Dirichlet series
| L(s) = 1 | + 6.72e5·4-s + 1.62e9·13-s + 2.04e11·16-s + 4.10e10·17-s + 4.54e11·23-s + 7.15e12·25-s − 1.04e13·29-s − 9.16e13·43-s + 1.97e15·49-s + 1.09e15·52-s + 9.00e14·53-s + 9.65e14·61-s + 3.68e16·64-s + 2.76e16·68-s + 2.53e16·79-s + 3.05e17·92-s + 4.81e18·100-s − 5.44e16·101-s − 4.24e17·103-s − 2.33e17·107-s + 1.88e18·113-s − 7.00e18·116-s + 3.79e18·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 5.13·4-s + 0.552·13-s + 11.9·16-s + 1.42·17-s + 1.20·23-s + 9.37·25-s − 3.86·29-s − 1.19·43-s + 8.47·49-s + 2.83·52-s + 1.98·53-s + 0.644·61-s + 16.3·64-s + 7.32·68-s + 1.87·79-s + 6.20·92-s + 48.1·100-s − 0.500·101-s − 3.30·103-s − 1.31·107-s + 6.65·113-s − 19.8·116-s + 7.50·121-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
| \(L(9)\) | \(\approx\) | \(2160.287923\) |
| \(L(\frac12)\) | \(\approx\) | \(2160.287923\) |
| \(L(\frac{19}{2})\) | 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 | \( 1 - 1624882760 T - 461104377686396288 p T^{2} + \)\(13\!\cdots\!00\)\( p^{3} T^{3} + \)\(50\!\cdots\!36\)\( p^{6} T^{4} - \)\(22\!\cdots\!72\)\( p^{10} T^{5} + \)\(81\!\cdots\!76\)\( p^{16} T^{6} + \)\(20\!\cdots\!48\)\( p^{20} T^{7} - \)\(13\!\cdots\!48\)\( p^{25} T^{8} - \)\(24\!\cdots\!88\)\( p^{32} T^{9} + \)\(33\!\cdots\!28\)\( p^{40} T^{10} - \)\(24\!\cdots\!88\)\( p^{49} T^{11} - \)\(13\!\cdots\!48\)\( p^{59} T^{12} + \)\(20\!\cdots\!48\)\( p^{71} T^{13} + \)\(81\!\cdots\!76\)\( p^{84} T^{14} - \)\(22\!\cdots\!72\)\( p^{95} T^{15} + \)\(50\!\cdots\!36\)\( p^{108} T^{16} + \)\(13\!\cdots\!00\)\( p^{122} T^{17} - 461104377686396288 p^{137} T^{18} - 1624882760 p^{153} T^{19} + p^{170} T^{20} \) | |
| good | 2 | \( 1 - 672701 T^{2} + 7750720343 p^{5} T^{4} - 4133948469362827 p^{4} T^{6} + 55497010741402338137 p^{8} T^{8} - \)\(15\!\cdots\!21\)\( p^{14} T^{10} + \)\(15\!\cdots\!33\)\( p^{18} T^{12} - \)\(21\!\cdots\!31\)\( p^{28} T^{14} + \)\(21\!\cdots\!19\)\( p^{35} T^{16} - \)\(53\!\cdots\!83\)\( p^{44} T^{18} + \)\(10\!\cdots\!01\)\( p^{50} T^{20} - \)\(53\!\cdots\!83\)\( p^{78} T^{22} + \)\(21\!\cdots\!19\)\( p^{103} T^{24} - \)\(21\!\cdots\!31\)\( p^{130} T^{26} + \)\(15\!\cdots\!33\)\( p^{154} T^{28} - \)\(15\!\cdots\!21\)\( p^{184} T^{30} + 55497010741402338137 p^{212} T^{32} - 4133948469362827 p^{242} T^{34} + 7750720343 p^{277} T^{36} - 672701 p^{306} T^{38} + p^{340} T^{40} \) |
