Dirichlet series
| L(s) = 1 | + 2.09e5·2-s − 6.50e10·4-s + 2.37e11·5-s − 2.63e13·7-s − 1.95e16·8-s + 4.98e16·10-s + 1.49e18·11-s − 3.43e19·13-s − 5.52e18·14-s + 5.39e20·16-s + 7.62e21·17-s − 1.60e22·19-s − 1.54e22·20-s + 3.14e23·22-s + 5.65e23·23-s − 1.46e25·25-s − 7.20e24·26-s + 1.71e24·28-s − 1.55e25·29-s + 1.54e26·31-s + 5.65e26·32-s + 1.60e27·34-s − 6.25e24·35-s − 3.07e27·37-s − 3.37e27·38-s − 4.63e27·40-s − 6.69e27·41-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 1.89·4-s + 0.139·5-s − 0.0427·7-s − 3.06·8-s + 0.157·10-s + 0.892·11-s − 1.10·13-s − 0.0483·14-s + 0.456·16-s + 2.23·17-s − 0.673·19-s − 0.263·20-s + 1.01·22-s + 0.836·23-s − 5.03·25-s − 1.24·26-s + 0.0809·28-s − 0.397·29-s + 1.23·31-s + 2.58·32-s + 2.53·34-s − 0.00595·35-s − 1.10·37-s − 0.762·38-s − 0.426·40-s − 0.399·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(36-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{36}\right)^{s/2} \, \Gamma_{\C}(s+35/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{36}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.15119\times 10^{27}\) |
| Root analytic conductor: | \(14.4743\) |
| Motivic weight: | \(35\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{36} ,\ ( \ : [35/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(18)\) | \(\approx\) | \(0.006753041911\) |
| \(L(\frac12)\) | \(\approx\) | \(0.006753041911\) |
| \(L(\frac{37}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 209817 T + 54542390637 p T^{2} - 531926391261645 p^{5} T^{3} + 47141615049735550335 p^{7} T^{4} - \)\(34\!\cdots\!87\)\( p^{11} T^{5} + \)\(35\!\cdots\!45\)\( p^{16} T^{6} - \)\(22\!\cdots\!93\)\( p^{23} T^{7} + \)\(16\!\cdots\!77\)\( p^{32} T^{8} - \)\(92\!\cdots\!01\)\( p^{41} T^{9} + \)\(12\!\cdots\!33\)\( p^{50} T^{10} + \)\(21\!\cdots\!99\)\( p^{59} T^{11} + \)\(98\!\cdots\!45\)\( p^{68} T^{12} + \)\(21\!\cdots\!99\)\( p^{94} T^{13} + \)\(12\!\cdots\!33\)\( p^{120} T^{14} - \)\(92\!\cdots\!01\)\( p^{146} T^{15} + \)\(16\!\cdots\!77\)\( p^{172} T^{16} - \)\(22\!\cdots\!93\)\( p^{198} T^{17} + \)\(35\!\cdots\!45\)\( p^{226} T^{18} - \)\(34\!\cdots\!87\)\( p^{256} T^{19} + 47141615049735550335 p^{287} T^{20} - 531926391261645 p^{320} T^{21} + 54542390637 p^{351} T^{22} - 209817 p^{385} T^{23} + p^{420} T^{24} \) |
| 5 | \( 1 - 9503619732 p^{2} T + \)\(29\!\cdots\!54\)\( p T^{2} - \)\(28\!\cdots\!56\)\( p^{2} T^{3} + \)\(82\!\cdots\!41\)\( p^{3} T^{4} + \)\(15\!\cdots\!04\)\( p^{6} T^{5} + \)\(12\!\cdots\!78\)\( p^{8} T^{6} + \)\(12\!\cdots\!04\)\( p^{12} T^{7} + \)\(10\!\cdots\!18\)\( p^{16} T^{8} + \)\(38\!\cdots\!72\)\( p^{21} T^{9} + \)\(16\!\cdots\!78\)\( p^{25} T^{10} + \)\(78\!\cdots\!72\)\( p^{30} T^{11} + \)\(49\!\cdots\!97\)\( p^{35} T^{12} + \)\(78\!\cdots\!72\)\( p^{65} T^{13} + \)\(16\!\cdots\!78\)\( p^{95} T^{14} + \)\(38\!\cdots\!72\)\( p^{126} T^{15} + \)\(10\!\cdots\!18\)\( p^{156} T^{16} + \)\(12\!\cdots\!04\)\( p^{187} T^{17} + \)\(12\!\cdots\!78\)\( p^{218} T^{18} + \)\(15\!\cdots\!04\)\( p^{251} T^{19} + \)\(82\!\cdots\!41\)\( p^{283} T^{20} - \)\(28\!\cdots\!56\)\( p^{317} T^{21} + \)\(29\!\cdots\!54\)\( p^{351} T^{22} - 9503619732 p^{387} T^{23} + p^{420} T^{24} \) | |
