578956137010p21T19+2757626355714p24T20−167755306p27T21−2168759p30T22+374p33T23+p36T24 p^{15} T^{17} - 1822411905315756507 p^{18} T^{18} - 328578956137010 p^{21} T^{19} + 2757626355714 p^{24} T^{20} - 167755306 p^{27} T^{21} - 2168759 p^{30} T^{22} + 374 p^{33} T^{23} + p^{36} T^{24}
97 1+330T−4456453T2−2135265822T3+11088575098674T4+5879616897388588T5−18416879164626935101T6− 1 + 330 T - 4456453 T^{2} - 2135265822 T^{3} + 11088575098674 T^{4} + 5879616897388588 T^{5} - 18416879164626935101 T^{6} - 94 ⁣⋯ ⁣1094\!\cdots\!10T7+ T^{7} + 23 ⁣⋯ ⁣9323\!\cdots\!93T8+ T^{8} + 91 ⁣⋯ ⁣5891\!\cdots\!58T9− T^{9} - 24 ⁣⋯ ⁣1824\!\cdots\!18T10− T^{10} - 36 ⁣⋯ ⁣3436\!\cdots\!34T11+ T^{11} + 22 ⁣⋯ ⁣2922\!\cdots\!29T12− T^{12} - 36 ⁣⋯ ⁣3436\!\cdots\!34p3T13− p^{3} T^{13} - 24 ⁣⋯ ⁣1824\!\cdots\!18p6T14+ p^{6} T^{14} + 91 ⁣⋯ ⁣5891\!\cdots\!58p9T15+ p^{9} T^{15} + 23 ⁣⋯ ⁣9323\!\cdots\!93p12T16− p^{12} T^{16} - 94 ⁣⋯ ⁣1094\!\cdots\!10p15T17−18416879164626935101p18T18+5879616897388588p21T19+11088575098674p24T20−2135265822p27T21−4456453p30T22+330p33T23+p36T24 p^{15} T^{17} - 18416879164626935101 p^{18} T^{18} + 5879616897388588 p^{21} T^{19} + 11088575098674 p^{24} T^{20} - 2135265822 p^{27} T^{21} - 4456453 p^{30} T^{22} + 330 p^{33} T^{23} + p^{36} T^{24}
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   L(s)=∏p ∏j=124(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−3.53237886113411524340728771355, −3.37679876248029380780868077252, −3.29967630988982248644577459690, −3.24233964973641692981141942281, −3.08443732134004839722248129114, −2.81992668026463742374165397079, −2.72526529883301546338933083387, −2.64352631567657573806976756859, −2.61336623161907359077961984679, −2.57762586093683013153948980981, −2.32778708963850150562451539333, −2.23225478123826717330491205753, −2.01929257029529925835906499601, −1.99898084331206432604088252998, −1.70580105612999719203534041685, −1.67998580616428711872636420873, −1.33985306769273469520210488402, −0.903329723735886571259340563529, −0.882273031208986115956641445603, −0.858427081618626560424266431864, −0.75887645925424578169659197639, −0.63667427378560946888801301842, −0.56684348546714581482050626667, −0.48604128536228963792787890561, −0.02068279098380190777965778585, 0.02068279098380190777965778585, 0.48604128536228963792787890561, 0.56684348546714581482050626667, 0.63667427378560946888801301842, 0.75887645925424578169659197639, 0.858427081618626560424266431864, 0.882273031208986115956641445603, 0.903329723735886571259340563529, 1.33985306769273469520210488402, 1.67998580616428711872636420873, 1.70580105612999719203534041685, 1.99898084331206432604088252998, 2.01929257029529925835906499601, 2.23225478123826717330491205753, 2.32778708963850150562451539333, 2.57762586093683013153948980981, 2.61336623161907359077961984679, 2.64352631567657573806976756859, 2.72526529883301546338933083387, 2.81992668026463742374165397079, 3.08443732134004839722248129114, 3.24233964973641692981141942281, 3.29967630988982248644577459690, 3.37679876248029380780868077252, 3.53237886113411524340728771355

Graph of the ZZ-function along the critical line

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