Dirichlet series
| L(s) = 1 | + 5.89e5·4-s + 2.16e9·13-s + 1.61e11·16-s + 4.18e10·17-s − 2.22e11·23-s + 5.07e12·25-s − 3.52e12·29-s − 2.01e14·43-s + 2.35e15·49-s + 1.27e15·52-s − 2.42e15·53-s − 1.40e14·61-s + 2.86e16·64-s + 2.47e16·68-s − 3.18e16·79-s − 1.30e17·92-s + 2.99e18·100-s − 2.76e17·101-s − 3.52e17·103-s + 1.48e16·107-s − 9.08e17·113-s − 2.07e18·116-s + 4.95e18·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4.49·4-s + 0.735·13-s + 9.37·16-s + 1.45·17-s − 0.591·23-s + 6.65·25-s − 1.30·29-s − 2.62·43-s + 10.1·49-s + 3.31·52-s − 5.36·53-s − 0.0939·61-s + 12.7·64-s + 6.55·68-s − 2.35·79-s − 2.66·92-s + 29.9·100-s − 2.54·101-s − 2.74·103-s + 0.0836·107-s − 3.21·113-s − 5.88·116-s + 9.80·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{40} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.20023\times 10^{46}\) |
| Root analytic conductor: | \(14.6413\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{40} \cdot 13^{20} ,\ ( \ : [17/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(9)\) | \(\approx\) | \(522.9160988\) |
| \(L(\frac12)\) | \(\approx\) | \(522.9160988\) |
| \(L(\frac{19}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - 2164505000 T + 671308041805452416 p T^{2} - \)\(22\!\cdots\!28\)\( p^{3} T^{3} + \)\(53\!\cdots\!12\)\( p^{6} T^{4} - \)\(45\!\cdots\!88\)\( p^{10} T^{5} + \)\(58\!\cdots\!44\)\( p^{15} T^{6} - \)\(51\!\cdots\!96\)\( p^{20} T^{7} + \)\(43\!\cdots\!20\)\( p^{25} T^{8} - \)\(20\!\cdots\!84\)\( p^{32} T^{9} + \)\(93\!\cdots\!60\)\( p^{40} T^{10} - \)\(20\!\cdots\!84\)\( p^{49} T^{11} + \)\(43\!\cdots\!20\)\( p^{59} T^{12} - \)\(51\!\cdots\!96\)\( p^{71} T^{13} + \)\(58\!\cdots\!44\)\( p^{83} T^{14} - \)\(45\!\cdots\!88\)\( p^{95} T^{15} + \)\(53\!\cdots\!12\)\( p^{108} T^{16} - \)\(22\!\cdots\!28\)\( p^{122} T^{17} + 671308041805452416 p^{137} T^{18} - 2164505000 p^{153} T^{19} + p^{170} T^{20} \) | |
| good | 2 | \( 1 - 589823 T^{2} + 11677953259 p^{4} T^{4} - 2741238686352457 p^{4} T^{6} + 16972062276290469865 p^{9} T^{8} - \)\(38\!\cdots\!45\)\( p^{12} T^{10} + \)\(25\!\cdots\!43\)\( p^{20} T^{12} - \)\(80\!\cdots\!45\)\( p^{29} T^{14} + \)\(47\!\cdots\!15\)\( p^{37} T^{16} - \)\(51\!\cdots\!89\)\( p^{44} T^{18} + \)\(26\!\cdots\!39\)\( p^{52} T^{20} - \)\(51\!\cdots\!89\)\( p^{78} T^{22} + \)\(47\!\cdots\!15\)\( p^{105} T^{24} - \)\(80\!\cdots\!45\)\( p^{131} T^{26} + \)\(25\!\cdots\!43\)\( p^{156} T^{28} - \)\(38\!\cdots\!45\)\( p^{182} T^{30} + 16972062276290469865 p^{213} T^{32} - 2741238686352457 p^{242} T^{34} + 11677953259 p^{276} T^{36} - 589823 p^{306} T^{38} + p^{340} T^{40} \) |