| 7 | \( 1 + 26315460937956 T + \)\(16\!\cdots\!46\)\( T^{2} - \)\(24\!\cdots\!92\)\( p T^{3} + \)\(27\!\cdots\!41\)\( p^{2} T^{4} - \)\(17\!\cdots\!00\)\( p^{4} T^{5} + \)\(70\!\cdots\!22\)\( p^{6} T^{6} - \)\(67\!\cdots\!04\)\( p^{8} T^{7} + \)\(15\!\cdots\!26\)\( p^{10} T^{8} - \)\(16\!\cdots\!80\)\( p^{12} T^{9} + \)\(30\!\cdots\!14\)\( p^{14} T^{10} - \)\(47\!\cdots\!08\)\( p^{17} T^{11} + \)\(10\!\cdots\!01\)\( p^{20} T^{12} - \)\(47\!\cdots\!08\)\( p^{52} T^{13} + \)\(30\!\cdots\!14\)\( p^{84} T^{14} - \)\(16\!\cdots\!80\)\( p^{117} T^{15} + \)\(15\!\cdots\!26\)\( p^{150} T^{16} - \)\(67\!\cdots\!04\)\( p^{183} T^{17} + \)\(70\!\cdots\!22\)\( p^{216} T^{18} - \)\(17\!\cdots\!00\)\( p^{249} T^{19} + \)\(27\!\cdots\!41\)\( p^{282} T^{20} - \)\(24\!\cdots\!92\)\( p^{316} T^{21} + \)\(16\!\cdots\!46\)\( p^{350} T^{22} + 26315460937956 p^{385} T^{23} + p^{420} T^{24} \) | |
| 11 | \( 1 - 1496551124826816372 T + \)\(19\!\cdots\!42\)\( T^{2} - \)\(15\!\cdots\!76\)\( p T^{3} + \)\(13\!\cdots\!41\)\( p^{2} T^{4} - \)\(22\!\cdots\!44\)\( p^{3} T^{5} + \)\(50\!\cdots\!22\)\( p^{5} T^{6} + \)\(25\!\cdots\!80\)\( p^{7} T^{7} + \)\(11\!\cdots\!02\)\( p^{9} T^{8} + \)\(14\!\cdots\!16\)\( p^{11} T^{9} + \)\(21\!\cdots\!82\)\( p^{13} T^{10} + \)\(43\!\cdots\!12\)\( p^{15} T^{11} + \)\(40\!\cdots\!19\)\( p^{17} T^{12} + \)\(43\!\cdots\!12\)\( p^{50} T^{13} + \)\(21\!\cdots\!82\)\( p^{83} T^{14} + \)\(14\!\cdots\!16\)\( p^{116} T^{15} + \)\(11\!\cdots\!02\)\( p^{149} T^{16} + \)\(25\!\cdots\!80\)\( p^{182} T^{17} + \)\(50\!\cdots\!22\)\( p^{215} T^{18} - \)\(22\!\cdots\!44\)\( p^{248} T^{19} + \)\(13\!\cdots\!41\)\( p^{282} T^{20} - \)\(15\!\cdots\!76\)\( p^{316} T^{21} + \)\(19\!\cdots\!42\)\( p^{350} T^{22} - 1496551124826816372 p^{385} T^{23} + p^{420} T^{24} \) | |
| 13 | \( 1 + 34328912703023996472 T + \)\(55\!\cdots\!52\)\( T^{2} + \)\(10\!\cdots\!84\)\( p T^{3} + \)\(13\!\cdots\!86\)\( T^{4} + \)\(17\!\cdots\!12\)\( p T^{5} + \)\(13\!\cdots\!04\)\( p^{2} T^{6} + \)\(82\!\cdots\!60\)\( p^{3} T^{7} + \)\(84\!\cdots\!87\)\( p^{4} T^{8} - \)\(70\!\cdots\!44\)\( p^{6} T^{9} + \)\(25\!\cdots\!00\)\( p^{8} T^{10} - \)\(17\!\cdots\!56\)\( p^{10} T^{11} + \)\(78\!\cdots\!68\)\( p^{12} T^{12} - \)\(17\!\cdots\!56\)\( p^{45} T^{13} + \)\(25\!\cdots\!00\)\( p^{78} T^{14} - \)\(70\!\cdots\!44\)\( p^{111} T^{15} + \)\(84\!\cdots\!87\)\( p^{144} T^{16} + \)\(82\!\cdots\!60\)\( p^{178} T^{17} + \)\(13\!\cdots\!04\)\( p^{212} T^{18} + \)\(17\!\cdots\!12\)\( p^{246} T^{19} + \)\(13\!\cdots\!86\)\( p^{280} T^{20} + \)\(10\!\cdots\!84\)\( p^{316} T^{21} + \)\(55\!\cdots\!52\)\( p^{350} T^{22} + 34328912703023996472 p^{385} T^{23} + p^{420} T^{24} \) | |