| 5 | \( 1 - 7151952658544 T^{2} + \)\(50\!\cdots\!26\)\( p T^{4} - \)\(23\!\cdots\!36\)\( p^{2} T^{6} + \)\(31\!\cdots\!13\)\( p^{5} T^{8} - \)\(87\!\cdots\!72\)\( p^{6} T^{10} + \)\(16\!\cdots\!72\)\( p^{10} T^{12} - \)\(26\!\cdots\!28\)\( p^{14} T^{14} + \)\(39\!\cdots\!34\)\( p^{18} T^{16} - \)\(21\!\cdots\!96\)\( p^{24} T^{18} + \)\(67\!\cdots\!44\)\( p^{26} T^{20} - \)\(21\!\cdots\!96\)\( p^{58} T^{22} + \)\(39\!\cdots\!34\)\( p^{86} T^{24} - \)\(26\!\cdots\!28\)\( p^{116} T^{26} + \)\(16\!\cdots\!72\)\( p^{146} T^{28} - \)\(87\!\cdots\!72\)\( p^{176} T^{30} + \)\(31\!\cdots\!13\)\( p^{209} T^{32} - \)\(23\!\cdots\!36\)\( p^{240} T^{34} + \)\(50\!\cdots\!26\)\( p^{273} T^{36} - 7151952658544 p^{306} T^{38} + p^{340} T^{40} \) | |
| 7 | \( 1 - 1971468752597804 T^{2} + \)\(19\!\cdots\!42\)\( T^{4} - \)\(12\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!57\)\( p^{2} T^{8} - \)\(79\!\cdots\!20\)\( p^{4} T^{10} + \)\(44\!\cdots\!12\)\( p^{6} T^{12} - \)\(21\!\cdots\!80\)\( p^{8} T^{14} + \)\(18\!\cdots\!74\)\( p^{12} T^{16} - \)\(16\!\cdots\!76\)\( p^{16} T^{18} + \)\(14\!\cdots\!04\)\( p^{20} T^{20} - \)\(16\!\cdots\!76\)\( p^{50} T^{22} + \)\(18\!\cdots\!74\)\( p^{80} T^{24} - \)\(21\!\cdots\!80\)\( p^{110} T^{26} + \)\(44\!\cdots\!12\)\( p^{142} T^{28} - \)\(79\!\cdots\!20\)\( p^{174} T^{30} + \)\(11\!\cdots\!57\)\( p^{206} T^{32} - \)\(12\!\cdots\!64\)\( p^{238} T^{34} + \)\(19\!\cdots\!42\)\( p^{272} T^{36} - 1971468752597804 p^{306} T^{38} + p^{340} T^{40} \) | |
| 11 | \( 1 - 3792340197643271504 T^{2} + \)\(71\!\cdots\!14\)\( T^{4} - \)\(93\!\cdots\!08\)\( T^{6} + \)\(98\!\cdots\!89\)\( T^{8} - \)\(72\!\cdots\!88\)\( p^{2} T^{10} + \)\(47\!\cdots\!16\)\( p^{4} T^{12} - \)\(27\!\cdots\!00\)\( p^{6} T^{14} + \)\(14\!\cdots\!70\)\( p^{8} T^{16} - \)\(69\!\cdots\!80\)\( p^{10} T^{18} + \)\(30\!\cdots\!00\)\( p^{12} T^{20} - \)\(69\!\cdots\!80\)\( p^{44} T^{22} + \)\(14\!\cdots\!70\)\( p^{76} T^{24} - \)\(27\!\cdots\!00\)\( p^{108} T^{26} + \)\(47\!\cdots\!16\)\( p^{140} T^{28} - \)\(72\!\cdots\!88\)\( p^{172} T^{30} + \)\(98\!\cdots\!89\)\( p^{204} T^{32} - \)\(93\!\cdots\!08\)\( p^{238} T^{34} + \)\(71\!\cdots\!14\)\( p^{272} T^{36} - 3792340197643271504 p^{306} T^{38} + p^{340} T^{40} \) | |
| 17 | \( ( 1 - 20517503016 T + \)\(48\!\cdots\!34\)\( T^{2} - \)\(75\!\cdots\!28\)\( T^{3} + \)\(11\!\cdots\!45\)\( T^{4} - \)\(14\!\cdots\!68\)\( T^{5} + \)\(19\!\cdots\!36\)\( T^{6} - \)\(20\!\cdots\!84\)\( T^{7} + \)\(23\!\cdots\!50\)\( T^{8} - \)\(21\!\cdots\!60\)\( T^{9} + \)\(21\!\cdots\!00\)\( T^{10} - \)\(21\!\cdots\!60\)\( p^{17} T^{11} + \)\(23\!\cdots\!50\)\( p^{34} T^{12} - \)\(20\!