Dirichlet series
| L(s) = 1 | + 162·3-s − 384·4-s + 1.67e3·5-s − 1.30e3·7-s + 5.24e3·9-s + 1.03e4·11-s − 6.22e4·12-s + 2.71e5·15-s + 4.91e4·16-s + 1.73e5·17-s + 4.05e5·19-s − 6.42e5·20-s − 2.11e5·21-s + 1.58e5·23-s + 6.48e5·25-s − 5.68e5·27-s + 5.02e5·28-s − 4.35e6·29-s + 4.52e6·31-s + 1.66e6·33-s − 2.18e6·35-s − 2.01e6·36-s + 1.34e5·37-s − 1.29e7·43-s − 3.95e6·44-s + 8.77e6·45-s + 1.83e7·47-s + ⋯ |
| L(s) = 1 | + 2·3-s − 3/2·4-s + 2.67·5-s − 0.544·7-s + 0.798·9-s + 0.703·11-s − 3·12-s + 5.35·15-s + 3/4·16-s + 2.07·17-s + 3.11·19-s − 4.01·20-s − 1.08·21-s + 0.567·23-s + 1.66·25-s − 1.06·27-s + 0.817·28-s − 6.15·29-s + 4.89·31-s + 1.40·33-s − 1.45·35-s − 1.19·36-s + 0.0716·37-s − 3.79·43-s − 1.05·44-s + 2.13·45-s + 3.76·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.18444\times 10^{9}\) |
| Root analytic conductor: | \(2.38815\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 7^{12} ,\ ( \ : [4]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(17.43186572\) |
| \(L(\frac12)\) | \(\approx\) | \(17.43186572\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{3} \) |
| 7 | \( 1 + 1308 T + 383574 p T^{2} - 45603188 p^{2} T^{3} + 23480641215 p^{4} T^{4} - 21449321496 p^{7} T^{5} + 58138527948 p^{10} T^{6} - 21449321496 p^{15} T^{7} + 23480641215 p^{20} T^{8} - 45603188 p^{26} T^{9} + 383574 p^{33} T^{10} + 1308 p^{40} T^{11} + p^{48} T^{12} \) | |
| good | 3 | \( 1 - 2 p^{4} T + 7001 p T^{2} - 8170 p^{5} T^{3} + 16318919 p^{2} T^{4} - 419910472 p^{3} T^{5} + 17812167452 p^{3} T^{6} - 688521620308 p^{4} T^{7} + 64538199517165 p^{4} T^{8} - 840730489886618 p^{6} T^{9} + 10055587638936257 p^{8} T^{10} - 279408959784149422 p^{9} T^{11} + 9540034607392023382 p^{10} T^{12} - 279408959784149422 p^{17} T^{13} + 10055587638936257 p^{24} T^{14} - 840730489886618 p^{30} T^{15} + 64538199517165 p^{36} T^{16} - 688521620308 p^{44} T^{17} + 17812167452 p^{51} T^{18} - 419910472 p^{59} T^{19} + 16318919 p^{66} T^{20} - 8170 p^{77} T^{21} + 7001 p^{81} T^{22} - 2 p^{92} T^{23} + p^{96} T^{24} \) |
| 5 | \( 1 - 1674 T + 2153679 T^{2} - 2041588638 T^{3} + 1534221412419 T^{4} - 39333326844888 p^{2} T^{5} + 19961636689784216 p^{2} T^{6} - 378297034574480964 p^{4} T^{7} + 35437790320196057073 p^{5} T^{8} - \)\(45\!\cdots\!86\)\( p^{6} T^{9} + \)\(37\!\cdots\!01\)\( p^{6} T^{10} - \)\(11\!\cdots\!14\)\( p^{8} T^{11} + \)\(80\!\cdots\!74\)\( p^{8} T^{12} - \)\(11\!\cdots\!14\)\( p^{16} T^{13} + \)\(37\!\cdots\!01\)\( p^{22} T^{14} - \)\(45\!\cdots\!86\)\( p^{30} T^{15} + 35437790320196057073 p^{37} T^{16} - 378297034574480964 p^{44} T^{17} + 19961636689784216 p^{50} T^{18} - 39333326844888 p^{58} T^{19} + 1534221412419 p^{64} T^{20} - 2041588638 p^{72} T^{21} + 2153679 p^{80} T^{22} - 1674 p^{88} T^{23} + p^{96} T^{24} \) | |
