Dirichlet series
| L(s) = 1 | − 384·4-s − 1.67e3·5-s − 1.30e3·7-s − 1.03e4·11-s + 4.91e4·16-s − 1.73e5·17-s + 4.05e5·19-s + 6.42e5·20-s − 1.58e5·23-s + 6.48e5·25-s + 5.02e5·28-s + 4.35e6·29-s + 4.52e6·31-s + 2.18e6·35-s + 1.34e5·37-s − 1.29e7·43-s + 3.95e6·44-s − 1.83e7·47-s − 9.74e5·49-s + 1.65e7·53-s + 1.72e7·55-s − 3.11e7·59-s − 8.53e7·61-s + 4.19e6·64-s − 3.77e7·67-s + 6.65e7·68-s − 4.55e7·71-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 2.67·5-s − 0.544·7-s − 0.703·11-s + 3/4·16-s − 2.07·17-s + 3.11·19-s + 4.01·20-s − 0.567·23-s + 1.66·25-s + 0.817·28-s + 6.15·29-s + 4.89·31-s + 1.45·35-s + 0.0716·37-s − 3.79·43-s + 1.05·44-s − 3.76·47-s − 0.168·49-s + 2.09·53-s + 1.88·55-s − 2.57·59-s − 6.16·61-s + 1/4·64-s − 1.87·67-s + 3.11·68-s − 1.79·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.34522\times 10^{20}\) |
| Root analytic conductor: | \(7.16447\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [4]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(0.2051789521\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2051789521\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{7} T^{2} + p^{14} T^{4} )^{3} \) |
| 3 | \( 1 \) | |
| 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 | 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
−3.12012469958349985900307086800, −3.08130651787987469917427825487, −2.92607317997821183471522131382, −2.82551866219814506139769722517, −2.79104539501444267516342766927, −2.71565617502287762812657906137, −2.55634374576427428647406567908, −2.53413054789907055775704797234, −2.01588949162979932006981565848, −1.87675672215917714221941645689, −1.85468895581186582684485586077, −1.74963964696900749937389130797, −1.48808252559116928757337411909, −1.48441523885102451863380165294, −1.39047384203577809122285226054, −1.16834557231802891447604421874, −1.14891969960793943945817945061, −0.76562745724533828746641202975, −0.74744518181260869884404765744, −0.63757313253507804174950402851, −0.59533734169480750772385766093, −0.37885234224723430202022290019, −0.37287730936434336451018740807, −0.091475572987534488385582235195, −0.089587571819913779118041373307, 0.089587571819913779118041373307, 0.091475572987534488385582235195, 0.37287730936434336451018740807, 0.37885234224723430202022290019, 0.59533734169480750772385766093, 0.63757313253507804174950402851, 0.74744518181260869884404765744, 0.76562745724533828746641202975, 1.14891969960793943945817945061, 1.16834557231802891447604421874, 1.39047384203577809122285226054, 1.48441523885102451863380165294, 1.48808252559116928757337411909, 1.74963964696900749937389130797, 1.85468895581186582684485586077, 1.87675672215917714221941645689, 2.01588949162979932006981565848, 2.53413054789907055775704797234, 2.55634374576427428647406567908, 2.71565617502287762812657906137, 2.79104539501444267516342766927, 2.82551866219814506139769722517, 2.92607317997821183471522131382, 3.08130651787987469917427825487, 3.12012469958349985900307086800
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.