| 5 | \( 1 - 5076675918062 T^{2} + \)\(13\!\cdots\!73\)\( T^{4} - \)\(24\!\cdots\!14\)\( T^{6} + \)\(29\!\cdots\!52\)\( p^{3} T^{8} - \)\(72\!\cdots\!06\)\( p^{4} T^{10} + \)\(12\!\cdots\!67\)\( p^{8} T^{12} - \)\(20\!\cdots\!02\)\( p^{12} T^{14} + \)\(30\!\cdots\!07\)\( p^{16} T^{16} - \)\(42\!\cdots\!32\)\( p^{20} T^{18} + \)\(54\!\cdots\!36\)\( p^{24} T^{20} - \)\(42\!\cdots\!32\)\( p^{54} T^{22} + \)\(30\!\cdots\!07\)\( p^{84} T^{24} - \)\(20\!\cdots\!02\)\( p^{114} T^{26} + \)\(12\!\cdots\!67\)\( p^{144} T^{28} - \)\(72\!\cdots\!06\)\( p^{174} T^{30} + \)\(29\!\cdots\!52\)\( p^{207} T^{32} - \)\(24\!\cdots\!14\)\( p^{238} T^{34} + \)\(13\!\cdots\!73\)\( p^{272} T^{36} - 5076675918062 p^{306} T^{38} + p^{340} T^{40} \) | |
| 7 | \( 1 - 336131244456626 p T^{2} + \)\(39\!\cdots\!71\)\( p T^{4} - \)\(22\!\cdots\!74\)\( T^{6} + \)\(27\!\cdots\!72\)\( p^{2} T^{8} - \)\(27\!\cdots\!50\)\( p^{4} T^{10} + \)\(22\!\cdots\!47\)\( p^{6} T^{12} - \)\(34\!\cdots\!94\)\( p^{10} T^{14} + \)\(90\!\cdots\!43\)\( p^{16} T^{16} - \)\(51\!\cdots\!44\)\( p^{18} T^{18} + \)\(52\!\cdots\!44\)\( p^{22} T^{20} - \)\(51\!\cdots\!44\)\( p^{52} T^{22} + \)\(90\!\cdots\!43\)\( p^{84} T^{24} - \)\(34\!\cdots\!94\)\( p^{112} T^{26} + \)\(22\!\cdots\!47\)\( p^{142} T^{28} - \)\(27\!\cdots\!50\)\( p^{174} T^{30} + \)\(27\!\cdots\!72\)\( p^{206} T^{32} - \)\(22\!\cdots\!74\)\( p^{238} T^{34} + \)\(39\!\cdots\!71\)\( p^{273} T^{36} - 336131244456626 p^{307} T^{38} + p^{340} T^{40} \) | |
| 11 | \( 1 - 4954925397142683476 T^{2} + \)\(12\!\cdots\!34\)\( T^{4} - \)\(21\!\cdots\!96\)\( T^{6} + \)\(23\!\cdots\!81\)\( p^{2} T^{8} - \)\(20\!\cdots\!00\)\( p^{4} T^{10} + \)\(14\!\cdots\!80\)\( p^{6} T^{12} - \)\(88\!\cdots\!04\)\( p^{8} T^{14} + \)\(47\!\cdots\!82\)\( p^{10} T^{16} - \)\(22\!\cdots\!84\)\( p^{12} T^{18} + \)\(10\!\cdots\!64\)\( p^{14} T^{20} - \)\(22\!\cdots\!84\)\( p^{46} T^{22} + \)\(47\!\cdots\!82\)\( p^{78} T^{24} - \)\(88\!\cdots\!04\)\( p^{110} T^{26} + \)\(14\!\cdots\!80\)\( p^{142} T^{28} - \)\(20\!\cdots\!00\)\( p^{174} T^{30} + \)\(23\!\cdots\!81\)\( p^{206} T^{32} - \)\(21\!\cdots\!96\)\( p^{238} T^{34} + \)\(12\!\cdots\!34\)\( p^{272} T^{36} - 4954925397142683476 p^{306} T^{38} + p^{340} T^{40} \) | |