| 17 | \( 1 - \)\(76\!\cdots\!68\)\( T + \)\(32\!\cdots\!88\)\( p^{2} T^{2} - \)\(17\!\cdots\!32\)\( p^{2} T^{3} + \)\(77\!\cdots\!94\)\( p^{3} T^{4} - \)\(19\!\cdots\!28\)\( p^{4} T^{5} + \)\(67\!\cdots\!72\)\( p^{5} T^{6} - \)\(14\!\cdots\!24\)\( p^{6} T^{7} + \)\(42\!\cdots\!15\)\( p^{7} T^{8} - \)\(80\!\cdots\!68\)\( p^{8} T^{9} + \)\(21\!\cdots\!52\)\( p^{9} T^{10} - \)\(21\!\cdots\!96\)\( p^{11} T^{11} + \)\(94\!\cdots\!32\)\( p^{11} T^{12} - \)\(21\!\cdots\!96\)\( p^{46} T^{13} + \)\(21\!\cdots\!52\)\( p^{79} T^{14} - \)\(80\!\cdots\!68\)\( p^{113} T^{15} + \)\(42\!\cdots\!15\)\( p^{147} T^{16} - \)\(14\!\cdots\!24\)\( p^{181} T^{17} + \)\(67\!\cdots\!72\)\( p^{215} T^{18} - \)\(19\!\cdots\!28\)\( p^{249} T^{19} + \)\(77\!\cdots\!94\)\( p^{283} T^{20} - \)\(17\!\cdots\!32\)\( p^{317} T^{21} + \)\(32\!\cdots\!88\)\( p^{352} T^{22} - \)\(76\!\cdots\!68\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 19 | \( 1 + \)\(16\!\cdots\!28\)\( T + \)\(37\!\cdots\!24\)\( T^{2} + \)\(43\!\cdots\!40\)\( p T^{3} + \)\(75\!\cdots\!38\)\( T^{4} + \)\(18\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!68\)\( p T^{7} + \)\(30\!\cdots\!63\)\( p^{2} T^{8} + \)\(35\!\cdots\!12\)\( p^{3} T^{9} + \)\(67\!\cdots\!08\)\( p^{4} T^{10} + \)\(72\!\cdots\!72\)\( p^{5} T^{11} + \)\(12\!\cdots\!52\)\( p^{6} T^{12} + \)\(72\!\cdots\!72\)\( p^{40} T^{13} + \)\(67\!\cdots\!08\)\( p^{74} T^{14} + \)\(35\!\cdots\!12\)\( p^{108} T^{15} + \)\(30\!\cdots\!63\)\( p^{142} T^{16} + \)\(12\!\cdots\!68\)\( p^{176} T^{17} + \)\(10\!\cdots\!96\)\( p^{210} T^{18} + \)\(18\!\cdots\!32\)\( p^{245} T^{19} + \)\(75\!\cdots\!38\)\( p^{280} T^{20} + \)\(43\!\cdots\!40\)\( p^{316} T^{21} + \)\(37\!\cdots\!24\)\( p^{350} T^{22} + \)\(16\!\cdots\!28\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 23 | \( 1 - \)\(24\!\cdots\!40\)\( p T + \)\(25\!\cdots\!88\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!74\)\( T^{4} - \)\(12\!\cdots\!60\)\( p T^{5} + \)\(71\!\cdots\!60\)\( p^{2} T^{6} - \)\(24\!\cdots\!40\)\( p^{3} T^{7} + \)\(10\!\cdots\!15\)\( p^{4} T^{8} - \)\(35\!\cdots\!80\)\( p^{5} T^{9} + \)\(12\!\cdots\!72\)\( p^{6} T^{10} - \)\(39\!\cdots\!60\)\( p^{7} T^{11} + \)\(11\!\cdots\!84\)\( p^{8} T^{12} - \)\(39\!\cdots\!60\)\( p^{42} T^{13} + \)\(12\!\cdots\!72\)\( p^{76} T^{14} - \)\(35\!\cdots\!80\)\( p^{110} T^{15} + \)\(10\!\cdots\!15\)\( p^{144} T^{16} - \)\(24\!\cdots\!40\)\( p^{178} T^{17} + \)\(71\!\cdots\!60\)\( p^{212} T^{18} - \)\(12\!\cdots\!60\)\( p^{246} T^{19} + \)\(37\!\cdots\!74\)\( p^{280} T^{20} - \)\(18\!\cdots\!60\)\( p^{315} T^{21} + \)\(25\!\cdots\!88\)\( p^{350} T^{22} - \)\(24\!\cdots\!40\)\( p^{386} T^{23} + p^{420} T^{24} \) | |
| 29 | \( 1 + \)\(15\!\cdots\!40\)\( T + \)\(89\!\cdots\!44\)\( T^{2} + \)\(76\!\cdots\!80\)\( p T^{3} + \)\(47\!\cdots\!98\)\( p^{2} T^{4} - \)\(11\!\cdots\!20\)\( p^{3} T^{5} + \)\(17\!\cdots\!52\)\( p^{4} T^{6} - \)\(81\!\cdots\!40\)\( p^{5} T^{7} + \)\(52\!\cdots\!03\)\( p^{6} T^{8} - \)\(28\!\cdots\!20\)\( p^{7} T^{9} + \)\(12\!\cdots\!56\)\( p^{8} T^{10} - \)\(69\!\cdots\!80\)\( p^{9} T^{11} + \)\(25\!\cdots\!92\)\( p^{10} T^{12} - \)\(69\!\cdots\!80\)\( p^{44} T^{13} + \)\(12\!\cdots\!56\)\( p^{78} T^{14} - \)\(28\!\cdots\!20\)\( p^{112} T^{15} + \)\(52\!