Dirichlet series
| L(s) = 1 | − 285·5-s + 198·7-s + 1.79e4·11-s + 2.05e5·17-s + 7.43e4·19-s + 6.28e4·23-s − 6.91e5·25-s + 5.75e5·29-s + 1.44e6·31-s − 5.64e4·35-s − 2.05e6·37-s + 7.72e6·43-s − 1.20e7·47-s − 8.46e6·49-s + 5.50e6·53-s − 5.10e6·55-s − 7.51e6·59-s − 3.72e7·61-s − 3.68e7·67-s + 3.00e7·71-s + 9.50e7·73-s + 3.54e6·77-s + 8.51e6·79-s − 5.86e7·85-s − 8.30e7·89-s − 2.11e7·95-s − 6.72e8·101-s + ⋯ |
| L(s) = 1 | − 0.455·5-s + 0.0824·7-s + 1.22·11-s + 2.46·17-s + 0.570·19-s + 0.224·23-s − 1.77·25-s + 0.813·29-s + 1.56·31-s − 0.0376·35-s − 1.09·37-s + 2.25·43-s − 2.47·47-s − 1.46·49-s + 0.697·53-s − 0.558·55-s − 0.619·59-s − 2.68·61-s − 1.82·67-s + 1.18·71-s + 3.34·73-s + 0.100·77-s + 0.218·79-s − 1.12·85-s − 1.32·89-s − 0.260·95-s − 6.46·101-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+4)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.37020\times 10^{24}\) |
| Root analytic conductor: | \(10.1320\) |
| Motivic weight: | \(8\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [4]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{9}{2})\) | \(\approx\) | \(3.108273081\) |
| \(L(\frac12)\) | \(\approx\) | \(3.108273081\) |
| \(L(5)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 198 T + 8501949 T^{2} + 4349076514 p T^{3} + 168809648166 p^{3} T^{4} + 1246332270114 p^{6} T^{5} + 2613116425173 p^{10} T^{6} + 1246332270114 p^{14} T^{7} + 168809648166 p^{19} T^{8} + 4349076514 p^{25} T^{9} + 8501949 p^{32} T^{10} - 198 p^{40} T^{11} + p^{48} T^{12} \) | |
| good | 5 | \( 1 + 57 p T + 773148 T^{2} + 42526161 p T^{3} + 382895712834 T^{4} + 210666306118233 T^{5} + 84509030530123142 T^{6} + 5383721618415903483 p T^{7} - \)\(10\!\cdots\!46\)\( p^{2} T^{8} - \)\(32\!\cdots\!89\)\( p^{4} T^{9} - \)\(34\!\cdots\!96\)\( p^{4} T^{10} - \)\(15\!\cdots\!89\)\( p^{6} T^{11} - \)\(90\!\cdots\!46\)\( p^{6} T^{12} - \)\(15\!\cdots\!89\)\( p^{14} T^{13} - \)\(34\!\cdots\!96\)\( p^{20} T^{14} - \)\(32\!\cdots\!89\)\( p^{28} T^{15} - \)\(10\!\cdots\!46\)\( p^{34} T^{16} + 5383721618415903483 p^{41} T^{17} + 84509030530123142 p^{48} T^{18} + 210666306118233 p^{56} T^{19} + 382895712834 p^{64} T^{20} + 42526161 p^{73} T^{21} + 773148 p^{80} T^{22} + 57 p^{89} T^{23} + p^{96} T^{24} \) |