| 17 | \( ( 1 - 20943268590 T + \)\(41\!\cdots\!61\)\( T^{2} - \)\(76\!\cdots\!30\)\( T^{3} + \)\(81\!\cdots\!88\)\( T^{4} - \)\(11\!\cdots\!90\)\( T^{5} + \)\(98\!\cdots\!07\)\( T^{6} - \)\(78\!\cdots\!30\)\( T^{7} + \)\(87\!\cdots\!75\)\( T^{8} - \)\(25\!\cdots\!20\)\( T^{9} + \)\(70\!\cdots\!64\)\( T^{10} - \)\(25\!\cdots\!20\)\( p^{17} T^{11} + \)\(87\!\cdots\!75\)\( p^{34} T^{12} - \)\(78\!\cdots\!30\)\( p^{51} T^{13} + \)\(98\!\cdots\!07\)\( p^{68} T^{14} - \)\(11\!\cdots\!90\)\( p^{85} T^{15} + \)\(81\!\cdots\!88\)\( p^{102} T^{16} - \)\(76\!\cdots\!30\)\( p^{119} T^{17} + \)\(41\!\cdots\!61\)\( p^{136} T^{18} - 20943268590 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 19 | \( 1 - \)\(28\!\cdots\!68\)\( T^{2} + \)\(52\!\cdots\!82\)\( T^{4} - \)\(69\!\cdots\!64\)\( T^{6} + \)\(76\!\cdots\!33\)\( T^{8} - \)\(71\!\cdots\!84\)\( T^{10} + \)\(59\!\cdots\!28\)\( T^{12} - \)\(44\!\cdots\!80\)\( T^{14} + \)\(30\!\cdots\!22\)\( T^{16} - \)\(18\!\cdots\!88\)\( T^{18} + \)\(10\!\cdots\!96\)\( T^{20} - \)\(18\!\cdots\!88\)\( p^{34} T^{22} + \)\(30\!\cdots\!22\)\( p^{68} T^{24} - \)\(44\!\cdots\!80\)\( p^{102} T^{26} + \)\(59\!\cdots\!28\)\( p^{136} T^{28} - \)\(71\!\cdots\!84\)\( p^{170} T^{30} + \)\(76\!\cdots\!33\)\( p^{204} T^{32} - \)\(69\!\cdots\!64\)\( p^{238} T^{34} + \)\(52\!\cdots\!82\)\( p^{272} T^{36} - \)\(28\!\cdots\!68\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 23 | \( ( 1 + 111039405240 T + \)\(72\!\cdots\!46\)\( T^{2} + \)\(49\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!89\)\( T^{4} + \)\(50\!\cdots\!80\)\( T^{5} + \)\(65\!\cdots\!04\)\( T^{6} - \)\(15\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!58\)\( T^{8} - \)\(55\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!20\)\( T^{10} - \)\(55\!\cdots\!80\)\( p^{17} T^{11} + \)\(12\!\cdots\!58\)\( p^{34} T^{12} - \)\(15\!\cdots\!00\)\( p^{51} T^{13} + \)\(65\!\cdots\!04\)\( p^{68} T^{14} + \)\(50\!\cdots\!80\)\( p^{85} T^{15} + \)\(26\!\cdots\!89\)\( p^{102} T^{16} + \)\(49\!\cdots\!80\)\( p^{119} T^{17} + \)\(72\!\cdots\!46\)\( p^{136} T^{18} + 111039405240 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 29 | \( ( 1 + 1760995205784 T + \)\(21\!\cdots\!02\)\( T^{2} + \)\(42\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!01\)\( T^{4} + \)\(32\!\cdots\!24\)\( T^{5} + \)\(27\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!40\)\( T^{7} + \)\(16\!\cdots\!66\)\( T^{8} - \)\(86\!\cdots\!76\)\( T^{9} + \)\(11\!\cdots\!64\)\( T^{10} - \)\(86\!\cdots\!76\)\( p^{17} T^{11} + \)\(16\!\cdots\!66\)\( p^{34} T^{12} + \)\(11\!\cdots\!40\)\( p^{51} T^{13} + \)\(27\!\cdots\!28\)\( p^{68} T^{14} + \)\(32\!\cdots\!24\)\( p^{85} T^{15} + \)\(30\!\cdots\!01\)\( p^{102} T^{16} + \)\(42\!\cdots\!00\)\( p^{119} T^{17} + \)\(21\!