Dirichlet series
| L(s) = 1 | + 72·2-s + 2.59e3·4-s + 294·5-s − 104·7-s + 6.14e4·8-s + 4.54e3·9-s + 2.11e4·10-s − 6.76e3·13-s − 7.48e3·14-s + 1.05e6·16-s − 9.13e3·17-s + 3.26e5·18-s + 7.57e3·19-s + 7.62e5·20-s − 5.05e4·23-s + 4.32e4·25-s − 4.87e5·26-s − 2.69e5·28-s − 4.29e4·29-s − 1.73e4·31-s + 1.37e7·32-s − 6.57e5·34-s − 3.05e4·35-s + 1.17e7·36-s − 2.38e5·37-s + 5.45e5·38-s + 1.80e7·40-s + ⋯ |
| L(s) = 1 | + 9·2-s + 81/2·4-s + 2.35·5-s − 0.303·7-s + 120·8-s + 6.22·9-s + 21.1·10-s − 3.07·13-s − 2.72·14-s + 258.·16-s − 1.85·17-s + 56.0·18-s + 1.10·19-s + 95.2·20-s − 4.15·23-s + 2.76·25-s − 27.7·26-s − 12.2·28-s − 1.76·29-s − 0.582·31-s + 420.·32-s − 16.7·34-s − 0.713·35-s + 252.·36-s − 4.70·37-s + 9.94·38-s + 282.·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 37^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 37^{18}\right)^{s/2} \, \Gamma_{\C}(s+3)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{18} \cdot 37^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44248\times 10^{22}\) |
| Root analytic conductor: | \(4.12601\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{18} \cdot 37^{18} ,\ ( \ : [3]^{18} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(17.67025611\) |
| \(L(\frac12)\) | \(\approx\) | \(17.67025611\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T + p^{5} T^{2} )^{9} \) |
| 37 | \( 1 + 238242 T + 884143877 p T^{2} + 2233794867200 p^{2} T^{3} + 4439682603716448 p^{3} T^{4} + 7346619358852044800 p^{4} T^{5} + \)\(29\!\cdots\!60\)\( p^{6} T^{6} + \)\(11\!\cdots\!08\)\( p^{8} T^{7} + \)\(10\!\cdots\!56\)\( p^{11} T^{8} + \)\(10\!\cdots\!36\)\( p^{14} T^{9} + \)\(10\!\cdots\!56\)\( p^{17} T^{10} + \)\(11\!\cdots\!08\)\( p^{20} T^{11} + \)\(29\!\cdots\!60\)\( p^{24} T^{12} + 7346619358852044800 p^{28} T^{13} + 4439682603716448 p^{33} T^{14} + 2233794867200 p^{38} T^{15} + 884143877 p^{43} T^{16} + 238242 p^{48} T^{17} + p^{54} T^{18} \) | |
| good | 3 | \( 1 - 4540 T^{2} + 10352290 T^{4} - 5671568216 p T^{6} + 2606032382683 p^{2} T^{8} - 3122482825337906 p^{2} T^{10} + 1089699872094247334 p^{3} T^{12} - 1394705477228538880 p^{9} T^{14} + \)\(95\!\cdots\!03\)\( p^{5} T^{16} - \)\(24\!\cdots\!94\)\( p^{6} T^{18} + \)\(95\!\cdots\!03\)\( p^{17} T^{20} - 1394705477228538880 p^{33} T^{22} + 1089699872094247334 p^{39} T^{24} - 3122482825337906 p^{50} T^{26} + 2606032382683 p^{62} T^{28} - 5671568216 p^{73} T^{30} + 10352290 p^{84} T^{32} - 4540 p^{96} T^{34} + p^{108} T^{36} \) |
| 5 | \( 1 - 294 T + 43218 T^{2} - 6812838 T^{3} + 538757544 T^{4} + 2137910634 p T^{5} - 128774854458 p^{2} T^{6} + 4244071917702 p^{3} T^{7} + 685461742566 p^{7} T^{8} - 9027702980048706 p^{5} T^{9} + 266056120200527982 p^{6} T^{10} - 7612019869014724794 p^{7} T^{11} + 88073545913912659296 p^{8} T^{12} + \)\(13\!\cdots\!66\)\( p^{9} T^{13} - \)\(44\!\cdots\!82\)\( p^{10} T^{14} + \)\(58\!\cdots\!78\)\( p^{13} T^{15} + \)\(13\!\cdots\!21\)\( p^{16} T^{16} - \)\(21\!\cdots\!56\)\( p^{17} T^{17} + \)\(59\!\cdots\!08\)\( p^{18} T^{18} - \)\(21\!\cdots\!56\)\( p^{23} T^{19} + \)\(13\!\cdots\!21\)\( p^{28} T^{20} + \)\(58\!\cdots\!78\)\( p^{31} T^{21} - \)\(44\!\cdots\!82\)\( p^{34} T^{22} + \)\(13\!\cdots\!66\)\( p^{39} T^{23} + 88073545913912659296 p^{44} T^{24} - 7612019869014724794 p^{49} T^{25} + 266056120200527982 p^{54} T^{26} - 9027702980048706 p^{59} T^{27} + 685461742566 p^{67} T^{28} + 4244071917702 p^{69} T^{29} - 128774854458 p^{74} T^{30} + 2137910634 p^{79} T^{31} + 538757544 p^{84} T^{32} - 6812838 p^{90} T^{33} + 43218 p^{96} T^{34} - 294 p^{102} T^{35} + p^{108} T^{36} \) | |