| 11 | \( 1 - 1629 p T - 228436530 T^{2} + 9086672730411 T^{3} - 91396129870216860 T^{4} - \)\(34\!\cdots\!27\)\( T^{5} + \)\(14\!\cdots\!18\)\( T^{6} - \)\(25\!\cdots\!13\)\( T^{7} + \)\(54\!\cdots\!48\)\( T^{8} - \)\(87\!\cdots\!79\)\( T^{9} + \)\(35\!\cdots\!14\)\( T^{10} + \)\(22\!\cdots\!59\)\( T^{11} - \)\(53\!\cdots\!22\)\( T^{12} + \)\(22\!\cdots\!59\)\( p^{8} T^{13} + \)\(35\!\cdots\!14\)\( p^{16} T^{14} - \)\(87\!\cdots\!79\)\( p^{24} T^{15} + \)\(54\!\cdots\!48\)\( p^{32} T^{16} - \)\(25\!\cdots\!13\)\( p^{40} T^{17} + \)\(14\!\cdots\!18\)\( p^{48} T^{18} - \)\(34\!\cdots\!27\)\( p^{56} T^{19} - 91396129870216860 p^{64} T^{20} + 9086672730411 p^{72} T^{21} - 228436530 p^{80} T^{22} - 1629 p^{89} T^{23} + p^{96} T^{24} \) | |
| 13 | \( 1 - 4680855171 T^{2} + 11303530637810821443 T^{4} - \)\(18\!\cdots\!56\)\( T^{6} + \)\(24\!\cdots\!01\)\( T^{8} - \)\(26\!\cdots\!25\)\( T^{10} + \)\(23\!\cdots\!98\)\( T^{12} - \)\(26\!\cdots\!25\)\( p^{16} T^{14} + \)\(24\!\cdots\!01\)\( p^{32} T^{16} - \)\(18\!\cdots\!56\)\( p^{48} T^{18} + 11303530637810821443 p^{64} T^{20} - 4680855171 p^{80} T^{22} + p^{96} T^{24} \) | |
| 17 | \( 1 - 205782 T + 32244830382 T^{2} - 3730708280511468 T^{3} + \)\(28\!\cdots\!73\)\( T^{4} - \)\(12\!\cdots\!92\)\( T^{5} - \)\(42\!\cdots\!50\)\( T^{6} + \)\(16\!\cdots\!58\)\( T^{7} - \)\(15\!\cdots\!06\)\( T^{8} + \)\(80\!\cdots\!30\)\( T^{9} + \)\(26\!\cdots\!82\)\( T^{10} - \)\(99\!\cdots\!60\)\( T^{11} + \)\(10\!\cdots\!69\)\( T^{12} - \)\(99\!\cdots\!60\)\( p^{8} T^{13} + \)\(26\!\cdots\!82\)\( p^{16} T^{14} + \)\(80\!\cdots\!30\)\( p^{24} T^{15} - \)\(15\!\cdots\!06\)\( p^{32} T^{16} + \)\(16\!\cdots\!58\)\( p^{40} T^{17} - \)\(42\!\cdots\!50\)\( p^{48} T^{18} - \)\(12\!\cdots\!92\)\( p^{56} T^{19} + \)\(28\!\cdots\!73\)\( p^{64} T^{20} - 3730708280511468 p^{72} T^{21} + 32244830382 p^{80} T^{22} - 205782 p^{88} T^{23} + p^{96} T^{24} \) | |
| 19 | \( 1 - 74313 T + 42805567800 T^{2} - 3044214245327301 T^{3} + \)\(64\!\cdots\!10\)\( T^{4} + \)\(13\!\cdots\!11\)\( T^{5} - \)\(20\!\cdots\!94\)\( T^{6} + \)\(16\!\cdots\!05\)\( T^{7} - \)\(13\!\cdots\!38\)\( T^{8} + \)\(11\!\cdots\!99\)\( p T^{9} + \)\(19\!\cdots\!48\)\( T^{10} - \)\(94\!\cdots\!83\)\( T^{11} + \)\(96\!\cdots\!66\)\( T^{12} - \)\(94\!\cdots\!83\)\( p^{8} T^{13} + \)\(19\!\cdots\!48\)\( p^{16} T^{14} + \)\(11\!\cdots\!99\)\( p^{25} T^{15} - \)\(13\!\cdots\!38\)\( p^{32} T^{16} + \)\(16\!\cdots\!05\)\( p^{40} T^{17} - \)\(20\!\cdots\!94\)\( p^{48} T^{18} + \)\(13\!\cdots\!11\)\( p^{56} T^{19} + \)\(64\!\cdots\!10\)\( p^{64} T^{20} - 3044214245327301 p^{72} T^{21} + 42805567800 p^{80} T^{22} - 74313 p^{88} T^{23} + p^{96} T^{24} \) | |