\cdots\!03\)\( p^{146} T^{16} - \)\(81\!\cdots\!40\)\( p^{180} T^{17} + \)\(17\!\cdots\!52\)\( p^{214} T^{18} - \)\(11\!\cdots\!20\)\( p^{248} T^{19} + \)\(47\!\cdots\!98\)\( p^{282} T^{20} + \)\(76\!\cdots\!80\)\( p^{316} T^{21} + \)\(89\!\cdots\!44\)\( p^{350} T^{22} + \)\(15\!\cdots\!40\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 31 | \( 1 - \)\(15\!\cdots\!84\)\( T + \)\(61\!\cdots\!46\)\( T^{2} - \)\(19\!\cdots\!72\)\( p T^{3} + \)\(18\!\cdots\!49\)\( p^{2} T^{4} - \)\(34\!\cdots\!12\)\( p^{3} T^{5} + \)\(36\!\cdots\!02\)\( p^{4} T^{6} - \)\(73\!\cdots\!68\)\( p^{5} T^{7} + \)\(47\!\cdots\!70\)\( p^{6} T^{8} + \)\(12\!\cdots\!20\)\( p^{7} T^{9} + \)\(43\!\cdots\!86\)\( p^{8} T^{10} + \)\(38\!\cdots\!36\)\( p^{9} T^{11} + \)\(42\!\cdots\!09\)\( p^{10} T^{12} + \)\(38\!\cdots\!36\)\( p^{44} T^{13} + \)\(43\!\cdots\!86\)\( p^{78} T^{14} + \)\(12\!\cdots\!20\)\( p^{112} T^{15} + \)\(47\!\cdots\!70\)\( p^{146} T^{16} - \)\(73\!\cdots\!68\)\( p^{180} T^{17} + \)\(36\!\cdots\!02\)\( p^{214} T^{18} - \)\(34\!\cdots\!12\)\( p^{248} T^{19} + \)\(18\!\cdots\!49\)\( p^{282} T^{20} - \)\(19\!\cdots\!72\)\( p^{316} T^{21} + \)\(61\!\cdots\!46\)\( p^{350} T^{22} - \)\(15\!\cdots\!84\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 37 | \( 1 + \)\(30\!\cdots\!72\)\( T + \)\(57\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!92\)\( T^{3} + \)\(16\!\cdots\!38\)\( T^{4} + \)\(56\!\cdots\!64\)\( T^{5} + \)\(33\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!56\)\( T^{7} + \)\(49\!\cdots\!51\)\( T^{8} + \)\(14\!\cdots\!96\)\( T^{9} + \)\(55\!\cdots\!84\)\( T^{10} + \)\(14\!\cdots\!28\)\( T^{11} + \)\(48\!\cdots\!20\)\( T^{12} + \)\(14\!\cdots\!28\)\( p^{35} T^{13} + \)\(55\!\cdots\!84\)\( p^{70} T^{14} + \)\(14\!\cdots\!96\)\( p^{105} T^{15} + \)\(49\!\cdots\!51\)\( p^{140} T^{16} + \)\(10\!\cdots\!56\)\( p^{175} T^{17} + \)\(33\!\cdots\!12\)\( p^{210} T^{18} + \)\(56\!\cdots\!64\)\( p^{245} T^{19} + \)\(16\!\cdots\!38\)\( p^{280} T^{20} + \)\(19\!\cdots\!92\)\( p^{315} T^{21} + \)\(57\!\cdots\!00\)\( p^{350} T^{22} + \)\(30\!\cdots\!72\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 41 | \( 1 + \)\(66\!\cdots\!12\)\( T + \)\(23\!\cdots\!72\)\( T^{2} + \)\(84\!\cdots\!36\)\( T^{3} + \)\(25\!\cdots\!46\)\( T^{4} + \)\(13\!\cdots\!84\)\( T^{5} + \)\(17\!\cdots\!92\)\( T^{6} - \)\(40\!\cdots\!60\)\( T^{7} + \)\(85\!\cdots\!27\)\( T^{8} - \)\(39\!\cdots\!56\)\( T^{9} + \)\(31\!\cdots\!52\)\( T^{10} - \)\(18\!\cdots\!72\)\( T^{11} + \)\(97\!\cdots\!84\)\( T^{12} - \)\(18\!\cdots\!72\)\( p^{35} T^{13} + \)\(31\!\cdots\!52\)\( p^{70} T^{14} - \)\(39\!\cdots\!56\)\( p^{105} T^{15} + \)\(85\!\cdots\!27\)\( p^{140} T^{16} - \)\(40\!\cdots\!60\)\( p^{175} T^{17} + \)\(17\!\cdots\!92\)\( p^{210} T^{18} + \)\(13\!\cdots\!84\)\( p^{245} T^{19} + \)\(25\!\cdots\!46\)\( p^{280} T^{20} + \)\(84\!\cdots\!36\)\( p^{315} T^{21} + \)\(23\!\cdots\!72\)\( p^{350} T^{22} + \)\(66\!\cdots\!12\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 43 | \( 1 - \)\(44\!