| 11 | \( 1 - 10302 T - 545397909 T^{2} + 15359099422182 T^{3} + 89055245929877391 T^{4} - \)\(70\!\cdots\!16\)\( T^{5} + \)\(37\!\cdots\!28\)\( p T^{6} + \)\(18\!\cdots\!60\)\( T^{7} - \)\(25\!\cdots\!75\)\( T^{8} - \)\(28\!\cdots\!14\)\( T^{9} + \)\(64\!\cdots\!69\)\( p^{2} T^{10} + \)\(19\!\cdots\!58\)\( T^{11} - \)\(18\!\cdots\!70\)\( T^{12} + \)\(19\!\cdots\!58\)\( p^{8} T^{13} + \)\(64\!\cdots\!69\)\( p^{18} T^{14} - \)\(28\!\cdots\!14\)\( p^{24} T^{15} - \)\(25\!\cdots\!75\)\( p^{32} T^{16} + \)\(18\!\cdots\!60\)\( p^{40} T^{17} + \)\(37\!\cdots\!28\)\( p^{49} T^{18} - \)\(70\!\cdots\!16\)\( p^{56} T^{19} + 89055245929877391 p^{64} T^{20} + 15359099422182 p^{72} T^{21} - 545397909 p^{80} T^{22} - 10302 p^{88} T^{23} + p^{96} T^{24} \) | |
| 13 | \( 1 - 5176217148 T^{2} + 14197931443386365442 T^{4} - \)\(26\!\cdots\!92\)\( T^{6} + \)\(37\!\cdots\!59\)\( T^{8} - \)\(41\!\cdots\!96\)\( T^{10} + \)\(37\!\cdots\!92\)\( T^{12} - \)\(41\!\cdots\!96\)\( p^{16} T^{14} + \)\(37\!\cdots\!59\)\( p^{32} T^{16} - \)\(26\!\cdots\!92\)\( p^{48} T^{18} + 14197931443386365442 p^{64} T^{20} - 5176217148 p^{80} T^{22} + p^{96} T^{24} \) | |
| 17 | \( 1 - 173178 T + 1524384399 p T^{2} - 2756588792771790 T^{3} + \)\(17\!\cdots\!39\)\( T^{4} - \)\(13\!\cdots\!80\)\( T^{5} - \)\(75\!\cdots\!92\)\( p T^{6} + \)\(22\!\cdots\!24\)\( T^{7} - \)\(19\!\cdots\!55\)\( T^{8} + \)\(90\!\cdots\!90\)\( T^{9} + \)\(24\!\cdots\!01\)\( T^{10} - \)\(97\!\cdots\!58\)\( T^{11} + \)\(11\!\cdots\!30\)\( T^{12} - \)\(97\!\cdots\!58\)\( p^{8} T^{13} + \)\(24\!\cdots\!01\)\( p^{16} T^{14} + \)\(90\!\cdots\!90\)\( p^{24} T^{15} - \)\(19\!\cdots\!55\)\( p^{32} T^{16} + \)\(22\!\cdots\!24\)\( p^{40} T^{17} - \)\(75\!\cdots\!92\)\( p^{49} T^{18} - \)\(13\!\cdots\!80\)\( p^{56} T^{19} + \)\(17\!\cdots\!39\)\( p^{64} T^{20} - 2756588792771790 p^{72} T^{21} + 1524384399 p^{81} T^{22} - 173178 p^{88} T^{23} + p^{96} T^{24} \) | |
| 19 | \( 1 - 405978 T + 142141866315 T^{2} - 35402291464997286 T^{3} + \)\(79\!\cdots\!07\)\( T^{4} - \)\(15\!\cdots\!64\)\( T^{5} + \)\(25\!\cdots\!72\)\( T^{6} - \)\(38\!\cdots\!68\)\( T^{7} + \)\(53\!\cdots\!33\)\( T^{8} - \)\(67\!\cdots\!54\)\( T^{9} + \)\(80\!\cdots\!93\)\( T^{10} - \)\(97\!\cdots\!90\)\( T^{11} + \)\(12\!\cdots\!22\)\( T^{12} - \)\(97\!\cdots\!90\)\( p^{8} T^{13} + \)\(80\!\cdots\!93\)\( p^{16} T^{14} - \)\(67\!\cdots\!54\)\( p^{24} T^{15} + \)\(53\!\cdots\!33\)\( p^{32} T^{16} - \)\(38\!\cdots\!68\)\( p^{40} T^{17} + \)\(25\!\cdots\!72\)\( p^{48} T^{18} - \)\(15\!\cdots\!64\)\( p^{56} T^{19} + \)\(79\!\cdots\!07\)\( p^{64} T^{20} - 35402291464997286 p^{72} T^{21} + 142141866315 p^{80} T^{22} - 405978 p^{88} T^{23} + p^{96} T^{24} \) | |