| 7 | \( ( 1 + 52 T + 332851 T^{2} - 7022286 p T^{3} + 66413306865 T^{4} - 12509125993730 T^{5} + 12270595500468971 T^{6} - 1921205705748472090 T^{7} + \)\(17\!\cdots\!00\)\( T^{8} - \)\(26\!\cdots\!64\)\( T^{9} + \)\(17\!\cdots\!00\)\( p^{6} T^{10} - 1921205705748472090 p^{12} T^{11} + 12270595500468971 p^{18} T^{12} - 12509125993730 p^{24} T^{13} + 66413306865 p^{30} T^{14} - 7022286 p^{37} T^{15} + 332851 p^{42} T^{16} + 52 p^{48} T^{17} + p^{54} T^{18} )^{2} \) | |
| 11 | \( 1 - 16886164 T^{2} + 136745405249674 T^{4} - \)\(70\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!19\)\( T^{8} - \)\(73\!\cdots\!02\)\( T^{10} + \)\(16\!\cdots\!90\)\( T^{12} - \)\(30\!\cdots\!16\)\( T^{14} + \)\(49\!\cdots\!61\)\( T^{16} - \)\(84\!\cdots\!06\)\( T^{18} + \)\(49\!\cdots\!61\)\( p^{12} T^{20} - \)\(30\!\cdots\!16\)\( p^{24} T^{22} + \)\(16\!\cdots\!90\)\( p^{36} T^{24} - \)\(73\!\cdots\!02\)\( p^{48} T^{26} + \)\(26\!\cdots\!19\)\( p^{60} T^{28} - \)\(70\!\cdots\!20\)\( p^{72} T^{30} + 136745405249674 p^{84} T^{32} - 16886164 p^{96} T^{34} + p^{108} T^{36} \) | |
| 13 | \( 1 + 6766 T + 22889378 T^{2} + 47786787038 T^{3} + 79872162853384 T^{4} + 247984874389351646 T^{5} + \)\(99\!\cdots\!06\)\( T^{6} + \)\(24\!\cdots\!06\)\( p T^{7} + \)\(91\!\cdots\!62\)\( T^{8} + \)\(22\!\cdots\!98\)\( T^{9} + \)\(48\!\cdots\!50\)\( T^{10} + \)\(93\!\cdots\!62\)\( T^{11} + \)\(22\!\cdots\!12\)\( T^{12} + \)\(69\!\cdots\!34\)\( T^{13} + \)\(19\!\cdots\!42\)\( T^{14} + \)\(45\!\cdots\!18\)\( T^{15} + \)\(88\!\cdots\!37\)\( T^{16} + \)\(16\!\cdots\!76\)\( T^{17} + \)\(36\!\cdots\!16\)\( T^{18} + \)\(16\!\cdots\!76\)\( p^{6} T^{19} + \)\(88\!\cdots\!37\)\( p^{12} T^{20} + \)\(45\!\cdots\!18\)\( p^{18} T^{21} + \)\(19\!\cdots\!42\)\( p^{24} T^{22} + \)\(69\!\cdots\!34\)\( p^{30} T^{23} + \)\(22\!\cdots\!12\)\( p^{36} T^{24} + \)\(93\!\cdots\!62\)\( p^{42} T^{25} + \)\(48\!\cdots\!50\)\( p^{48} T^{26} + \)\(22\!\cdots\!98\)\( p^{54} T^{27} + \)\(91\!\cdots\!62\)\( p^{60} T^{28} + \)\(24\!\cdots\!06\)\( p^{67} T^{29} + \)\(99\!\cdots\!06\)\( p^{72} T^{30} + 247984874389351646 p^{78} T^{31} + 79872162853384 p^{84} T^{32} + 47786787038 p^{90} T^{33} + 22889378 p^{96} T^{34} + 6766 p^{102} T^{35} + p^{108} T^{36} \) | |