\cdots\!02\)\( p^{136} T^{18} + 1760995205784 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 31 | \( 1 - \)\(20\!\cdots\!36\)\( T^{2} + \)\(23\!\cdots\!82\)\( T^{4} - \)\(18\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!33\)\( T^{8} - \)\(54\!\cdots\!84\)\( T^{10} + \)\(22\!\cdots\!40\)\( T^{12} - \)\(81\!\cdots\!56\)\( T^{14} + \)\(25\!\cdots\!82\)\( T^{16} - \)\(69\!\cdots\!96\)\( T^{18} + \)\(16\!\cdots\!72\)\( T^{20} - \)\(69\!\cdots\!96\)\( p^{34} T^{22} + \)\(25\!\cdots\!82\)\( p^{68} T^{24} - \)\(81\!\cdots\!56\)\( p^{102} T^{26} + \)\(22\!\cdots\!40\)\( p^{136} T^{28} - \)\(54\!\cdots\!84\)\( p^{170} T^{30} + \)\(11\!\cdots\!33\)\( p^{204} T^{32} - \)\(18\!\cdots\!12\)\( p^{238} T^{34} + \)\(23\!\cdots\!82\)\( p^{272} T^{36} - \)\(20\!\cdots\!36\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 37 | \( 1 - \)\(56\!\cdots\!06\)\( T^{2} + \)\(15\!\cdots\!45\)\( T^{4} - \)\(27\!\cdots\!86\)\( T^{6} + \)\(37\!\cdots\!32\)\( T^{8} - \)\(38\!\cdots\!38\)\( T^{10} + \)\(33\!\cdots\!75\)\( T^{12} - \)\(24\!\cdots\!98\)\( T^{14} + \)\(15\!\cdots\!83\)\( T^{16} - \)\(86\!\cdots\!80\)\( T^{18} + \)\(42\!\cdots\!48\)\( T^{20} - \)\(86\!\cdots\!80\)\( p^{34} T^{22} + \)\(15\!\cdots\!83\)\( p^{68} T^{24} - \)\(24\!\cdots\!98\)\( p^{102} T^{26} + \)\(33\!\cdots\!75\)\( p^{136} T^{28} - \)\(38\!\cdots\!38\)\( p^{170} T^{30} + \)\(37\!\cdots\!32\)\( p^{204} T^{32} - \)\(27\!\cdots\!86\)\( p^{238} T^{34} + \)\(15\!\cdots\!45\)\( p^{272} T^{36} - \)\(56\!\cdots\!06\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 41 | \( 1 - \)\(31\!\cdots\!28\)\( T^{2} + \)\(50\!\cdots\!74\)\( T^{4} - \)\(54\!\cdots\!20\)\( T^{6} + \)\(43\!\cdots\!33\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{10} + \)\(14\!\cdots\!68\)\( T^{12} - \)\(63\!\cdots\!20\)\( T^{14} + \)\(23\!\cdots\!02\)\( T^{16} - \)\(77\!\cdots\!92\)\( T^{18} + \)\(21\!\cdots\!84\)\( T^{20} - \)\(77\!\cdots\!92\)\( p^{34} T^{22} + \)\(23\!\cdots\!02\)\( p^{68} T^{24} - \)\(63\!\cdots\!20\)\( p^{102} T^{26} + \)\(14\!\cdots\!68\)\( p^{136} T^{28} - \)\(27\!\cdots\!20\)\( p^{170} T^{30} + \)\(43\!\cdots\!33\)\( p^{204} T^{32} - \)\(54\!\cdots\!20\)\( p^{238} T^{34} + \)\(50\!\cdots\!74\)\( p^{272} T^{36} - \)\(31\!\cdots\!28\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 43 | \( ( 1 + 100737371540080 T + \)\(36\!\cdots\!63\)\( T^{2} + \)\(32\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!84\)\( T^{4} + \)\(50\!\cdots\!60\)\( T^{5} + \)\(71\!\cdots\!81\)\( T^{6} + \)\(51\!\cdots\!00\)\( T^{7} + \)\(60\!