| 23 | \( 1 - 62832 T - 267130501926 T^{2} + 2408529172043712 T^{3} + \)\(34\!\cdots\!41\)\( T^{4} + \)\(78\!\cdots\!16\)\( T^{5} - \)\(30\!\cdots\!30\)\( T^{6} + \)\(53\!\cdots\!08\)\( T^{7} + \)\(24\!\cdots\!18\)\( T^{8} - \)\(18\!\cdots\!88\)\( T^{9} - \)\(23\!\cdots\!78\)\( T^{10} + \)\(86\!\cdots\!16\)\( T^{11} + \)\(21\!\cdots\!93\)\( T^{12} + \)\(86\!\cdots\!16\)\( p^{8} T^{13} - \)\(23\!\cdots\!78\)\( p^{16} T^{14} - \)\(18\!\cdots\!88\)\( p^{24} T^{15} + \)\(24\!\cdots\!18\)\( p^{32} T^{16} + \)\(53\!\cdots\!08\)\( p^{40} T^{17} - \)\(30\!\cdots\!30\)\( p^{48} T^{18} + \)\(78\!\cdots\!16\)\( p^{56} T^{19} + \)\(34\!\cdots\!41\)\( p^{64} T^{20} + 2408529172043712 p^{72} T^{21} - 267130501926 p^{80} T^{22} - 62832 p^{88} T^{23} + p^{96} T^{24} \) | |
| 29 | \( ( 1 - 287727 T + 1872284339769 T^{2} - 775009483994138304 T^{3} + \)\(17\!\cdots\!47\)\( T^{4} - \)\(75\!\cdots\!97\)\( T^{5} + \)\(10\!\cdots\!98\)\( T^{6} - \)\(75\!\cdots\!97\)\( p^{8} T^{7} + \)\(17\!\cdots\!47\)\( p^{16} T^{8} - 775009483994138304 p^{24} T^{9} + 1872284339769 p^{32} T^{10} - 287727 p^{40} T^{11} + p^{48} T^{12} )^{2} \) | |
| 31 | \( 1 - 1442952 T + 2753998863693 T^{2} - 2972426343990906600 T^{3} + \)\(45\!\cdots\!47\)\( T^{4} - \)\(43\!\cdots\!68\)\( T^{5} + \)\(45\!\cdots\!68\)\( T^{6} - \)\(38\!\cdots\!56\)\( T^{7} + \)\(33\!\cdots\!61\)\( T^{8} - \)\(81\!\cdots\!00\)\( p T^{9} + \)\(14\!\cdots\!59\)\( T^{10} - \)\(44\!\cdots\!44\)\( p T^{11} + \)\(91\!\cdots\!98\)\( T^{12} - \)\(44\!\cdots\!44\)\( p^{9} T^{13} + \)\(14\!\cdots\!59\)\( p^{16} T^{14} - \)\(81\!\cdots\!00\)\( p^{25} T^{15} + \)\(33\!\cdots\!61\)\( p^{32} T^{16} - \)\(38\!\cdots\!56\)\( p^{40} T^{17} + \)\(45\!\cdots\!68\)\( p^{48} T^{18} - \)\(43\!\cdots\!68\)\( p^{56} T^{19} + \)\(45\!\cdots\!47\)\( p^{64} T^{20} - 2972426343990906600 p^{72} T^{21} + 2753998863693 p^{80} T^{22} - 1442952 p^{88} T^{23} + p^{96} T^{24} \) | |
| 37 | \( 1 + 2058621 T - 12851151325950 T^{2} - 24300204894085231549 T^{3} + \)\(96\!\cdots\!96\)\( T^{4} + \)\(14\!\cdots\!17\)\( T^{5} - \)\(58\!\cdots\!58\)\( T^{6} - \)\(55\!\cdots\!85\)\( T^{7} + \)\(31\!\cdots\!92\)\( T^{8} + \)\(14\!\cdots\!77\)\( T^{9} - \)\(14\!\cdots\!02\)\( T^{10} - \)\(19\!\cdots\!77\)\( T^{11} + \)\(54\!\cdots\!90\)\( T^{12} - \)\(19\!\cdots\!77\)\( p^{8} T^{13} - \)\(14\!\cdots\!02\)\( p^{16} T^{14} + \)\(14\!\cdots\!77\)\( p^{24} T^{15} + \)\(31\!\cdots\!92\)\( p^{32} T^{16} - \)\(55\!\cdots\!85\)\( p^{40} T^{17} - \)\(58\!\cdots\!58\)\( p^{48} T^{18} + \)\(14\!\cdots\!17\)\( p^{56} T^{19} + \)\(96\!\cdots\!96\)\( p^{64} T^{20} - 24300204894085231549 p^{72} T^{21} - 12851151325950 p^{80} T^{22} + 2058621 p^{88} T^{23} + p^{96} T^{24} \) | |