| 23 | \( 1 - 158934 T - 212751088053 T^{2} - 16910206248252354 T^{3} + \)\(33\!\cdots\!15\)\( T^{4} + \)\(65\!\cdots\!68\)\( T^{5} - \)\(18\!\cdots\!00\)\( T^{6} - \)\(13\!\cdots\!20\)\( T^{7} - \)\(38\!\cdots\!11\)\( T^{8} + \)\(11\!\cdots\!78\)\( T^{9} + \)\(31\!\cdots\!37\)\( T^{10} - \)\(47\!\cdots\!98\)\( T^{11} - \)\(30\!\cdots\!62\)\( T^{12} - \)\(47\!\cdots\!98\)\( p^{8} T^{13} + \)\(31\!\cdots\!37\)\( p^{16} T^{14} + \)\(11\!\cdots\!78\)\( p^{24} T^{15} - \)\(38\!\cdots\!11\)\( p^{32} T^{16} - \)\(13\!\cdots\!20\)\( p^{40} T^{17} - \)\(18\!\cdots\!00\)\( p^{48} T^{18} + \)\(65\!\cdots\!68\)\( p^{56} T^{19} + \)\(33\!\cdots\!15\)\( p^{64} T^{20} - 16910206248252354 p^{72} T^{21} - 212751088053 p^{80} T^{22} - 158934 p^{88} T^{23} + p^{96} T^{24} \) | |
| 29 | \( ( 1 + 2177628 T + 3821244451098 T^{2} + 4505827811503662732 T^{3} + \)\(46\!\cdots\!67\)\( T^{4} + \)\(38\!\cdots\!08\)\( T^{5} + \)\(29\!\cdots\!08\)\( T^{6} + \)\(38\!\cdots\!08\)\( p^{8} T^{7} + \)\(46\!\cdots\!67\)\( p^{16} T^{8} + 4505827811503662732 p^{24} T^{9} + 3821244451098 p^{32} T^{10} + 2177628 p^{40} T^{11} + p^{48} T^{12} )^{2} \) | |
| 31 | \( 1 - 4520250 T + 14538126047019 T^{2} - 34929053714865759750 T^{3} + \)\(71\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!16\)\( T^{6} - \)\(28\!\cdots\!48\)\( T^{7} + \)\(37\!\cdots\!09\)\( T^{8} - \)\(45\!\cdots\!14\)\( T^{9} + \)\(50\!\cdots\!17\)\( T^{10} - \)\(52\!\cdots\!86\)\( T^{11} + \)\(49\!\cdots\!62\)\( T^{12} - \)\(52\!\cdots\!86\)\( p^{8} T^{13} + \)\(50\!\cdots\!17\)\( p^{16} T^{14} - \)\(45\!\cdots\!14\)\( p^{24} T^{15} + \)\(37\!\cdots\!09\)\( p^{32} T^{16} - \)\(28\!\cdots\!48\)\( p^{40} T^{17} + \)\(20\!\cdots\!16\)\( p^{48} T^{18} - \)\(12\!\cdots\!96\)\( p^{56} T^{19} + \)\(71\!\cdots\!23\)\( p^{64} T^{20} - 34929053714865759750 p^{72} T^{21} + 14538126047019 p^{80} T^{22} - 4520250 p^{88} T^{23} + p^{96} T^{24} \) | |
| 37 | \( 1 - 134214 T - 13999782080625 T^{2} - 6710709959056216666 T^{3} + \)\(98\!\cdots\!55\)\( T^{4} + \)\(78\!\cdots\!00\)\( T^{5} - \)\(49\!\cdots\!20\)\( T^{6} - \)\(28\!\cdots\!80\)\( T^{7} + \)\(22\!\cdots\!85\)\( T^{8} + \)\(29\!\cdots\!74\)\( T^{9} - \)\(10\!\cdots\!55\)\( T^{10} + \)\(11\!\cdots\!06\)\( T^{11} + \)\(42\!\cdots\!02\)\( T^{12} + \)\(11\!\cdots\!06\)\( p^{8} T^{13} - \)\(10\!\cdots\!55\)\( p^{16} T^{14} + \)\(29\!\cdots\!74\)\( p^{24} T^{15} + \)\(22\!\cdots\!85\)\( p^{32} T^{16} - \)\(28\!\cdots\!80\)\( p^{40} T^{17} - \)\(49\!\cdots\!20\)\( p^{48} T^{18} + \)\(78\!\cdots\!00\)\( p^{56} T^{19} + \)\(98\!\cdots\!55\)\( p^{64} T^{20} - 6710709959056216666 p^{72} T^{21} - 13999782080625 p^{80} T^{22} - 134214 p^{88} T^{23} + p^{96} T^{24} \) | |