\cdots\!43\)\( T^{8} + \)\(38\!\cdots\!20\)\( T^{9} + \)\(40\!\cdots\!12\)\( T^{10} + \)\(38\!\cdots\!20\)\( p^{17} T^{11} + \)\(60\!\cdots\!43\)\( p^{34} T^{12} + \)\(51\!\cdots\!00\)\( p^{51} T^{13} + \)\(71\!\cdots\!81\)\( p^{68} T^{14} + \)\(50\!\cdots\!60\)\( p^{85} T^{15} + \)\(63\!\cdots\!84\)\( p^{102} T^{16} + \)\(32\!\cdots\!20\)\( p^{119} T^{17} + \)\(36\!\cdots\!63\)\( p^{136} T^{18} + 100737371540080 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 47 | \( 1 - \)\(33\!\cdots\!34\)\( T^{2} + \)\(55\!\cdots\!69\)\( T^{4} - \)\(60\!\cdots\!90\)\( T^{6} + \)\(49\!\cdots\!60\)\( T^{8} - \)\(31\!\cdots\!42\)\( T^{10} + \)\(16\!\cdots\!39\)\( T^{12} - \)\(72\!\cdots\!10\)\( T^{14} + \)\(27\!\cdots\!35\)\( T^{16} - \)\(90\!\cdots\!04\)\( T^{18} + \)\(25\!\cdots\!52\)\( T^{20} - \)\(90\!\cdots\!04\)\( p^{34} T^{22} + \)\(27\!\cdots\!35\)\( p^{68} T^{24} - \)\(72\!\cdots\!10\)\( p^{102} T^{26} + \)\(16\!\cdots\!39\)\( p^{136} T^{28} - \)\(31\!\cdots\!42\)\( p^{170} T^{30} + \)\(49\!\cdots\!60\)\( p^{204} T^{32} - \)\(60\!\cdots\!90\)\( p^{238} T^{34} + \)\(55\!\cdots\!69\)\( p^{272} T^{36} - \)\(33\!\cdots\!34\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 53 | \( ( 1 + 1214749246599660 T + \)\(19\!\cdots\!94\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(15\!\cdots\!81\)\( T^{4} + \)\(18\!\cdots\!00\)\( p T^{5} + \)\(67\!\cdots\!40\)\( T^{6} + \)\(35\!\cdots\!20\)\( T^{7} + \)\(20\!\cdots\!06\)\( T^{8} + \)\(96\!\cdots\!00\)\( T^{9} + \)\(48\!\cdots\!12\)\( T^{10} + \)\(96\!\cdots\!00\)\( p^{17} T^{11} + \)\(20\!\cdots\!06\)\( p^{34} T^{12} + \)\(35\!\cdots\!20\)\( p^{51} T^{13} + \)\(67\!\cdots\!40\)\( p^{68} T^{14} + \)\(18\!\cdots\!00\)\( p^{86} T^{15} + \)\(15\!\cdots\!81\)\( p^{102} T^{16} + \)\(16\!\cdots\!40\)\( p^{119} T^{17} + \)\(19\!\cdots\!94\)\( p^{136} T^{18} + 1214749246599660 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 59 | \( 1 - \)\(91\!\cdots\!32\)\( T^{2} + \)\(41\!\cdots\!14\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!25\)\( T^{8} - \)\(48\!\cdots\!16\)\( T^{10} + \)\(70\!\cdots\!32\)\( T^{12} - \)\(90\!\cdots\!60\)\( T^{14} + \)\(10\!\cdots\!30\)\( T^{16} - \)\(12\!\cdots\!56\)\( T^{18} + \)\(16\!\cdots\!64\)\( T^{20} - \)\(12\!\cdots\!56\)\( p^{34} T^{22} + \)\(10\!\cdots\!30\)\( p^{68} T^{24} - \)\(90\!\cdots\!60\)\( p^{102} T^{26} + \)\(70\!\cdots\!32\)\( p^{136} T^{28} - \)\(48\!\cdots\!16\)\( p^{170} T^{30} + \)\(27\!\cdots\!25\)\( p^{204} T^{32} - \)\(12\!\cdots\!00\)\( p^{238} T^{34} + \)\(41\!\cdots\!14\)\( p^{272} T^{36} - \)\(91\!