\cdots\!12\)\( T + \)\(89\!\cdots\!92\)\( T^{2} - \)\(38\!\cdots\!52\)\( T^{3} + \)\(43\!\cdots\!26\)\( T^{4} - \)\(17\!\cdots\!96\)\( T^{5} + \)\(14\!\cdots\!16\)\( T^{6} - \)\(55\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!87\)\( T^{8} - \)\(13\!\cdots\!64\)\( T^{9} + \)\(75\!\cdots\!20\)\( T^{10} - \)\(24\!\cdots\!96\)\( T^{11} + \)\(12\!\cdots\!68\)\( T^{12} - \)\(24\!\cdots\!96\)\( p^{35} T^{13} + \)\(75\!\cdots\!20\)\( p^{70} T^{14} - \)\(13\!\cdots\!64\)\( p^{105} T^{15} + \)\(37\!\cdots\!87\)\( p^{140} T^{16} - \)\(55\!\cdots\!40\)\( p^{175} T^{17} + \)\(14\!\cdots\!16\)\( p^{210} T^{18} - \)\(17\!\cdots\!96\)\( p^{245} T^{19} + \)\(43\!\cdots\!26\)\( p^{280} T^{20} - \)\(38\!\cdots\!52\)\( p^{315} T^{21} + \)\(89\!\cdots\!92\)\( p^{350} T^{22} - \)\(44\!\cdots\!12\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 47 | \( 1 - \)\(20\!\cdots\!16\)\( T + \)\(17\!\cdots\!12\)\( T^{2} - \)\(24\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!62\)\( T^{4} - \)\(18\!\cdots\!32\)\( T^{5} + \)\(10\!\cdots\!84\)\( T^{6} - \)\(89\!\cdots\!80\)\( T^{7} + \)\(51\!\cdots\!87\)\( T^{8} - \)\(34\!\cdots\!88\)\( T^{9} + \)\(21\!\cdots\!12\)\( T^{10} - \)\(11\!\cdots\!00\)\( T^{11} + \)\(75\!\cdots\!92\)\( T^{12} - \)\(11\!\cdots\!00\)\( p^{35} T^{13} + \)\(21\!\cdots\!12\)\( p^{70} T^{14} - \)\(34\!\cdots\!88\)\( p^{105} T^{15} + \)\(51\!\cdots\!87\)\( p^{140} T^{16} - \)\(89\!\cdots\!80\)\( p^{175} T^{17} + \)\(10\!\cdots\!84\)\( p^{210} T^{18} - \)\(18\!\cdots\!32\)\( p^{245} T^{19} + \)\(16\!\cdots\!62\)\( p^{280} T^{20} - \)\(24\!\cdots\!32\)\( p^{315} T^{21} + \)\(17\!\cdots\!12\)\( p^{350} T^{22} - \)\(20\!\cdots\!16\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 53 | \( 1 - \)\(26\!\cdots\!80\)\( T + \)\(20\!\cdots\!82\)\( T^{2} - \)\(47\!\cdots\!88\)\( T^{3} + \)\(19\!\cdots\!49\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!58\)\( T^{6} - \)\(21\!\cdots\!48\)\( T^{7} + \)\(51\!\cdots\!34\)\( T^{8} - \)\(83\!\cdots\!20\)\( T^{9} + \)\(16\!\cdots\!98\)\( T^{10} - \)\(23\!\cdots\!96\)\( T^{11} + \)\(42\!\cdots\!09\)\( T^{12} - \)\(23\!\cdots\!96\)\( p^{35} T^{13} + \)\(16\!\cdots\!98\)\( p^{70} T^{14} - \)\(83\!\cdots\!20\)\( p^{105} T^{15} + \)\(51\!\cdots\!34\)\( p^{140} T^{16} - \)\(21\!\cdots\!48\)\( p^{175} T^{17} + \)\(11\!\cdots\!58\)\( p^{210} T^{18} - \)\(40\!\cdots\!60\)\( p^{245} T^{19} + \)\(19\!\cdots\!49\)\( p^{280} T^{20} - \)\(47\!\cdots\!88\)\( p^{315} T^{21} + \)\(20\!\cdots\!82\)\( p^{350} T^{22} - \)\(26\!\cdots\!80\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 59 | \( 1 - \)\(32\!\cdots\!20\)\( T + \)\(95\!\cdots\!64\)\( T^{2} - \)\(17\!\cdots\!60\)\( T^{3} + \)\(30\!\cdots\!98\)\( T^{4} - \)\(39\!\cdots\!60\)\( T^{5} + \)\(49\!\cdots\!52\)\( T^{6} - \)\(51\!\cdots\!20\)\( T^{7} + \)\(54\!\cdots\!23\)\( T^{8} - \)\(52\!\cdots\!60\)\( T^{9} + \)\(57\!\cdots\!36\)\( T^{10} - \)\(55\!\cdots\!40\)\( T^{11} + \)\(58\!\cdots\!52\)\( T^{12} - \)\(55\!\cdots\!40\)\( p^{35} T^{13} + \)\(57\!\cdots\!36\)\( p^{70} T^{14} - \)\(52\!