| 41 | \( 1 - 32692668523068 T^{2} + \)\(46\!\cdots\!14\)\( T^{4} - \)\(39\!\cdots\!40\)\( T^{6} + \)\(32\!\cdots\!67\)\( T^{8} - \)\(35\!\cdots\!80\)\( T^{10} + \)\(33\!\cdots\!68\)\( T^{12} - \)\(35\!\cdots\!80\)\( p^{16} T^{14} + \)\(32\!\cdots\!67\)\( p^{32} T^{16} - \)\(39\!\cdots\!40\)\( p^{48} T^{18} + \)\(46\!\cdots\!14\)\( p^{64} T^{20} - 32692668523068 p^{80} T^{22} + p^{96} T^{24} \) | |
| 43 | \( ( 1 + 6480948 T + 50579820669522 T^{2} + \)\(23\!\cdots\!96\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} + \)\(45\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!76\)\( T^{6} + \)\(45\!\cdots\!60\)\( p^{8} T^{7} + \)\(12\!\cdots\!91\)\( p^{16} T^{8} + \)\(23\!\cdots\!96\)\( p^{24} T^{9} + 50579820669522 p^{32} T^{10} + 6480948 p^{40} T^{11} + p^{48} T^{12} )^{2} \) | |
| 47 | \( 1 - 18385002 T + 5062779198165 p T^{2} - \)\(23\!\cdots\!74\)\( T^{3} + \)\(18\!\cdots\!71\)\( T^{4} - \)\(13\!\cdots\!96\)\( T^{5} + \)\(82\!\cdots\!24\)\( T^{6} - \)\(48\!\cdots\!48\)\( T^{7} + \)\(26\!\cdots\!13\)\( T^{8} - \)\(14\!\cdots\!66\)\( T^{9} + \)\(74\!\cdots\!93\)\( T^{10} - \)\(38\!\cdots\!42\)\( T^{11} + \)\(18\!\cdots\!66\)\( T^{12} - \)\(38\!\cdots\!42\)\( p^{8} T^{13} + \)\(74\!\cdots\!93\)\( p^{16} T^{14} - \)\(14\!\cdots\!66\)\( p^{24} T^{15} + \)\(26\!\cdots\!13\)\( p^{32} T^{16} - \)\(48\!\cdots\!48\)\( p^{40} T^{17} + \)\(82\!\cdots\!24\)\( p^{48} T^{18} - \)\(13\!\cdots\!96\)\( p^{56} T^{19} + \)\(18\!\cdots\!71\)\( p^{64} T^{20} - \)\(23\!\cdots\!74\)\( p^{72} T^{21} + 5062779198165 p^{81} T^{22} - 18385002 p^{88} T^{23} + p^{96} T^{24} \) | |
| 53 | \( 1 + 16540506 T - 168608890068369 T^{2} - \)\(28\!\cdots\!46\)\( T^{3} + \)\(36\!\cdots\!79\)\( T^{4} + \)\(41\!\cdots\!12\)\( T^{5} - \)\(47\!\cdots\!20\)\( T^{6} - \)\(32\!\cdots\!16\)\( T^{7} + \)\(56\!\cdots\!33\)\( T^{8} + \)\(20\!\cdots\!34\)\( T^{9} - \)\(48\!\cdots\!19\)\( T^{10} - \)\(43\!\cdots\!22\)\( T^{11} + \)\(34\!\cdots\!30\)\( T^{12} - \)\(43\!\cdots\!22\)\( p^{8} T^{13} - \)\(48\!\cdots\!19\)\( p^{16} T^{14} + \)\(20\!\cdots\!34\)\( p^{24} T^{15} + \)\(56\!\cdots\!33\)\( p^{32} T^{16} - \)\(32\!\cdots\!16\)\( p^{40} T^{17} - \)\(47\!\cdots\!20\)\( p^{48} T^{18} + \)\(41\!\cdots\!12\)\( p^{56} T^{19} + \)\(36\!\cdots\!79\)\( p^{64} T^{20} - \)\(28\!\cdots\!46\)\( p^{72} T^{21} - 168608890068369 p^{80} T^{22} + 16540506 p^{88} T^{23} + p^{96} T^{24} \) | |