| 17 | \( 1 + 9134 T + 41714978 T^{2} + 49248428390 T^{3} - 1529721213963031 T^{4} - 8691827922498777192 T^{5} - \)\(14\!\cdots\!60\)\( T^{6} + \)\(16\!\cdots\!16\)\( T^{7} + \)\(14\!\cdots\!16\)\( T^{8} + \)\(41\!\cdots\!84\)\( T^{9} - \)\(31\!\cdots\!80\)\( T^{10} - \)\(27\!\cdots\!92\)\( T^{11} - \)\(90\!\cdots\!52\)\( T^{12} + \)\(19\!\cdots\!84\)\( T^{13} + \)\(23\!\cdots\!12\)\( T^{14} + \)\(13\!\cdots\!20\)\( T^{15} + \)\(23\!\cdots\!10\)\( T^{16} - \)\(16\!\cdots\!92\)\( T^{17} - \)\(10\!\cdots\!48\)\( T^{18} - \)\(16\!\cdots\!92\)\( p^{6} T^{19} + \)\(23\!\cdots\!10\)\( p^{12} T^{20} + \)\(13\!\cdots\!20\)\( p^{18} T^{21} + \)\(23\!\cdots\!12\)\( p^{24} T^{22} + \)\(19\!\cdots\!84\)\( p^{30} T^{23} - \)\(90\!\cdots\!52\)\( p^{36} T^{24} - \)\(27\!\cdots\!92\)\( p^{42} T^{25} - \)\(31\!\cdots\!80\)\( p^{48} T^{26} + \)\(41\!\cdots\!84\)\( p^{54} T^{27} + \)\(14\!\cdots\!16\)\( p^{60} T^{28} + \)\(16\!\cdots\!16\)\( p^{66} T^{29} - \)\(14\!\cdots\!60\)\( p^{72} T^{30} - 8691827922498777192 p^{78} T^{31} - 1529721213963031 p^{84} T^{32} + 49248428390 p^{90} T^{33} + 41714978 p^{96} T^{34} + 9134 p^{102} T^{35} + p^{108} T^{36} \) | |
| 19 | \( 1 - 7578 T + 28713042 T^{2} - 235870919202 T^{3} + 5302906983411705 T^{4} - 36773488843963071128 T^{5} + \)\(15\!\cdots\!76\)\( T^{6} - \)\(13\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!00\)\( T^{8} - \)\(66\!\cdots\!88\)\( T^{9} + \)\(30\!\cdots\!24\)\( T^{10} - \)\(13\!\cdots\!52\)\( p T^{11} + \)\(36\!\cdots\!68\)\( T^{12} - \)\(22\!\cdots\!88\)\( T^{13} + \)\(10\!\cdots\!16\)\( T^{14} - \)\(88\!\cdots\!16\)\( T^{15} + \)\(11\!\cdots\!30\)\( T^{16} - \)\(73\!\cdots\!80\)\( T^{17} + \)\(35\!\cdots\!12\)\( T^{18} - \)\(73\!\cdots\!80\)\( p^{6} T^{19} + \)\(11\!\cdots\!30\)\( p^{12} T^{20} - \)\(88\!\cdots\!16\)\( p^{18} T^{21} + \)\(10\!\cdots\!16\)\( p^{24} T^{22} - \)\(22\!\cdots\!88\)\( p^{30} T^{23} + \)\(36\!\cdots\!68\)\( p^{36} T^{24} - \)\(13\!\cdots\!52\)\( p^{43} T^{25} + \)\(30\!\cdots\!24\)\( p^{48} T^{26} - \)\(66\!\cdots\!88\)\( p^{54} T^{27} + \)\(12\!\cdots\!00\)\( p^{60} T^{28} - \)\(13\!\cdots\!24\)\( p^{66} T^{29} + \)\(15\!\cdots\!76\)\( p^{72} T^{30} - 36773488843963071128 p^{78} T^{31} + 5302906983411705 p^{84} T^{32} - 235870919202 p^{90} T^{33} + 28713042 p^{96} T^{34} - 7578 p^{102} T^{35} + p^{108} T^{36} \) | |
| 23 | \( 1 + 50578 T + 1279067042 T^{2} + 23967892638234 T^{3} + 359682134788405236 T^{4} + \)\(39\!\cdots\!22\)\( T^{5} + \)\(26\!\cdots\!82\)\( T^{6} - \)\(11\!\cdots\!14\)\( p T^{7} - \)\(39\!\cdots\!66\)\( T^{8} - \)\(65\!\cdots\!38\)\( T^{9} - \)\(24\!\cdots\!50\)\( p T^{10} + \)\(22\!\cdots\!62\)\( T^{11} + \)\(10\!\cdots\!60\)\( T^{12} + \)\(21\!\cdots\!10\)\( T^{13} + \)\(30\!\cdots\!86\)\( T^{14} + \)\(30\!\cdots\!26\)\( T^{15} + \)\(20\!\cdots\!53\)\( T^{16} + \)\(61\!\cdots\!64\)\( T^{17} - \)\(39\!\cdots\!88\)\( T^{18} + \)\(61\!\cdots\!64\)\( p^{6} T^{19} + \)\(20\!\cdots\!53\)\( p^{12} T^{20} + \)\(30\!\cdots\!26\)\( p^{18} T^{21} + \)\(30\!\cdots\!86\)\( p^{24} T^{22} + \)\(21\!\cdots\!10\)\( p^{30} T^{23} + \)\(10\!\cdots\!60\)\( p^{36} T^{24} + \)\(22\!\cdots\!62\)\( p^{42} T^{25} - \)\(24\!\cdots\!50\)\( p^{49} T^{26} - \)\(65\!\cdots\!38\)\( p^{54} T^{27} - \)\(39\!\cdots\!66\)\( p^{60} T^{28} - \)\(11\!\cdots\!14\)\( p^{67} T^{29} + \)\(26\!\cdots\!82\)\( p^{72} T^{30} + \)\(39\!\cdots\!22\)\( p^{78} T^{31} + 359682134788405236 p^{84} T^{32} + 23967892638234 p^{90} T^{33} + 1279067042 p^{96} T^{34} + 50578 p^{102} T^{35} + p^{108} T^{36} \) | |