\cdots\!60\)\( p^{105} T^{15} + \)\(54\!\cdots\!23\)\( p^{140} T^{16} - \)\(51\!\cdots\!20\)\( p^{175} T^{17} + \)\(49\!\cdots\!52\)\( p^{210} T^{18} - \)\(39\!\cdots\!60\)\( p^{245} T^{19} + \)\(30\!\cdots\!98\)\( p^{280} T^{20} - \)\(17\!\cdots\!60\)\( p^{315} T^{21} + \)\(95\!\cdots\!64\)\( p^{350} T^{22} - \)\(32\!\cdots\!20\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 61 | \( 1 + \)\(11\!\cdots\!28\)\( T + \)\(19\!\cdots\!48\)\( T^{2} + \)\(12\!\cdots\!44\)\( T^{3} + \)\(18\!\cdots\!86\)\( T^{4} + \)\(77\!\cdots\!16\)\( T^{5} + \)\(11\!\cdots\!04\)\( T^{6} + \)\(41\!\cdots\!84\)\( T^{7} + \)\(58\!\cdots\!15\)\( T^{8} + \)\(19\!\cdots\!16\)\( T^{9} + \)\(23\!\cdots\!52\)\( T^{10} + \)\(69\!\cdots\!68\)\( T^{11} + \)\(78\!\cdots\!92\)\( T^{12} + \)\(69\!\cdots\!68\)\( p^{35} T^{13} + \)\(23\!\cdots\!52\)\( p^{70} T^{14} + \)\(19\!\cdots\!16\)\( p^{105} T^{15} + \)\(58\!\cdots\!15\)\( p^{140} T^{16} + \)\(41\!\cdots\!84\)\( p^{175} T^{17} + \)\(11\!\cdots\!04\)\( p^{210} T^{18} + \)\(77\!\cdots\!16\)\( p^{245} T^{19} + \)\(18\!\cdots\!86\)\( p^{280} T^{20} + \)\(12\!\cdots\!44\)\( p^{315} T^{21} + \)\(19\!\cdots\!48\)\( p^{350} T^{22} + \)\(11\!\cdots\!28\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 67 | \( 1 + \)\(62\!\cdots\!32\)\( T + \)\(62\!\cdots\!16\)\( T^{2} + \)\(28\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!54\)\( T^{4} + \)\(62\!\cdots\!16\)\( T^{5} + \)\(37\!\cdots\!32\)\( T^{6} + \)\(14\!\cdots\!32\)\( p T^{7} + \)\(55\!\cdots\!31\)\( T^{8} + \)\(10\!\cdots\!24\)\( T^{9} + \)\(63\!\cdots\!16\)\( T^{10} + \)\(10\!\cdots\!76\)\( T^{11} + \)\(58\!\cdots\!56\)\( T^{12} + \)\(10\!\cdots\!76\)\( p^{35} T^{13} + \)\(63\!\cdots\!16\)\( p^{70} T^{14} + \)\(10\!\cdots\!24\)\( p^{105} T^{15} + \)\(55\!\cdots\!31\)\( p^{140} T^{16} + \)\(14\!\cdots\!32\)\( p^{176} T^{17} + \)\(37\!\cdots\!32\)\( p^{210} T^{18} + \)\(62\!\cdots\!16\)\( p^{245} T^{19} + \)\(18\!\cdots\!54\)\( p^{280} T^{20} + \)\(28\!\cdots\!40\)\( p^{315} T^{21} + \)\(62\!\cdots\!16\)\( p^{350} T^{22} + \)\(62\!\cdots\!32\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 71 | \( 1 + \)\(37\!\cdots\!48\)\( T + \)\(38\!\cdots\!92\)\( T^{2} + \)\(33\!\cdots\!44\)\( T^{3} + \)\(73\!\cdots\!66\)\( T^{4} + \)\(11\!\cdots\!76\)\( T^{5} + \)\(95\!\cdots\!72\)\( T^{6} + \)\(20\!\cdots\!40\)\( T^{7} + \)\(99\!\cdots\!07\)\( T^{8} + \)\(21\!\cdots\!16\)\( T^{9} + \)\(84\!\cdots\!72\)\( T^{10} + \)\(17\!\cdots\!12\)\( T^{11} + \)\(58\!\cdots\!24\)\( T^{12} + \)\(17\!\cdots\!12\)\( p^{35} T^{13} + \)\(84\!\cdots\!72\)\( p^{70} T^{14} + \)\(21\!\cdots\!16\)\( p^{105} T^{15} + \)\(99\!\cdots\!07\)\( p^{140} T^{16} + \)\(20\!\cdots\!40\)\( p^{175} T^{17} + \)\(95\!\cdots\!72\)\( p^{210} T^{18} + \)\(11\!\cdots\!76\)\( p^{245} T^{19} + \)\(73\!\cdots\!66\)\( p^{280} T^{20} + \)\(33\!\cdots\!44\)\( p^{315} T^{21} + \)\(38\!\cdots\!92\)\( p^{350} T^{22} + \)\(37\!\cdots\!48\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 73 | \( 1 + \)\(45\!\cdots\!76\)\( T + \)\(12\!\cdots\!34\)\( T^{2} + \)\(33\!