| 59 | \( 1 - 31163922 T + 1001043368969307 T^{2} - \)\(21\!\cdots\!38\)\( T^{3} + \)\(42\!\cdots\!35\)\( T^{4} - \)\(68\!\cdots\!76\)\( T^{5} + \)\(10\!\cdots\!84\)\( T^{6} - \)\(14\!\cdots\!16\)\( T^{7} + \)\(20\!\cdots\!25\)\( T^{8} - \)\(26\!\cdots\!78\)\( T^{9} + \)\(34\!\cdots\!37\)\( T^{10} - \)\(44\!\cdots\!02\)\( T^{11} + \)\(53\!\cdots\!02\)\( T^{12} - \)\(44\!\cdots\!02\)\( p^{8} T^{13} + \)\(34\!\cdots\!37\)\( p^{16} T^{14} - \)\(26\!\cdots\!78\)\( p^{24} T^{15} + \)\(20\!\cdots\!25\)\( p^{32} T^{16} - \)\(14\!\cdots\!16\)\( p^{40} T^{17} + \)\(10\!\cdots\!84\)\( p^{48} T^{18} - \)\(68\!\cdots\!76\)\( p^{56} T^{19} + \)\(42\!\cdots\!35\)\( p^{64} T^{20} - \)\(21\!\cdots\!38\)\( p^{72} T^{21} + 1001043368969307 p^{80} T^{22} - 31163922 p^{88} T^{23} + p^{96} T^{24} \) | |
| 61 | \( 1 + 85390158 T + 3917593658539647 T^{2} + \)\(12\!\cdots\!22\)\( T^{3} + \)\(31\!\cdots\!87\)\( T^{4} + \)\(65\!\cdots\!76\)\( T^{5} + \)\(11\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!16\)\( T^{7} + \)\(26\!\cdots\!17\)\( T^{8} + \)\(38\!\cdots\!78\)\( T^{9} + \)\(56\!\cdots\!85\)\( T^{10} + \)\(82\!\cdots\!50\)\( T^{11} + \)\(11\!\cdots\!02\)\( T^{12} + \)\(82\!\cdots\!50\)\( p^{8} T^{13} + \)\(56\!\cdots\!85\)\( p^{16} T^{14} + \)\(38\!\cdots\!78\)\( p^{24} T^{15} + \)\(26\!\cdots\!17\)\( p^{32} T^{16} + \)\(18\!\cdots\!16\)\( p^{40} T^{17} + \)\(11\!\cdots\!60\)\( p^{48} T^{18} + \)\(65\!\cdots\!76\)\( p^{56} T^{19} + \)\(31\!\cdots\!87\)\( p^{64} T^{20} + \)\(12\!\cdots\!22\)\( p^{72} T^{21} + 3917593658539647 p^{80} T^{22} + 85390158 p^{88} T^{23} + p^{96} T^{24} \) | |
| 67 | \( 1 + 37750362 T - 861772180843077 T^{2} - \)\(49\!\cdots\!66\)\( T^{3} + \)\(44\!\cdots\!11\)\( T^{4} + \)\(39\!\cdots\!20\)\( T^{5} - \)\(56\!\cdots\!60\)\( T^{6} - \)\(21\!\cdots\!12\)\( T^{7} - \)\(79\!\cdots\!15\)\( T^{8} + \)\(76\!\cdots\!42\)\( T^{9} + \)\(81\!\cdots\!61\)\( T^{10} - \)\(12\!\cdots\!18\)\( T^{11} - \)\(41\!\cdots\!82\)\( T^{12} - \)\(12\!\cdots\!18\)\( p^{8} T^{13} + \)\(81\!\cdots\!61\)\( p^{16} T^{14} + \)\(76\!\cdots\!42\)\( p^{24} T^{15} - \)\(79\!\cdots\!15\)\( p^{32} T^{16} - \)\(21\!\cdots\!12\)\( p^{40} T^{17} - \)\(56\!\cdots\!60\)\( p^{48} T^{18} + \)\(39\!\cdots\!20\)\( p^{56} T^{19} + \)\(44\!\cdots\!11\)\( p^{64} T^{20} - \)\(49\!\cdots\!66\)\( p^{72} T^{21} - 861772180843077 p^{80} T^{22} + 37750362 p^{88} T^{23} + p^{96} T^{24} \) | |