\cdots\!32\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 61 | \( ( 1 + 70353931526176 T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(63\!\cdots\!40\)\( T^{3} + \)\(12\!\cdots\!93\)\( T^{4} - \)\(13\!\cdots\!44\)\( T^{5} + \)\(60\!\cdots\!16\)\( T^{6} - \)\(84\!\cdots\!72\)\( T^{7} + \)\(21\!\cdots\!18\)\( T^{8} - \)\(29\!\cdots\!96\)\( T^{9} + \)\(55\!\cdots\!80\)\( T^{10} - \)\(29\!\cdots\!96\)\( p^{17} T^{11} + \)\(21\!\cdots\!18\)\( p^{34} T^{12} - \)\(84\!\cdots\!72\)\( p^{51} T^{13} + \)\(60\!\cdots\!16\)\( p^{68} T^{14} - \)\(13\!\cdots\!44\)\( p^{85} T^{15} + \)\(12\!\cdots\!93\)\( p^{102} T^{16} - \)\(63\!\cdots\!40\)\( p^{119} T^{17} + \)\(15\!\cdots\!18\)\( p^{136} T^{18} + 70353931526176 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 67 | \( 1 - \)\(12\!\cdots\!32\)\( T^{2} + \)\(79\!\cdots\!02\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{6} + \)\(90\!\cdots\!05\)\( T^{8} - \)\(20\!\cdots\!92\)\( T^{10} + \)\(36\!\cdots\!76\)\( T^{12} - \)\(54\!\cdots\!40\)\( T^{14} + \)\(71\!\cdots\!94\)\( T^{16} - \)\(84\!\cdots\!32\)\( T^{18} + \)\(95\!\cdots\!84\)\( T^{20} - \)\(84\!\cdots\!32\)\( p^{34} T^{22} + \)\(71\!\cdots\!94\)\( p^{68} T^{24} - \)\(54\!\cdots\!40\)\( p^{102} T^{26} + \)\(36\!\cdots\!76\)\( p^{136} T^{28} - \)\(20\!\cdots\!92\)\( p^{170} T^{30} + \)\(90\!\cdots\!05\)\( p^{204} T^{32} - \)\(31\!\cdots\!80\)\( p^{238} T^{34} + \)\(79\!\cdots\!02\)\( p^{272} T^{36} - \)\(12\!\cdots\!32\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 71 | \( 1 - \)\(17\!\cdots\!74\)\( T^{2} + \)\(16\!\cdots\!65\)\( T^{4} - \)\(10\!\cdots\!18\)\( T^{6} + \)\(57\!\cdots\!48\)\( T^{8} - \)\(26\!\cdots\!22\)\( T^{10} + \)\(10\!\cdots\!07\)\( T^{12} - \)\(39\!\cdots\!26\)\( T^{14} + \)\(13\!\cdots\!15\)\( T^{16} - \)\(42\!\cdots\!60\)\( T^{18} + \)\(12\!\cdots\!48\)\( T^{20} - \)\(42\!\cdots\!60\)\( p^{34} T^{22} + \)\(13\!\cdots\!15\)\( p^{68} T^{24} - \)\(39\!\cdots\!26\)\( p^{102} T^{26} + \)\(10\!\cdots\!07\)\( p^{136} T^{28} - \)\(26\!\cdots\!22\)\( p^{170} T^{30} + \)\(57\!\cdots\!48\)\( p^{204} T^{32} - \)\(10\!\cdots\!18\)\( p^{238} T^{34} + \)\(16\!\cdots\!65\)\( p^{272} T^{36} - \)\(17\!\cdots\!74\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 73 | \( 1 - \)\(26\!\cdots\!44\)\( T^{2} + \)\(36\!\cdots\!74\)\( T^{4} - \)\(29\!\cdots\!00\)\( T^{6} + \)\(15\!\cdots\!65\)\( T^{8} - \)\(38\!\cdots\!32\)\( T^{10} - \)\(64\!\cdots\!96\)\( T^{12} + \)\(69\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!70\)\( T^{16} - \)\(49\!