\cdots\!60\)\( T^{3} + \)\(62\!\cdots\!89\)\( T^{4} + \)\(70\!\cdots\!68\)\( T^{5} + \)\(18\!\cdots\!38\)\( T^{6} - \)\(13\!\cdots\!12\)\( T^{7} + \)\(38\!\cdots\!46\)\( T^{8} - \)\(11\!\cdots\!48\)\( T^{9} + \)\(58\!\cdots\!14\)\( T^{10} - \)\(33\!\cdots\!88\)\( T^{11} + \)\(88\!\cdots\!81\)\( T^{12} - \)\(33\!\cdots\!88\)\( p^{35} T^{13} + \)\(58\!\cdots\!14\)\( p^{70} T^{14} - \)\(11\!\cdots\!48\)\( p^{105} T^{15} + \)\(38\!\cdots\!46\)\( p^{140} T^{16} - \)\(13\!\cdots\!12\)\( p^{175} T^{17} + \)\(18\!\cdots\!38\)\( p^{210} T^{18} + \)\(70\!\cdots\!68\)\( p^{245} T^{19} + \)\(62\!\cdots\!89\)\( p^{280} T^{20} + \)\(33\!\cdots\!60\)\( p^{315} T^{21} + \)\(12\!\cdots\!34\)\( p^{350} T^{22} + \)\(45\!\cdots\!76\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 79 | \( 1 - \)\(14\!\cdots\!96\)\( T + \)\(15\!\cdots\!88\)\( T^{2} - \)\(12\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!18\)\( T^{4} - \)\(35\!\cdots\!64\)\( T^{5} + \)\(44\!\cdots\!92\)\( T^{6} + \)\(97\!\cdots\!20\)\( T^{7} + \)\(14\!\cdots\!03\)\( T^{8} + \)\(12\!\cdots\!20\)\( T^{9} + \)\(37\!\cdots\!04\)\( T^{10} + \)\(53\!\cdots\!80\)\( T^{11} + \)\(94\!\cdots\!92\)\( T^{12} + \)\(53\!\cdots\!80\)\( p^{35} T^{13} + \)\(37\!\cdots\!04\)\( p^{70} T^{14} + \)\(12\!\cdots\!20\)\( p^{105} T^{15} + \)\(14\!\cdots\!03\)\( p^{140} T^{16} + \)\(97\!\cdots\!20\)\( p^{175} T^{17} + \)\(44\!\cdots\!92\)\( p^{210} T^{18} - \)\(35\!\cdots\!64\)\( p^{245} T^{19} + \)\(10\!\cdots\!18\)\( p^{280} T^{20} - \)\(12\!\cdots\!04\)\( p^{315} T^{21} + \)\(15\!\cdots\!88\)\( p^{350} T^{22} - \)\(14\!\cdots\!96\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 83 | \( 1 + \)\(13\!\cdots\!52\)\( T + \)\(13\!\cdots\!30\)\( T^{2} + \)\(89\!\cdots\!24\)\( T^{3} + \)\(78\!\cdots\!41\)\( p^{2} T^{4} + \)\(27\!\cdots\!92\)\( T^{5} + \)\(12\!\cdots\!14\)\( T^{6} + \)\(49\!\cdots\!80\)\( T^{7} + \)\(17\!\cdots\!58\)\( T^{8} + \)\(52\!\cdots\!08\)\( T^{9} + \)\(14\!\cdots\!30\)\( T^{10} + \)\(35\!\cdots\!00\)\( T^{11} + \)\(12\!\cdots\!45\)\( T^{12} + \)\(35\!\cdots\!00\)\( p^{35} T^{13} + \)\(14\!\cdots\!30\)\( p^{70} T^{14} + \)\(52\!\cdots\!08\)\( p^{105} T^{15} + \)\(17\!\cdots\!58\)\( p^{140} T^{16} + \)\(49\!\cdots\!80\)\( p^{175} T^{17} + \)\(12\!\cdots\!14\)\( p^{210} T^{18} + \)\(27\!\cdots\!92\)\( p^{245} T^{19} + \)\(78\!\cdots\!41\)\( p^{282} T^{20} + \)\(89\!\cdots\!24\)\( p^{315} T^{21} + \)\(13\!\cdots\!30\)\( p^{350} T^{22} + \)\(13\!\cdots\!52\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 89 | \( 1 + \)\(20\!\cdots\!44\)\( T + \)\(11\!\cdots\!68\)\( T^{2} + \)\(23\!\cdots\!08\)\( T^{3} + \)\(72\!\cdots\!66\)\( T^{4} + \)\(12\!\cdots\!88\)\( T^{5} + \)\(29\!\cdots\!88\)\( T^{6} + \)\(47\!\cdots\!80\)\( T^{7} + \)\(89\!\cdots\!47\)\( T^{8} + \)\(12\!\cdots\!08\)\( T^{9} + \)\(20\!\cdots\!28\)\( T^{10} + \)\(27\!\cdots\!04\)\( T^{11} + \)\(39\!\cdots\!04\)\( T^{12} + \)\(27\!\cdots\!04\)\( p^{35} T^{13} + \)\(20\!\cdots\!28\)\( p^{70} T^{14} + \)\(12\!\cdots\!08\)\( p^{105} T^{15} + \)\(89\!\cdots\!47\)\( p^{140} T^{16} + \)\(47\!\cdots\!80\)\( p^{175} T^{17} + \)\(29\!\cdots\!88\)\( p^{210} T^{18} + \)\(12\!\cdots\!88\)\( p^{245} T^{19} + \)\(72\!\cdots\!66\)\( p^{280} T^{20} + \)\(23\!\cdots\!08\)\( p^{315} T^{21} + \)\(11\!\cdots\!68\)\( p^{350} T^{22} + \)\(20\!\cdots\!44\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