\cdots\!84\)\( p^{51} T^{13} + \)\(19\!\cdots\!36\)\( p^{68} T^{14} - \)\(14\!\cdots\!68\)\( p^{85} T^{15} + \)\(11\!\cdots\!45\)\( p^{102} T^{16} - \)\(75\!\cdots\!28\)\( p^{119} T^{17} + \)\(48\!\cdots\!34\)\( p^{136} T^{18} - 20517503016 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 19 | \( 1 - \)\(50\!\cdots\!20\)\( T^{2} + \)\(12\!\cdots\!86\)\( T^{4} - \)\(20\!\cdots\!64\)\( T^{6} + \)\(26\!\cdots\!77\)\( T^{8} - \)\(27\!\cdots\!40\)\( T^{10} + \)\(25\!\cdots\!20\)\( T^{12} - \)\(20\!\cdots\!40\)\( T^{14} + \)\(14\!\cdots\!98\)\( T^{16} - \)\(94\!\cdots\!36\)\( T^{18} + \)\(54\!\cdots\!36\)\( T^{20} - \)\(94\!\cdots\!36\)\( p^{34} T^{22} + \)\(14\!\cdots\!98\)\( p^{68} T^{24} - \)\(20\!\cdots\!40\)\( p^{102} T^{26} + \)\(25\!\cdots\!20\)\( p^{136} T^{28} - \)\(27\!\cdots\!40\)\( p^{170} T^{30} + \)\(26\!\cdots\!77\)\( p^{204} T^{32} - \)\(20\!\cdots\!64\)\( p^{238} T^{34} + \)\(12\!\cdots\!86\)\( p^{272} T^{36} - \)\(50\!\cdots\!20\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 23 | \( ( 1 - 227161274964 T + \)\(59\!\cdots\!38\)\( T^{2} - \)\(19\!\cdots\!56\)\( T^{3} + \)\(22\!\cdots\!89\)\( T^{4} - \)\(77\!\cdots\!60\)\( T^{5} + \)\(60\!\cdots\!64\)\( T^{6} - \)\(20\!\cdots\!76\)\( T^{7} + \)\(12\!\cdots\!02\)\( T^{8} - \)\(37\!\cdots\!72\)\( T^{9} + \)\(19\!\cdots\!96\)\( T^{10} - \)\(37\!\cdots\!72\)\( p^{17} T^{11} + \)\(12\!\cdots\!02\)\( p^{34} T^{12} - \)\(20\!\cdots\!76\)\( p^{51} T^{13} + \)\(60\!\cdots\!64\)\( p^{68} T^{14} - \)\(77\!\cdots\!60\)\( p^{85} T^{15} + \)\(22\!\cdots\!89\)\( p^{102} T^{16} - \)\(19\!\cdots\!56\)\( p^{119} T^{17} + \)\(59\!\cdots\!38\)\( p^{136} T^{18} - 227161274964 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 29 | \( ( 1 + 5209903427364 T + \)\(51\!\cdots\!78\)\( T^{2} + \)\(23\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!13\)\( T^{4} + \)\(50\!\cdots\!16\)\( T^{5} + \)\(22\!\cdots\!80\)\( T^{6} + \)\(71\!\cdots\!76\)\( T^{7} + \)\(25\!\cdots\!26\)\( T^{8} + \)\(71\!\cdots\!68\)\( T^{9} + \)\(21\!\cdots\!04\)\( T^{10} + \)\(71\!\cdots\!68\)\( p^{17} T^{11} + \)\(25\!\cdots\!26\)\( p^{34} T^{12} + \)\(71\!\cdots\!76\)\( p^{51} T^{13} + \)\(22\!\cdots\!80\)\( p^{68} T^{14} + \)\(50\!\cdots\!16\)\( p^{85} T^{15} + \)\(13\!\cdots\!13\)\( p^{102} T^{16} + \)\(23\!\cdots\!12\)\( p^{119} T^{17} + \)\(51\!\cdots\!78\)\( p^{136} T^{18} + 5209903427364 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 31 | \( 1 - \)\(22\!\cdots\!84\)\( T^{2} + \)\(26\!