\cdots\!04\)\( T^{18} + \)\(32\!\cdots\!52\)\( T^{20} - \)\(49\!\cdots\!04\)\( p^{34} T^{22} + \)\(18\!\cdots\!70\)\( p^{68} T^{24} + \)\(69\!\cdots\!40\)\( p^{102} T^{26} - \)\(64\!\cdots\!96\)\( p^{136} T^{28} - \)\(38\!\cdots\!32\)\( p^{170} T^{30} + \)\(15\!\cdots\!65\)\( p^{204} T^{32} - \)\(29\!\cdots\!00\)\( p^{238} T^{34} + \)\(36\!\cdots\!74\)\( p^{272} T^{36} - \)\(26\!\cdots\!44\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 79 | \( ( 1 + 15907264829481376 T + \)\(16\!\cdots\!18\)\( T^{2} + \)\(21\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!57\)\( T^{4} + \)\(13\!\cdots\!92\)\( T^{5} + \)\(52\!\cdots\!52\)\( T^{6} + \)\(51\!\cdots\!52\)\( T^{7} + \)\(15\!\cdots\!46\)\( T^{8} + \)\(13\!\cdots\!36\)\( T^{9} + \)\(33\!\cdots\!12\)\( T^{10} + \)\(13\!\cdots\!36\)\( p^{17} T^{11} + \)\(15\!\cdots\!46\)\( p^{34} T^{12} + \)\(51\!\cdots\!52\)\( p^{51} T^{13} + \)\(52\!\cdots\!52\)\( p^{68} T^{14} + \)\(13\!\cdots\!92\)\( p^{85} T^{15} + \)\(11\!\cdots\!57\)\( p^{102} T^{16} + \)\(21\!\cdots\!72\)\( p^{119} T^{17} + \)\(16\!\cdots\!18\)\( p^{136} T^{18} + 15907264829481376 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 83 | \( 1 - \)\(43\!\cdots\!44\)\( T^{2} + \)\(99\!\cdots\!06\)\( T^{4} - \)\(15\!\cdots\!56\)\( T^{6} + \)\(18\!\cdots\!69\)\( T^{8} - \)\(17\!\cdots\!68\)\( T^{10} + \)\(13\!\cdots\!52\)\( T^{12} - \)\(94\!\cdots\!36\)\( T^{14} + \)\(55\!\cdots\!90\)\( T^{16} - \)\(28\!\cdots\!04\)\( T^{18} + \)\(12\!\cdots\!64\)\( T^{20} - \)\(28\!\cdots\!04\)\( p^{34} T^{22} + \)\(55\!\cdots\!90\)\( p^{68} T^{24} - \)\(94\!\cdots\!36\)\( p^{102} T^{26} + \)\(13\!\cdots\!52\)\( p^{136} T^{28} - \)\(17\!\cdots\!68\)\( p^{170} T^{30} + \)\(18\!\cdots\!69\)\( p^{204} T^{32} - \)\(15\!\cdots\!56\)\( p^{238} T^{34} + \)\(99\!\cdots\!06\)\( p^{272} T^{36} - \)\(43\!\cdots\!44\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 89 | \( 1 - \)\(15\!\cdots\!56\)\( T^{2} + \)\(12\!\cdots\!66\)\( T^{4} - \)\(63\!\cdots\!80\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} - \)\(75\!\cdots\!32\)\( T^{10} + \)\(19\!\cdots\!92\)\( T^{12} - \)\(41\!\cdots\!68\)\( T^{14} + \)\(77\!\cdots\!26\)\( T^{16} - \)\(12\!\cdots\!48\)\( T^{18} + \)\(18\!\cdots\!76\)\( T^{20} - \)\(12\!\cdots\!48\)\( p^{34} T^{22} + \)\(77\!\cdots\!26\)\( p^{68} T^{24} - \)\(41\!\cdots\!68\)\( p^{102} T^{26} + \)\(19\!\cdots\!92\)\( p^{136} T^{28} - \)\(75\!\cdots\!32\)\( p^{170} T^{30} + \)\(24\!\cdots\!01\)\( p^{204} T^{32} - \)\(63\!\cdots\!80\)\( p^{238} T^{34} + \)\(12\!\cdots\!66\)\( p^{272} T^{36} - \)\(15\!