| 97 | \( 1 - \)\(99\!\cdots\!12\)\( T + \)\(15\!\cdots\!70\)\( T^{2} - \)\(51\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!33\)\( T^{4} - \)\(61\!\cdots\!04\)\( T^{5} + \)\(93\!\cdots\!02\)\( T^{6} - \)\(45\!\cdots\!56\)\( T^{7} + \)\(50\!\cdots\!46\)\( T^{8} - \)\(24\!\cdots\!56\)\( T^{9} + \)\(22\!\cdots\!74\)\( T^{10} - \)\(10\!\cdots\!48\)\( T^{11} + \)\(84\!\cdots\!65\)\( T^{12} - \)\(10\!\cdots\!48\)\( p^{35} T^{13} + \)\(22\!\cdots\!74\)\( p^{70} T^{14} - \)\(24\!\cdots\!56\)\( p^{105} T^{15} + \)\(50\!\cdots\!46\)\( p^{140} T^{16} - \)\(45\!\cdots\!56\)\( p^{175} T^{17} + \)\(93\!\cdots\!02\)\( p^{210} T^{18} - \)\(61\!\cdots\!04\)\( p^{245} T^{19} + \)\(13\!\cdots\!33\)\( p^{280} T^{20} - \)\(51\!\cdots\!92\)\( p^{315} T^{21} + \)\(15\!\cdots\!70\)\( p^{350} T^{22} - \)\(99\!\cdots\!12\)\( p^{385} T^{23} + p^{420} T^{24} \) | |
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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
−2.43686181804724100456617228418, −2.40827019214045415026625673689, −2.17951891994058385054071318180, −2.15448418888013386696596000452, −1.87530851670248066749643660844, −1.84229868756018476590639403520, −1.80988103261669377993417430346, −1.77879355094577774814129576632, −1.67431961446879246124275650124, −1.48272735571308010754580439552, −1.40070258285978895063415874923, −1.35903262417185741620968000173, −1.25311588377025962670172886486, −1.24067054709024994420005684079, −1.02055361522408400542874652942, −0.857279508298490667761424676109, −0.76002912145519492622265942013, −0.57706358116454816632393805648, −0.53644447438409677586176483177, −0.53547302413803411922510684449, −0.42055474733901089992958201241, −0.32231165486550934864415493393, −0.31065778252743407987232942804, −0.22468734539219486593939223030, −0.00225818215184871612951753953, 0.00225818215184871612951753953, 0.22468734539219486593939223030, 0.31065778252743407987232942804, 0.32231165486550934864415493393, 0.42055474733901089992958201241, 0.53547302413803411922510684449, 0.53644447438409677586176483177, 0.57706358116454816632393805648, 0.76002912145519492622265942013, 0.857279508298490667761424676109, 1.02055361522408400542874652942, 1.24067054709024994420005684079, 1.25311588377025962670172886486, 1.35903262417185741620968000173, 1.40070258285978895063415874923, 1.48272735571308010754580439552, 1.67431961446879246124275650124, 1.77879355094577774814129576632, 1.80988103261669377993417430346, 1.84229868756018476590639403520, 1.87530851670248066749643660844, 2.15448418888013386696596000452, 2.17951891994058385054071318180, 2.40827019214045415026625673689, 2.43686181804724100456617228418
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.