| 29 | \( 1 + 42950 T + 922351250 T^{2} + 23269642035782 T^{3} + 1164366851255681324 T^{4} + \)\(30\!\cdots\!86\)\( T^{5} + \)\(51\!\cdots\!62\)\( T^{6} + \)\(69\!\cdots\!34\)\( T^{7} + \)\(11\!\cdots\!74\)\( T^{8} + \)\(15\!\cdots\!82\)\( T^{9} - \)\(44\!\cdots\!02\)\( T^{10} - \)\(31\!\cdots\!42\)\( T^{11} - \)\(12\!\cdots\!16\)\( T^{12} - \)\(22\!\cdots\!30\)\( T^{13} - \)\(39\!\cdots\!54\)\( T^{14} - \)\(15\!\cdots\!02\)\( T^{15} - \)\(37\!\cdots\!19\)\( T^{16} - \)\(76\!\cdots\!12\)\( T^{17} - \)\(13\!\cdots\!24\)\( T^{18} - \)\(76\!\cdots\!12\)\( p^{6} T^{19} - \)\(37\!\cdots\!19\)\( p^{12} T^{20} - \)\(15\!\cdots\!02\)\( p^{18} T^{21} - \)\(39\!\cdots\!54\)\( p^{24} T^{22} - \)\(22\!\cdots\!30\)\( p^{30} T^{23} - \)\(12\!\cdots\!16\)\( p^{36} T^{24} - \)\(31\!\cdots\!42\)\( p^{42} T^{25} - \)\(44\!\cdots\!02\)\( p^{48} T^{26} + \)\(15\!\cdots\!82\)\( p^{54} T^{27} + \)\(11\!\cdots\!74\)\( p^{60} T^{28} + \)\(69\!\cdots\!34\)\( p^{66} T^{29} + \)\(51\!\cdots\!62\)\( p^{72} T^{30} + \)\(30\!\cdots\!86\)\( p^{78} T^{31} + 1164366851255681324 p^{84} T^{32} + 23269642035782 p^{90} T^{33} + 922351250 p^{96} T^{34} + 42950 p^{102} T^{35} + p^{108} T^{36} \) | |
| 31 | \( 1 + 17358 T + 150650082 T^{2} - 7720494579434 T^{3} + 114053813196225924 p T^{4} + \)\(85\!\cdots\!74\)\( T^{5} + \)\(97\!\cdots\!62\)\( T^{6} - \)\(12\!\cdots\!22\)\( T^{7} + \)\(55\!\cdots\!34\)\( T^{8} + \)\(18\!\cdots\!38\)\( T^{9} + \)\(26\!\cdots\!74\)\( T^{10} + \)\(27\!\cdots\!98\)\( T^{11} + \)\(50\!\cdots\!64\)\( T^{12} + \)\(23\!\cdots\!86\)\( T^{13} + \)\(41\!\cdots\!98\)\( T^{14} + \)\(10\!\cdots\!14\)\( T^{15} + \)\(35\!\cdots\!81\)\( T^{16} + \)\(17\!\cdots\!20\)\( T^{17} + \)\(40\!\cdots\!76\)\( T^{18} + \)\(17\!\cdots\!20\)\( p^{6} T^{19} + \)\(35\!\cdots\!81\)\( p^{12} T^{20} + \)\(10\!\cdots\!14\)\( p^{18} T^{21} + \)\(41\!\cdots\!98\)\( p^{24} T^{22} + \)\(23\!\cdots\!86\)\( p^{30} T^{23} + \)\(50\!\cdots\!64\)\( p^{36} T^{24} + \)\(27\!\cdots\!98\)\( p^{42} T^{25} + \)\(26\!\cdots\!74\)\( p^{48} T^{26} + \)\(18\!\cdots\!38\)\( p^{54} T^{27} + \)\(55\!\cdots\!34\)\( p^{60} T^{28} - \)\(12\!\cdots\!22\)\( p^{66} T^{29} + \)\(97\!\cdots\!62\)\( p^{72} T^{30} + \)\(85\!\cdots\!74\)\( p^{78} T^{31} + 114053813196225924 p^{85} T^{32} - 7720494579434 p^{90} T^{33} + 150650082 p^{96} T^{34} + 17358 p^{102} T^{35} + p^{108} T^{36} \) | |
| 41 | \( 1 - 59731767440 T^{2} + \)\(17\!\cdots\!94\)\( T^{4} - \)\(34\!\cdots\!04\)\( T^{6} + \)\(48\!\cdots\!79\)\( T^{8} - \)\(53\!\cdots\!74\)\( T^{10} + \)\(47\!\cdots\!94\)\( T^{12} - \)\(35\!\cdots\!48\)\( T^{14} + \)\(21\!\cdots\!17\)\( T^{16} - \)\(11\!\cdots\!38\)\( T^{18} + \)\(21\!\cdots\!17\)\( p^{12} T^{20} - \)\(35\!\cdots\!48\)\( p^{24} T^{22} + \)\(47\!\cdots\!94\)\( p^{36} T^{24} - \)\(53\!\cdots\!74\)\( p^{48} T^{26} + \)\(48\!\cdots\!79\)\( p^{60} T^{28} - \)\(34\!\cdots\!04\)\( p^{72} T^{30} + \)\(17\!\cdots\!94\)\( p^{84} T^{32} - 59731767440 p^{96} T^{34} + p^{108} T^{36} \) | |