\cdots\!18\)\( T^{4} - \)\(21\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} - \)\(62\!\cdots\!76\)\( T^{10} + \)\(25\!\cdots\!48\)\( T^{12} - \)\(90\!\cdots\!76\)\( T^{14} + \)\(27\!\cdots\!78\)\( T^{16} - \)\(75\!\cdots\!20\)\( T^{18} + \)\(18\!\cdots\!68\)\( T^{20} - \)\(75\!\cdots\!20\)\( p^{34} T^{22} + \)\(27\!\cdots\!78\)\( p^{68} T^{24} - \)\(90\!\cdots\!76\)\( p^{102} T^{26} + \)\(25\!\cdots\!48\)\( p^{136} T^{28} - \)\(62\!\cdots\!76\)\( p^{170} T^{30} + \)\(12\!\cdots\!61\)\( p^{204} T^{32} - \)\(21\!\cdots\!44\)\( p^{238} T^{34} + \)\(26\!\cdots\!18\)\( p^{272} T^{36} - \)\(22\!\cdots\!84\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 37 | \( 1 - \)\(22\!\cdots\!00\)\( T^{2} + \)\(36\!\cdots\!66\)\( T^{4} - \)\(43\!\cdots\!80\)\( T^{6} + \)\(43\!\cdots\!53\)\( T^{8} - \)\(36\!\cdots\!92\)\( T^{10} + \)\(27\!\cdots\!24\)\( T^{12} - \)\(17\!\cdots\!16\)\( T^{14} + \)\(10\!\cdots\!74\)\( T^{16} - \)\(15\!\cdots\!56\)\( p T^{18} + \)\(26\!\cdots\!64\)\( T^{20} - \)\(15\!\cdots\!56\)\( p^{35} T^{22} + \)\(10\!\cdots\!74\)\( p^{68} T^{24} - \)\(17\!\cdots\!16\)\( p^{102} T^{26} + \)\(27\!\cdots\!24\)\( p^{136} T^{28} - \)\(36\!\cdots\!92\)\( p^{170} T^{30} + \)\(43\!\cdots\!53\)\( p^{204} T^{32} - \)\(43\!\cdots\!80\)\( p^{238} T^{34} + \)\(36\!\cdots\!66\)\( p^{272} T^{36} - \)\(22\!\cdots\!00\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 41 | \( 1 - \)\(21\!\cdots\!64\)\( T^{2} + \)\(23\!\cdots\!18\)\( T^{4} - \)\(40\!\cdots\!56\)\( p T^{6} + \)\(89\!\cdots\!57\)\( T^{8} - \)\(39\!\cdots\!80\)\( T^{10} + \)\(14\!\cdots\!80\)\( T^{12} - \)\(49\!\cdots\!20\)\( T^{14} + \)\(15\!\cdots\!22\)\( T^{16} - \)\(44\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!44\)\( T^{20} - \)\(44\!\cdots\!00\)\( p^{34} T^{22} + \)\(15\!\cdots\!22\)\( p^{68} T^{24} - \)\(49\!\cdots\!20\)\( p^{102} T^{26} + \)\(14\!\cdots\!80\)\( p^{136} T^{28} - \)\(39\!\cdots\!80\)\( p^{170} T^{30} + \)\(89\!\cdots\!57\)\( p^{204} T^{32} - \)\(40\!\cdots\!56\)\( p^{239} T^{34} + \)\(23\!\cdots\!18\)\( p^{272} T^{36} - \)\(21\!\cdots\!64\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 43 | \( ( 1 + 45820270029688 T + \)\(29\!\cdots\!78\)\( T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(43\!\cdots\!49\)\( T^{4} + \)\(28\!\cdots\!52\)\( T^{5} + \)\(46\!\cdots\!12\)\( T^{6} + \)\(29\!\cdots\!36\)\( T^{7} + \)\(39\!\cdots\!90\)\( T^{8} + \)\(22\!\cdots\!32\)\( T^{9} + \)\(26\!\cdots\!20\)\( T^{10} + \)\(22\!\cdots\!32\)\( p^{17} T^{11} + \)\(39\!\cdots\!90\)\( p^{34} T^{12} + \)\(29\!