| 71 | \( ( 1 - 22753212 T + 1448264425828434 T^{2} - \)\(53\!\cdots\!88\)\( T^{3} + \)\(15\!\cdots\!55\)\( T^{4} - \)\(47\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!80\)\( T^{6} - \)\(47\!\cdots\!76\)\( p^{8} T^{7} + \)\(15\!\cdots\!55\)\( p^{16} T^{8} - \)\(53\!\cdots\!88\)\( p^{24} T^{9} + 1448264425828434 p^{32} T^{10} - 22753212 p^{40} T^{11} + p^{48} T^{12} )^{2} \) | |
| 73 | \( 1 - 9414786 T + 2857947168565167 T^{2} - \)\(26\!\cdots\!10\)\( T^{3} + \)\(48\!\cdots\!67\)\( T^{4} - \)\(49\!\cdots\!56\)\( T^{5} + \)\(44\!\cdots\!84\)\( T^{6} - \)\(65\!\cdots\!16\)\( T^{7} + \)\(21\!\cdots\!77\)\( T^{8} - \)\(70\!\cdots\!50\)\( T^{9} - \)\(62\!\cdots\!43\)\( T^{10} - \)\(63\!\cdots\!06\)\( T^{11} - \)\(13\!\cdots\!58\)\( T^{12} - \)\(63\!\cdots\!06\)\( p^{8} T^{13} - \)\(62\!\cdots\!43\)\( p^{16} T^{14} - \)\(70\!\cdots\!50\)\( p^{24} T^{15} + \)\(21\!\cdots\!77\)\( p^{32} T^{16} - \)\(65\!\cdots\!16\)\( p^{40} T^{17} + \)\(44\!\cdots\!84\)\( p^{48} T^{18} - \)\(49\!\cdots\!56\)\( p^{56} T^{19} + \)\(48\!\cdots\!67\)\( p^{64} T^{20} - \)\(26\!\cdots\!10\)\( p^{72} T^{21} + 2857947168565167 p^{80} T^{22} - 9414786 p^{88} T^{23} + p^{96} T^{24} \) | |
| 79 | \( 1 - 59730294 T - 1773934577959317 T^{2} - \)\(11\!\cdots\!02\)\( T^{3} + \)\(18\!\cdots\!31\)\( T^{4} + \)\(17\!\cdots\!64\)\( T^{5} - \)\(57\!\cdots\!40\)\( T^{6} - \)\(25\!\cdots\!28\)\( T^{7} + \)\(17\!\cdots\!25\)\( T^{8} + \)\(21\!\cdots\!58\)\( T^{9} + \)\(18\!\cdots\!09\)\( T^{10} - \)\(46\!\cdots\!10\)\( T^{11} - \)\(18\!\cdots\!38\)\( T^{12} - \)\(46\!\cdots\!10\)\( p^{8} T^{13} + \)\(18\!\cdots\!09\)\( p^{16} T^{14} + \)\(21\!\cdots\!58\)\( p^{24} T^{15} + \)\(17\!\cdots\!25\)\( p^{32} T^{16} - \)\(25\!\cdots\!28\)\( p^{40} T^{17} - \)\(57\!\cdots\!40\)\( p^{48} T^{18} + \)\(17\!\cdots\!64\)\( p^{56} T^{19} + \)\(18\!\cdots\!31\)\( p^{64} T^{20} - \)\(11\!\cdots\!02\)\( p^{72} T^{21} - 1773934577959317 p^{80} T^{22} - 59730294 p^{88} T^{23} + p^{96} T^{24} \) | |
| 83 | \( 1 - 17146153575309900 T^{2} + \)\(14\!\cdots\!90\)\( T^{4} - \)\(79\!\cdots\!20\)\( T^{6} + \)\(32\!\cdots\!15\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{10} + \)\(25\!\cdots\!32\)\( T^{12} - \)\(10\!\cdots\!00\)\( p^{16} T^{14} + \)\(32\!\cdots\!15\)\( p^{32} T^{16} - \)\(79\!\cdots\!20\)\( p^{48} T^{18} + \)\(14\!\cdots\!90\)\( p^{64} T^{20} - 17146153575309900 p^{80} T^{22} + p^{96} T^{24} \) | |