| 43 | \( 1 + 65470 T + 2143160450 T^{2} + 881091433212118 T^{3} + \)\(16\!\cdots\!49\)\( T^{4} + \)\(70\!\cdots\!36\)\( T^{5} + \)\(49\!\cdots\!32\)\( T^{6} + \)\(11\!\cdots\!68\)\( T^{7} + \)\(13\!\cdots\!36\)\( T^{8} + \)\(62\!\cdots\!36\)\( T^{9} + \)\(54\!\cdots\!24\)\( T^{10} + \)\(83\!\cdots\!36\)\( T^{11} + \)\(76\!\cdots\!24\)\( T^{12} + \)\(42\!\cdots\!76\)\( T^{13} + \)\(37\!\cdots\!48\)\( T^{14} + \)\(41\!\cdots\!12\)\( T^{15} + \)\(34\!\cdots\!34\)\( T^{16} + \)\(23\!\cdots\!32\)\( T^{17} + \)\(17\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!32\)\( p^{6} T^{19} + \)\(34\!\cdots\!34\)\( p^{12} T^{20} + \)\(41\!\cdots\!12\)\( p^{18} T^{21} + \)\(37\!\cdots\!48\)\( p^{24} T^{22} + \)\(42\!\cdots\!76\)\( p^{30} T^{23} + \)\(76\!\cdots\!24\)\( p^{36} T^{24} + \)\(83\!\cdots\!36\)\( p^{42} T^{25} + \)\(54\!\cdots\!24\)\( p^{48} T^{26} + \)\(62\!\cdots\!36\)\( p^{54} T^{27} + \)\(13\!\cdots\!36\)\( p^{60} T^{28} + \)\(11\!\cdots\!68\)\( p^{66} T^{29} + \)\(49\!\cdots\!32\)\( p^{72} T^{30} + \)\(70\!\cdots\!36\)\( p^{78} T^{31} + \)\(16\!\cdots\!49\)\( p^{84} T^{32} + 881091433212118 p^{90} T^{33} + 2143160450 p^{96} T^{34} + 65470 p^{102} T^{35} + p^{108} T^{36} \) | |
| 47 | \( ( 1 - 116096 T + 74480860225 T^{2} - 8957100232100350 T^{3} + \)\(26\!\cdots\!31\)\( T^{4} - \)\(30\!\cdots\!74\)\( T^{5} + \)\(12\!\cdots\!61\)\( p T^{6} - \)\(62\!\cdots\!90\)\( T^{7} + \)\(92\!\cdots\!76\)\( T^{8} - \)\(82\!\cdots\!44\)\( T^{9} + \)\(92\!\cdots\!76\)\( p^{6} T^{10} - \)\(62\!\cdots\!90\)\( p^{12} T^{11} + \)\(12\!\cdots\!61\)\( p^{19} T^{12} - \)\(30\!\cdots\!74\)\( p^{24} T^{13} + \)\(26\!\cdots\!31\)\( p^{30} T^{14} - 8957100232100350 p^{36} T^{15} + 74480860225 p^{42} T^{16} - 116096 p^{48} T^{17} + p^{54} T^{18} )^{2} \) | |
| 53 | \( ( 1 - 24986 T + 133008808329 T^{2} - 1781531755894292 T^{3} + \)\(79\!\cdots\!25\)\( T^{4} + \)\(21\!\cdots\!46\)\( T^{5} + \)\(29\!\cdots\!43\)\( T^{6} + \)\(54\!\cdots\!08\)\( T^{7} + \)\(80\!\cdots\!86\)\( T^{8} + \)\(36\!\cdots\!64\)\( p T^{9} + \)\(80\!\cdots\!86\)\( p^{6} T^{10} + \)\(54\!\cdots\!08\)\( p^{12} T^{11} + \)\(29\!\cdots\!43\)\( p^{18} T^{12} + \)\(21\!\cdots\!46\)\( p^{24} T^{13} + \)\(79\!\cdots\!25\)\( p^{30} T^{14} - 1781531755894292 p^{36} T^{15} + 133008808329 p^{42} T^{16} - 24986 p^{48} T^{17} + p^{54} T^{18} )^{2} \) | |
| 59 | \( 1 + 181570 T + 16483832450 T^{2} - 8930874840017366 T^{3} - \)\(42\!\cdots\!87\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{5} + \)\(89\!\cdots\!28\)\( T^{6} + \)\(41\!\cdots\!52\)\( T^{7} - \)\(30\!\cdots\!92\)\( T^{8} - \)\(22\!\cdots\!40\)\( T^{9} - \)\(39\!\cdots\!96\)\( T^{10} - \)\(18\!\cdots\!96\)\( T^{11} + \)\(23\!\cdots\!44\)\( T^{12} + \)\(29\!\cdots\!08\)\( T^{13} - \)\(19\!\cdots\!28\)\( T^{14} - \)\(19\!\cdots\!96\)\( T^{15} - \)\(27\!\cdots\!70\)\( T^{16} + \)\(69\!\cdots\!80\)\( T^{17} + \)\(18\!\cdots\!24\)\( T^{18} + \)\(69\!\cdots\!80\)\( p^{6} T^{19} - \)\(27\!\cdots\!70\)\( p^{12} T^{20} - \)\(19\!\cdots\!96\)\( p^{18} T^{21} - \)\(19\!\cdots\!28\)\( p^{24} T^{22} + \)\(29\!\cdots\!08\)\( p^{30} T^{23} + \)\(23\!\cdots\!44\)\( p^{36} T^{24} - \)\(18\!\cdots\!96\)\( p^{42} T^{25} - \)\(39\!\cdots\!96\)\( p^{48} T^{26} - \)\(22\!\cdots\!40\)\( p^{54} T^{27} - \)\(30\!\cdots\!92\)\( p^{60} T^{28} + \)\(41\!\cdots\!52\)\( p^{66} T^{29} + \)\(89\!\cdots\!28\)\( p^{72} T^{30} - \)\(11\!\cdots\!40\)\( p^{78} T^{31} - \)\(42\!\cdots\!87\)\( p^{84} T^{32} - 8930874840017366 p^{90} T^{33} + 16483832450 p^{96} T^{34} + 181570 p^{102} T^{35} + p^{108} T^{36} \) | |