\cdots\!36\)\( p^{51} T^{13} + \)\(46\!\cdots\!12\)\( p^{68} T^{14} + \)\(28\!\cdots\!52\)\( p^{85} T^{15} + \)\(43\!\cdots\!49\)\( p^{102} T^{16} + \)\(16\!\cdots\!68\)\( p^{119} T^{17} + \)\(29\!\cdots\!78\)\( p^{136} T^{18} + 45820270029688 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 47 | \( 1 - \)\(32\!\cdots\!60\)\( T^{2} + \)\(50\!\cdots\!10\)\( T^{4} - \)\(53\!\cdots\!12\)\( T^{6} + \)\(41\!\cdots\!05\)\( T^{8} - \)\(25\!\cdots\!40\)\( T^{10} + \)\(13\!\cdots\!44\)\( T^{12} - \)\(58\!\cdots\!84\)\( T^{14} + \)\(21\!\cdots\!62\)\( T^{16} - \)\(71\!\cdots\!64\)\( T^{18} + \)\(20\!\cdots\!56\)\( T^{20} - \)\(71\!\cdots\!64\)\( p^{34} T^{22} + \)\(21\!\cdots\!62\)\( p^{68} T^{24} - \)\(58\!\cdots\!84\)\( p^{102} T^{26} + \)\(13\!\cdots\!44\)\( p^{136} T^{28} - \)\(25\!\cdots\!40\)\( p^{170} T^{30} + \)\(41\!\cdots\!05\)\( p^{204} T^{32} - \)\(53\!\cdots\!12\)\( p^{238} T^{34} + \)\(50\!\cdots\!10\)\( p^{272} T^{36} - \)\(32\!\cdots\!60\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 53 | \( ( 1 - 450073844007144 T + \)\(10\!\cdots\!38\)\( T^{2} - \)\(35\!\cdots\!80\)\( T^{3} + \)\(54\!\cdots\!49\)\( T^{4} - \)\(15\!\cdots\!36\)\( T^{5} + \)\(19\!\cdots\!72\)\( T^{6} - \)\(48\!\cdots\!56\)\( T^{7} + \)\(55\!\cdots\!98\)\( T^{8} - \)\(12\!\cdots\!80\)\( T^{9} + \)\(12\!\cdots\!88\)\( T^{10} - \)\(12\!\cdots\!80\)\( p^{17} T^{11} + \)\(55\!\cdots\!98\)\( p^{34} T^{12} - \)\(48\!\cdots\!56\)\( p^{51} T^{13} + \)\(19\!\cdots\!72\)\( p^{68} T^{14} - \)\(15\!\cdots\!36\)\( p^{85} T^{15} + \)\(54\!\cdots\!49\)\( p^{102} T^{16} - \)\(35\!\cdots\!80\)\( p^{119} T^{17} + \)\(10\!\cdots\!38\)\( p^{136} T^{18} - 450073844007144 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 59 | \( 1 - \)\(17\!\cdots\!44\)\( T^{2} + \)\(15\!\cdots\!14\)\( T^{4} - \)\(84\!\cdots\!48\)\( T^{6} + \)\(34\!\cdots\!61\)\( T^{8} - \)\(11\!\cdots\!96\)\( T^{10} + \)\(29\!\cdots\!04\)\( T^{12} - \)\(64\!\cdots\!12\)\( T^{14} + \)\(12\!\cdots\!06\)\( T^{16} - \)\(19\!\cdots\!96\)\( T^{18} + \)\(26\!\cdots\!00\)\( T^{20} - \)\(19\!\cdots\!96\)\( p^{34} T^{22} + \)\(12\!\cdots\!06\)\( p^{68} T^{24} - \)\(64\!\cdots\!12\)\( p^{102} T^{26} + \)\(29\!\cdots\!04\)\( p^{136} T^{28} - \)\(11\!\cdots\!96\)\( p^{170} T^{30} + \)\(34\!\cdots\!61\)\( p^{204} T^{32} - \)\(84\!\cdots\!48\)\( p^{238} T^{34} + \)\(15\!\cdots\!14\)\( p^{272} T^{36} - \)\(17\!\cdots\!44\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 61 | \( ( 1 - 482871878328080 T + \)\(96\!