| 89 | \( 1 - 323014482 T + 65302730360370015 T^{2} - \)\(98\!\cdots\!74\)\( T^{3} + \)\(12\!\cdots\!27\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{5} + \)\(13\!\cdots\!92\)\( T^{6} - \)\(11\!\cdots\!48\)\( T^{7} + \)\(96\!\cdots\!85\)\( T^{8} - \)\(74\!\cdots\!66\)\( T^{9} + \)\(54\!\cdots\!41\)\( T^{10} - \)\(37\!\cdots\!02\)\( T^{11} + \)\(24\!\cdots\!78\)\( T^{12} - \)\(37\!\cdots\!02\)\( p^{8} T^{13} + \)\(54\!\cdots\!41\)\( p^{16} T^{14} - \)\(74\!\cdots\!66\)\( p^{24} T^{15} + \)\(96\!\cdots\!85\)\( p^{32} T^{16} - \)\(11\!\cdots\!48\)\( p^{40} T^{17} + \)\(13\!\cdots\!92\)\( p^{48} T^{18} - \)\(13\!\cdots\!20\)\( p^{56} T^{19} + \)\(12\!\cdots\!27\)\( p^{64} T^{20} - \)\(98\!\cdots\!74\)\( p^{72} T^{21} + 65302730360370015 p^{80} T^{22} - 323014482 p^{88} T^{23} + p^{96} T^{24} \) | |
| 97 | \( 1 - 61751299677392700 T^{2} + \)\(18\!\cdots\!42\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{6} + \)\(50\!\cdots\!11\)\( T^{8} - \)\(54\!\cdots\!00\)\( T^{10} + \)\(47\!\cdots\!88\)\( T^{12} - \)\(54\!\cdots\!00\)\( p^{16} T^{14} + \)\(50\!\cdots\!11\)\( p^{32} T^{16} - \)\(35\!\cdots\!00\)\( p^{48} T^{18} + \)\(18\!\cdots\!42\)\( p^{64} T^{20} - 61751299677392700 p^{80} T^{22} + p^{96} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.87364600215280282889458498943, −5.68052740294452192670686554369, −5.33662490308688030010680763013, −5.19418556946566651612247671047, −5.13002166106116936651835095912, −4.91270221888839629060186733502, −4.74810937183079476484236899639, −4.66502797981603064789947437624, −3.97621731013400630944291647151, −3.97585129536755568317086085830, −3.51481821687397014717700950217, −3.49135773693973568131115411674, −3.43444791805540276375881814571, −3.19550590132415873808056945169, −3.00590690460346104072954771131, −2.73764010261657042903696661412, −2.31545593309277633687512480533, −2.08622226188505217827437433837, −1.93637335508627641129831712451, −1.85311791017495019908303393003, −1.18450388211463989832713757557, −1.14305769407804586213452330542, −1.08515019810684212295390984246, −0.41162999514634069194585533385, −0.31842105264730258709154635307, 0.31842105264730258709154635307, 0.41162999514634069194585533385, 1.08515019810684212295390984246, 1.14305769407804586213452330542, 1.18450388211463989832713757557, 1.85311791017495019908303393003, 1.93637335508627641129831712451, 2.08622226188505217827437433837, 2.31545593309277633687512480533, 2.73764010261657042903696661412, 3.00590690460346104072954771131, 3.19550590132415873808056945169, 3.43444791805540276375881814571, 3.49135773693973568131115411674, 3.51481821687397014717700950217, 3.97585129536755568317086085830, 3.97621731013400630944291647151, 4.66502797981603064789947437624, 4.74810937183079476484236899639, 4.91270221888839629060186733502, 5.13002166106116936651835095912, 5.19418556946566651612247671047, 5.33662490308688030010680763013, 5.68052740294452192670686554369, 5.87364600215280282889458498943
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.