| 61 | \( 1 - 508802 T + 129439737602 T^{2} - 6011458845022482 T^{3} + \)\(19\!\cdots\!44\)\( T^{4} - \)\(21\!\cdots\!94\)\( T^{5} + \)\(10\!\cdots\!62\)\( T^{6} - \)\(24\!\cdots\!06\)\( T^{7} + \)\(33\!\cdots\!70\)\( T^{8} - \)\(70\!\cdots\!58\)\( T^{9} + \)\(28\!\cdots\!58\)\( T^{10} - \)\(84\!\cdots\!98\)\( T^{11} + \)\(23\!\cdots\!80\)\( T^{12} - \)\(59\!\cdots\!58\)\( T^{13} + \)\(13\!\cdots\!70\)\( T^{14} - \)\(20\!\cdots\!98\)\( T^{15} + \)\(45\!\cdots\!09\)\( T^{16} - \)\(17\!\cdots\!72\)\( T^{17} + \)\(51\!\cdots\!84\)\( T^{18} - \)\(17\!\cdots\!72\)\( p^{6} T^{19} + \)\(45\!\cdots\!09\)\( p^{12} T^{20} - \)\(20\!\cdots\!98\)\( p^{18} T^{21} + \)\(13\!\cdots\!70\)\( p^{24} T^{22} - \)\(59\!\cdots\!58\)\( p^{30} T^{23} + \)\(23\!\cdots\!80\)\( p^{36} T^{24} - \)\(84\!\cdots\!98\)\( p^{42} T^{25} + \)\(28\!\cdots\!58\)\( p^{48} T^{26} - \)\(70\!\cdots\!58\)\( p^{54} T^{27} + \)\(33\!\cdots\!70\)\( p^{60} T^{28} - \)\(24\!\cdots\!06\)\( p^{66} T^{29} + \)\(10\!\cdots\!62\)\( p^{72} T^{30} - \)\(21\!\cdots\!94\)\( p^{78} T^{31} + \)\(19\!\cdots\!44\)\( p^{84} T^{32} - 6011458845022482 p^{90} T^{33} + 129439737602 p^{96} T^{34} - 508802 p^{102} T^{35} + p^{108} T^{36} \) | |
| 67 | \( 1 - 638545373784 T^{2} + \)\(22\!\cdots\!20\)\( T^{4} - \)\(53\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(15\!\cdots\!26\)\( T^{10} + \)\(20\!\cdots\!04\)\( T^{12} - \)\(24\!\cdots\!08\)\( T^{14} + \)\(25\!\cdots\!25\)\( T^{16} - \)\(24\!\cdots\!08\)\( T^{18} + \)\(25\!\cdots\!25\)\( p^{12} T^{20} - \)\(24\!\cdots\!08\)\( p^{24} T^{22} + \)\(20\!\cdots\!04\)\( p^{36} T^{24} - \)\(15\!\cdots\!26\)\( p^{48} T^{26} + \)\(10\!\cdots\!70\)\( p^{60} T^{28} - \)\(53\!\cdots\!28\)\( p^{72} T^{30} + \)\(22\!\cdots\!20\)\( p^{84} T^{32} - 638545373784 p^{96} T^{34} + p^{108} T^{36} \) | |
| 71 | \( ( 1 + 101316 T + 591572964957 T^{2} + 16715640927077158 T^{3} + 37167728565961684527 p^{2} T^{4} + \)\(14\!\cdots\!58\)\( T^{5} + \)\(41\!\cdots\!15\)\( T^{6} - \)\(38\!\cdots\!18\)\( T^{7} + \)\(69\!\cdots\!88\)\( T^{8} - \)\(81\!\cdots\!24\)\( T^{9} + \)\(69\!\cdots\!88\)\( p^{6} T^{10} - \)\(38\!\cdots\!18\)\( p^{12} T^{11} + \)\(41\!\cdots\!15\)\( p^{18} T^{12} + \)\(14\!\cdots\!58\)\( p^{24} T^{13} + 37167728565961684527 p^{32} T^{14} + 16715640927077158 p^{36} T^{15} + 591572964957 p^{42} T^{16} + 101316 p^{48} T^{17} + p^{54} T^{18} )^{2} \) | |
| 73 | \( 1 - 696350545488 T^{2} + \)\(25\!\cdots\!94\)\( T^{4} - \)\(71\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!07\)\( T^{8} - \)\(41\!\cdots\!46\)\( T^{10} + \)\(82\!\cdots\!26\)\( T^{12} - \)\(15\!\cdots\!52\)\( T^{14} + \)\(25\!\cdots\!77\)\( T^{16} - \)\(40\!\cdots\!54\)\( T^{18} + \)\(25\!\cdots\!77\)\( p^{12} T^{20} - \)\(15\!\cdots\!52\)\( p^{24} T^{22} + \)\(82\!\cdots\!26\)\( p^{36} T^{24} - \)\(41\!\cdots\!46\)\( p^{48} T^{26} + \)\(18\!\cdots\!07\)\( p^{60} T^{28} - \)\(71\!\cdots\!72\)\( p^{72} T^{30} + \)\(25\!\cdots\!94\)\( p^{84} T^{32} - 696350545488 p^{96} T^{34} + p^{108} T^{36} \) | |