\cdots\!66\)\( T^{2} - \)\(14\!\cdots\!64\)\( T^{3} + \)\(47\!\cdots\!81\)\( T^{4} + \)\(93\!\cdots\!68\)\( T^{5} + \)\(16\!\cdots\!52\)\( T^{6} + \)\(60\!\cdots\!08\)\( T^{7} + \)\(50\!\cdots\!38\)\( T^{8} + \)\(17\!\cdots\!48\)\( T^{9} + \)\(12\!\cdots\!04\)\( T^{10} + \)\(17\!\cdots\!48\)\( p^{17} T^{11} + \)\(50\!\cdots\!38\)\( p^{34} T^{12} + \)\(60\!\cdots\!08\)\( p^{51} T^{13} + \)\(16\!\cdots\!52\)\( p^{68} T^{14} + \)\(93\!\cdots\!68\)\( p^{85} T^{15} + \)\(47\!\cdots\!81\)\( p^{102} T^{16} - \)\(14\!\cdots\!64\)\( p^{119} T^{17} + \)\(96\!\cdots\!66\)\( p^{136} T^{18} - 482871878328080 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 67 | \( 1 - \)\(78\!\cdots\!12\)\( T^{2} + \)\(36\!\cdots\!30\)\( T^{4} - \)\(12\!\cdots\!80\)\( T^{6} + \)\(32\!\cdots\!05\)\( T^{8} - \)\(71\!\cdots\!44\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(22\!\cdots\!08\)\( T^{14} + \)\(33\!\cdots\!54\)\( T^{16} - \)\(43\!\cdots\!72\)\( T^{18} + \)\(50\!\cdots\!76\)\( T^{20} - \)\(43\!\cdots\!72\)\( p^{34} T^{22} + \)\(33\!\cdots\!54\)\( p^{68} T^{24} - \)\(22\!\cdots\!08\)\( p^{102} T^{26} + \)\(13\!\cdots\!92\)\( p^{136} T^{28} - \)\(71\!\cdots\!44\)\( p^{170} T^{30} + \)\(32\!\cdots\!05\)\( p^{204} T^{32} - \)\(12\!\cdots\!80\)\( p^{238} T^{34} + \)\(36\!\cdots\!30\)\( p^{272} T^{36} - \)\(78\!\cdots\!12\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 71 | \( 1 - \)\(23\!\cdots\!84\)\( T^{2} + \)\(27\!\cdots\!38\)\( T^{4} - \)\(22\!\cdots\!44\)\( T^{6} + \)\(13\!\cdots\!81\)\( T^{8} - \)\(67\!\cdots\!84\)\( T^{10} + \)\(28\!\cdots\!16\)\( T^{12} - \)\(10\!\cdots\!64\)\( T^{14} + \)\(35\!\cdots\!18\)\( T^{16} - \)\(11\!\cdots\!84\)\( T^{18} + \)\(33\!\cdots\!12\)\( T^{20} - \)\(11\!\cdots\!84\)\( p^{34} T^{22} + \)\(35\!\cdots\!18\)\( p^{68} T^{24} - \)\(10\!\cdots\!64\)\( p^{102} T^{26} + \)\(28\!\cdots\!16\)\( p^{136} T^{28} - \)\(67\!\cdots\!84\)\( p^{170} T^{30} + \)\(13\!\cdots\!81\)\( p^{204} T^{32} - \)\(22\!\cdots\!44\)\( p^{238} T^{34} + \)\(27\!\cdots\!38\)\( p^{272} T^{36} - \)\(23\!\cdots\!84\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 73 | \( 1 - \)\(59\!\cdots\!48\)\( T^{2} + \)\(17\!\cdots\!42\)\( T^{4} - \)\(34\!\cdots\!12\)\( T^{6} + \)\(50\!\cdots\!41\)\( T^{8} - \)\(58\!\cdots\!52\)\( T^{10} + \)\(55\!\cdots\!24\)\( T^{12} - \)\(44\!\cdots\!08\)\( T^{14} + \)\(30\!\cdots\!18\)\( T^{16} - \)\(17\!\cdots\!40\)\( T^{18} + \)\(90\!\cdots\!88\)\( T^{20} - \)\(17\!\cdots\!40\)\( p^{34} T^{22} + \)\(30\!\cdots\!18\)\( p^{68} T^{24} - \)\(44\!