\cdots\!56\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 97 | \( 1 - \)\(69\!\cdots\!00\)\( T^{2} + \)\(24\!\cdots\!70\)\( T^{4} - \)\(56\!\cdots\!48\)\( T^{6} + \)\(99\!\cdots\!97\)\( T^{8} - \)\(14\!\cdots\!04\)\( T^{10} + \)\(16\!\cdots\!64\)\( T^{12} - \)\(15\!\cdots\!96\)\( T^{14} + \)\(13\!\cdots\!82\)\( T^{16} - \)\(98\!\cdots\!76\)\( T^{18} + \)\(62\!\cdots\!12\)\( T^{20} - \)\(98\!\cdots\!76\)\( p^{34} T^{22} + \)\(13\!\cdots\!82\)\( p^{68} T^{24} - \)\(15\!\cdots\!96\)\( p^{102} T^{26} + \)\(16\!\cdots\!64\)\( p^{136} T^{28} - \)\(14\!\cdots\!04\)\( p^{170} T^{30} + \)\(99\!\cdots\!97\)\( p^{204} T^{32} - \)\(56\!\cdots\!48\)\( p^{238} T^{34} + \)\(24\!\cdots\!70\)\( p^{272} T^{36} - \)\(69\!\cdots\!00\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.46854328037883527466895256405, −1.41628362196750292483063313484, −1.37253293336743833896028344916, −1.25805383195736808798020916907, −1.23184292545171741323385101554, −1.08890180153420734458816974957, −1.08364701332246012285289960474, −1.07352705765767399055059541168, −0.920629532161632050570368634210, −0.917160563908853261368206577883, −0.886350186289603031345133575439, −0.871387406405221875218970301809, −0.864537724784084352196068397160, −0.77723396928169153329316293006, −0.71115343102377523936423025221, −0.63938220914560612187847562801, −0.53931683023406858449713316298, −0.43823320010122072844471487895, −0.37221743805295188188897542935, −0.36636709825746561297114325356, −0.29800747692336071782693569608, −0.27946917749031745256654175294, −0.17866538907470766408204748000, −0.13884217423465275494073013506, −0.06077505914750507205120798972, 0.06077505914750507205120798972, 0.13884217423465275494073013506, 0.17866538907470766408204748000, 0.27946917749031745256654175294, 0.29800747692336071782693569608, 0.36636709825746561297114325356, 0.37221743805295188188897542935, 0.43823320010122072844471487895, 0.53931683023406858449713316298, 0.63938220914560612187847562801, 0.71115343102377523936423025221, 0.77723396928169153329316293006, 0.864537724784084352196068397160, 0.871387406405221875218970301809, 0.886350186289603031345133575439, 0.917160563908853261368206577883, 0.920629532161632050570368634210, 1.07352705765767399055059541168, 1.08364701332246012285289960474, 1.08890180153420734458816974957, 1.23184292545171741323385101554, 1.25805383195736808798020916907, 1.37253293336743833896028344916, 1.41628362196750292483063313484, 1.46854328037883527466895256405
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.