| 79 | \( 1 - 1752858 T + 1536255584082 T^{2} - 1048805682786333074 T^{3} + \)\(64\!\cdots\!12\)\( T^{4} - \)\(27\!\cdots\!98\)\( T^{5} + \)\(44\!\cdots\!38\)\( T^{6} + \)\(30\!\cdots\!62\)\( T^{7} - \)\(36\!\cdots\!62\)\( T^{8} + \)\(24\!\cdots\!82\)\( T^{9} - \)\(10\!\cdots\!14\)\( T^{10} + \)\(17\!\cdots\!62\)\( T^{11} + \)\(13\!\cdots\!60\)\( T^{12} - \)\(13\!\cdots\!90\)\( T^{13} + \)\(81\!\cdots\!22\)\( T^{14} - \)\(32\!\cdots\!90\)\( T^{15} + \)\(60\!\cdots\!25\)\( T^{16} + \)\(18\!\cdots\!36\)\( T^{17} - \)\(17\!\cdots\!64\)\( T^{18} + \)\(18\!\cdots\!36\)\( p^{6} T^{19} + \)\(60\!\cdots\!25\)\( p^{12} T^{20} - \)\(32\!\cdots\!90\)\( p^{18} T^{21} + \)\(81\!\cdots\!22\)\( p^{24} T^{22} - \)\(13\!\cdots\!90\)\( p^{30} T^{23} + \)\(13\!\cdots\!60\)\( p^{36} T^{24} + \)\(17\!\cdots\!62\)\( p^{42} T^{25} - \)\(10\!\cdots\!14\)\( p^{48} T^{26} + \)\(24\!\cdots\!82\)\( p^{54} T^{27} - \)\(36\!\cdots\!62\)\( p^{60} T^{28} + \)\(30\!\cdots\!62\)\( p^{66} T^{29} + \)\(44\!\cdots\!38\)\( p^{72} T^{30} - \)\(27\!\cdots\!98\)\( p^{78} T^{31} + \)\(64\!\cdots\!12\)\( p^{84} T^{32} - 1048805682786333074 p^{90} T^{33} + 1536255584082 p^{96} T^{34} - 1752858 p^{102} T^{35} + p^{108} T^{36} \) | |
| 83 | \( ( 1 - 1185808 T + 1362780195477 T^{2} - 810769303344198742 T^{3} + \)\(55\!\cdots\!11\)\( T^{4} - \)\(26\!\cdots\!46\)\( T^{5} + \)\(21\!\cdots\!99\)\( T^{6} - \)\(13\!\cdots\!34\)\( T^{7} + \)\(10\!\cdots\!80\)\( T^{8} - \)\(59\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!80\)\( p^{6} T^{10} - \)\(13\!\cdots\!34\)\( p^{12} T^{11} + \)\(21\!\cdots\!99\)\( p^{18} T^{12} - \)\(26\!\cdots\!46\)\( p^{24} T^{13} + \)\(55\!\cdots\!11\)\( p^{30} T^{14} - 810769303344198742 p^{36} T^{15} + 1362780195477 p^{42} T^{16} - 1185808 p^{48} T^{17} + p^{54} T^{18} )^{2} \) | |
| 89 | \( 1 - 1148346 T + 659349267858 T^{2} - 351361554154272154 T^{3} - \)\(30\!\cdots\!03\)\( T^{4} + \)\(28\!\cdots\!84\)\( T^{5} - \)\(57\!\cdots\!32\)\( T^{6} - \)\(10\!\cdots\!88\)\( T^{7} + \)\(26\!\cdots\!00\)\( T^{8} - \)\(90\!\cdots\!68\)\( T^{9} - \)\(91\!\cdots\!88\)\( T^{10} + \)\(66\!\cdots\!32\)\( T^{11} - \)\(65\!\cdots\!56\)\( T^{12} - \)\(65\!\cdots\!88\)\( T^{13} + \)\(20\!\cdots\!64\)\( T^{14} - \)\(25\!\cdots\!44\)\( T^{15} + \)\(60\!\cdots\!02\)\( T^{16} + \)\(11\!\cdots\!60\)\( T^{17} - \)\(82\!\cdots\!96\)\( T^{18} + \)\(11\!\cdots\!60\)\( p^{6} T^{19} + \)\(60\!\cdots\!02\)\( p^{12} T^{20} - \)\(25\!\cdots\!44\)\( p^{18} T^{21} + \)\(20\!\cdots\!64\)\( p^{24} T^{22} - \)\(65\!\cdots\!88\)\( p^{30} T^{23} - \)\(65\!\cdots\!56\)\( p^{36} T^{24} + \)\(66\!\cdots\!32\)\( p^{42} T^{25} - \)\(91\!\cdots\!88\)\( p^{48} T^{26} - \)\(90\!\cdots\!68\)\( p^{54} T^{27} + \)\(26\!\cdots\!00\)\( p^{60} T^{28} - \)\(10\!\cdots\!88\)\( p^{66} T^{29} - \)\(57\!\cdots\!32\)\( p^{72} T^{30} + \)\(28\!\cdots\!84\)\( p^{78} T^{31} - \)\(30\!\cdots\!03\)\( p^{84} T^{32} - 351361554154272154 p^{90} T^{33} + 659349267858 p^{96} T^{34} - 1148346 p^{102} T^{35} + p^{108} T^{36} \) | |