\cdots\!08\)\( p^{102} T^{26} + \)\(55\!\cdots\!24\)\( p^{136} T^{28} - \)\(58\!\cdots\!52\)\( p^{170} T^{30} + \)\(50\!\cdots\!41\)\( p^{204} T^{32} - \)\(34\!\cdots\!12\)\( p^{238} T^{34} + \)\(17\!\cdots\!42\)\( p^{272} T^{36} - \)\(59\!\cdots\!48\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 79 | \( ( 1 - 12657911656514336 T + \)\(56\!\cdots\!74\)\( T^{2} - \)\(93\!\cdots\!60\)\( T^{3} + \)\(25\!\cdots\!09\)\( T^{4} - \)\(35\!\cdots\!48\)\( T^{5} + \)\(78\!\cdots\!08\)\( T^{6} - \)\(99\!\cdots\!56\)\( T^{7} + \)\(19\!\cdots\!74\)\( T^{8} - \)\(22\!\cdots\!56\)\( T^{9} + \)\(37\!\cdots\!28\)\( T^{10} - \)\(22\!\cdots\!56\)\( p^{17} T^{11} + \)\(19\!\cdots\!74\)\( p^{34} T^{12} - \)\(99\!\cdots\!56\)\( p^{51} T^{13} + \)\(78\!\cdots\!08\)\( p^{68} T^{14} - \)\(35\!\cdots\!48\)\( p^{85} T^{15} + \)\(25\!\cdots\!09\)\( p^{102} T^{16} - \)\(93\!\cdots\!60\)\( p^{119} T^{17} + \)\(56\!\cdots\!74\)\( p^{136} T^{18} - 12657911656514336 p^{153} T^{19} + p^{170} T^{20} )^{2} \) | |
| 83 | \( 1 - \)\(44\!\cdots\!88\)\( p T^{2} + \)\(72\!\cdots\!54\)\( T^{4} - \)\(97\!\cdots\!96\)\( T^{6} + \)\(99\!\cdots\!13\)\( T^{8} - \)\(82\!\cdots\!52\)\( T^{10} + \)\(58\!\cdots\!92\)\( T^{12} - \)\(36\!\cdots\!60\)\( T^{14} + \)\(19\!\cdots\!94\)\( T^{16} - \)\(98\!\cdots\!92\)\( T^{18} + \)\(43\!\cdots\!72\)\( T^{20} - \)\(98\!\cdots\!92\)\( p^{34} T^{22} + \)\(19\!\cdots\!94\)\( p^{68} T^{24} - \)\(36\!\cdots\!60\)\( p^{102} T^{26} + \)\(58\!\cdots\!92\)\( p^{136} T^{28} - \)\(82\!\cdots\!52\)\( p^{170} T^{30} + \)\(99\!\cdots\!13\)\( p^{204} T^{32} - \)\(97\!\cdots\!96\)\( p^{238} T^{34} + \)\(72\!\cdots\!54\)\( p^{272} T^{36} - \)\(44\!\cdots\!88\)\( p^{307} T^{38} + p^{340} T^{40} \) | |
| 89 | \( 1 - \)\(14\!\cdots\!48\)\( T^{2} + \)\(10\!\cdots\!78\)\( T^{4} - \)\(54\!\cdots\!76\)\( T^{6} + \)\(21\!\cdots\!69\)\( T^{8} - \)\(65\!\cdots\!64\)\( T^{10} + \)\(17\!\cdots\!52\)\( T^{12} - \)\(37\!\cdots\!28\)\( T^{14} + \)\(72\!\cdots\!50\)\( T^{16} - \)\(12\!\cdots\!44\)\( T^{18} + \)\(17\!\cdots\!60\)\( T^{20} - \)\(12\!\cdots\!44\)\( p^{34} T^{22} + \)\(72\!\cdots\!50\)\( p^{68} T^{24} - \)\(37\!\cdots\!28\)\( p^{102} T^{26} + \)\(17\!\cdots\!52\)\( p^{136} T^{28} - \)\(65\!\cdots\!64\)\( p^{170} T^{30} + \)\(21\!\cdots\!69\)\( p^{204} T^{32} - \)\(54\!\cdots\!76\)\( p^{238} T^{34} + \)\(10\!\cdots\!78\)\( p^{272} T^{36} - \)\(14\!\cdots\!48\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