| 97 | \( 1 + 1670270 T + 1394900936450 T^{2} + 3213101342427685086 T^{3} + \)\(46\!\cdots\!73\)\( T^{4} + \)\(30\!\cdots\!84\)\( p T^{5} + \)\(35\!\cdots\!08\)\( T^{6} + \)\(54\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!44\)\( T^{8} + \)\(25\!\cdots\!28\)\( T^{9} + \)\(45\!\cdots\!08\)\( T^{10} + \)\(43\!\cdots\!76\)\( T^{11} + \)\(29\!\cdots\!28\)\( T^{12} + \)\(36\!\cdots\!60\)\( T^{13} + \)\(44\!\cdots\!44\)\( T^{14} + \)\(36\!\cdots\!48\)\( T^{15} + \)\(32\!\cdots\!18\)\( T^{16} + \)\(34\!\cdots\!48\)\( T^{17} + \)\(34\!\cdots\!52\)\( T^{18} + \)\(34\!\cdots\!48\)\( p^{6} T^{19} + \)\(32\!\cdots\!18\)\( p^{12} T^{20} + \)\(36\!\cdots\!48\)\( p^{18} T^{21} + \)\(44\!\cdots\!44\)\( p^{24} T^{22} + \)\(36\!\cdots\!60\)\( p^{30} T^{23} + \)\(29\!\cdots\!28\)\( p^{36} T^{24} + \)\(43\!\cdots\!76\)\( p^{42} T^{25} + \)\(45\!\cdots\!08\)\( p^{48} T^{26} + \)\(25\!\cdots\!28\)\( p^{54} T^{27} + \)\(35\!\cdots\!44\)\( p^{60} T^{28} + \)\(54\!\cdots\!80\)\( p^{66} T^{29} + \)\(35\!\cdots\!08\)\( p^{72} T^{30} + \)\(30\!\cdots\!84\)\( p^{79} T^{31} + \)\(46\!\cdots\!73\)\( p^{84} T^{32} + 3213101342427685086 p^{90} T^{33} + 1394900936450 p^{96} T^{34} + 1670270 p^{102} T^{35} + p^{108} T^{36} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.00073319057376862599507312658, −2.88969392077852864842538052875, −2.79411145770614489533780226415, −2.64036971669343740672631818299, −2.32103198070478924206741740524, −2.31950466953947370639913483703, −2.23950364147448680877445278994, −2.15145677302847185075150745519, −2.02127728128792558252500134353, −2.00681735795225927461007795011, −1.95733330728237873588503750478, −1.95227613070434604589527660790, −1.70654768934153231404873905307, −1.69403734275696581328741553371, −1.57885254323010488374371621720, −1.54191338570876468481560588825, −1.43351008086663334632076110251, −1.14339753065479092802852044211, −0.975399552779322818693552720406, −0.808157228971709554639971111862, −0.60099164986997014191544515719, −0.56910953192644054211298579341, −0.29762853732185441274430111212, −0.11294397825552101396264758071, −0.01681301371680465160136086857, 0.01681301371680465160136086857, 0.11294397825552101396264758071, 0.29762853732185441274430111212, 0.56910953192644054211298579341, 0.60099164986997014191544515719, 0.808157228971709554639971111862, 0.975399552779322818693552720406, 1.14339753065479092802852044211, 1.43351008086663334632076110251, 1.54191338570876468481560588825, 1.57885254323010488374371621720, 1.69403734275696581328741553371, 1.70654768934153231404873905307, 1.95227613070434604589527660790, 1.95733330728237873588503750478, 2.00681735795225927461007795011, 2.02127728128792558252500134353, 2.15145677302847185075150745519, 2.23950364147448680877445278994, 2.31950466953947370639913483703, 2.32103198070478924206741740524, 2.64036971669343740672631818299, 2.79411145770614489533780226415, 2.88969392077852864842538052875, 3.00073319057376862599507312658
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.