| 97 | \( 1 - \)\(54\!\cdots\!28\)\( T^{2} + \)\(15\!\cdots\!46\)\( T^{4} - \)\(31\!\cdots\!60\)\( T^{6} + \)\(48\!\cdots\!73\)\( T^{8} - \)\(61\!\cdots\!88\)\( T^{10} + \)\(64\!\cdots\!60\)\( T^{12} - \)\(59\!\cdots\!04\)\( T^{14} + \)\(47\!\cdots\!94\)\( T^{16} - \)\(33\!\cdots\!20\)\( T^{18} + \)\(21\!\cdots\!72\)\( T^{20} - \)\(33\!\cdots\!20\)\( p^{34} T^{22} + \)\(47\!\cdots\!94\)\( p^{68} T^{24} - \)\(59\!\cdots\!04\)\( p^{102} T^{26} + \)\(64\!\cdots\!60\)\( p^{136} T^{28} - \)\(61\!\cdots\!88\)\( p^{170} T^{30} + \)\(48\!\cdots\!73\)\( p^{204} T^{32} - \)\(31\!\cdots\!60\)\( p^{238} T^{34} + \)\(15\!\cdots\!46\)\( p^{272} T^{36} - \)\(54\!\cdots\!28\)\( p^{306} T^{38} + p^{340} T^{40} \) | |
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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
−1.44440518503015264518556332742, −1.34603312954957963596476673850, −1.30099285970115814838850802246, −1.30083778094276188755330007915, −1.19733921358664673252158977647, −1.18844489372166873062699047586, −1.05130471962539065655295789807, −1.00412509721397553999746252339, −0.995314147328199320756097966947, −0.992903132563807138708478410365, −0.910023361582296431454495756128, −0.793440781738261961394806951153, −0.75336353966246754911417492607, −0.74333735441238421956149505405, −0.72824344886159763016563151053, −0.65227199578307608741102579076, −0.63405217415889086323727185032, −0.45922531010250495754799401645, −0.43629586933223022209292638007, −0.37558324092739750001884975697, −0.36936462639060457180644696862, −0.28185515385489259107444161919, −0.27032405317383171847079027434, −0.10476965416573981333226133842, −0.06687018188001222489714491074, 0.06687018188001222489714491074, 0.10476965416573981333226133842, 0.27032405317383171847079027434, 0.28185515385489259107444161919, 0.36936462639060457180644696862, 0.37558324092739750001884975697, 0.43629586933223022209292638007, 0.45922531010250495754799401645, 0.63405217415889086323727185032, 0.65227199578307608741102579076, 0.72824344886159763016563151053, 0.74333735441238421956149505405, 0.75336353966246754911417492607, 0.793440781738261961394806951153, 0.910023361582296431454495756128, 0.992903132563807138708478410365, 0.995314147328199320756097966947, 1.00412509721397553999746252339, 1.05130471962539065655295789807, 1.18844489372166873062699047586, 1.19733921358664673252158977647, 1.30083778094276188755330007915, 1.30099285970115814838850802246, 1.34603312954957963596476673850, 1.44440518503015264518556332742
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.