| 41 | \( 1 - 51768779647524 T^{2} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(14\!\cdots\!44\)\( T^{6} + \)\(95\!\cdots\!15\)\( T^{8} - \)\(23\!\cdots\!32\)\( T^{10} - \)\(50\!\cdots\!56\)\( T^{12} - \)\(23\!\cdots\!32\)\( p^{16} T^{14} + \)\(95\!\cdots\!15\)\( p^{32} T^{16} - \)\(14\!\cdots\!44\)\( p^{48} T^{18} + \)\(11\!\cdots\!34\)\( p^{64} T^{20} - 51768779647524 p^{80} T^{22} + p^{96} T^{24} \) | |
| 43 | \( ( 1 - 3860661 T + 41440561775907 T^{2} - \)\(15\!\cdots\!68\)\( T^{3} + \)\(92\!\cdots\!09\)\( T^{4} - \)\(28\!\cdots\!79\)\( T^{5} + \)\(13\!\cdots\!06\)\( T^{6} - \)\(28\!\cdots\!79\)\( p^{8} T^{7} + \)\(92\!\cdots\!09\)\( p^{16} T^{8} - \)\(15\!\cdots\!68\)\( p^{24} T^{9} + 41440561775907 p^{32} T^{10} - 3860661 p^{40} T^{11} + p^{48} T^{12} )^{2} \) | |
| 47 | \( 1 + 12088194 T + 98045514715926 T^{2} + \)\(59\!\cdots\!16\)\( T^{3} + \)\(26\!\cdots\!53\)\( T^{4} + \)\(10\!\cdots\!44\)\( T^{5} + \)\(28\!\cdots\!58\)\( T^{6} + \)\(14\!\cdots\!62\)\( T^{7} + \)\(11\!\cdots\!78\)\( T^{8} + \)\(80\!\cdots\!42\)\( T^{9} + \)\(49\!\cdots\!74\)\( T^{10} + \)\(26\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!13\)\( T^{12} + \)\(26\!\cdots\!56\)\( p^{8} T^{13} + \)\(49\!\cdots\!74\)\( p^{16} T^{14} + \)\(80\!\cdots\!42\)\( p^{24} T^{15} + \)\(11\!\cdots\!78\)\( p^{32} T^{16} + \)\(14\!\cdots\!62\)\( p^{40} T^{17} + \)\(28\!\cdots\!58\)\( p^{48} T^{18} + \)\(10\!\cdots\!44\)\( p^{56} T^{19} + \)\(26\!\cdots\!53\)\( p^{64} T^{20} + \)\(59\!\cdots\!16\)\( p^{72} T^{21} + 98045514715926 p^{80} T^{22} + 12088194 p^{88} T^{23} + p^{96} T^{24} \) | |
| 53 | \( 1 - 5506743 T - 157455075543936 T^{2} + \)\(13\!\cdots\!05\)\( T^{3} + \)\(90\!\cdots\!54\)\( T^{4} - \)\(26\!\cdots\!59\)\( p T^{5} - \)\(12\!\cdots\!18\)\( T^{6} + \)\(10\!\cdots\!63\)\( T^{7} - \)\(26\!\cdots\!82\)\( T^{8} - \)\(55\!\cdots\!41\)\( T^{9} + \)\(34\!\cdots\!56\)\( T^{10} + \)\(14\!\cdots\!19\)\( T^{11} - \)\(26\!\cdots\!58\)\( T^{12} + \)\(14\!\cdots\!19\)\( p^{8} T^{13} + \)\(34\!\cdots\!56\)\( p^{16} T^{14} - \)\(55\!\cdots\!41\)\( p^{24} T^{15} - \)\(26\!\cdots\!82\)\( p^{32} T^{16} + \)\(10\!\cdots\!63\)\( p^{40} T^{17} - \)\(12\!\cdots\!18\)\( p^{48} T^{18} - \)\(26\!\cdots\!59\)\( p^{57} T^{19} + \)\(90\!\cdots\!54\)\( p^{64} T^{20} + \)\(13\!\cdots\!05\)\( p^{72} T^{21} - 157455075543936 p^{80} T^{22} - 5506743 p^{88} T^{23} + p^{96} T^{24} \) | |
| 59 | \( 1 + 7511901 T + 219421003168794 T^{2} + \)\(15\!\cdots\!27\)\( T^{3} + \)\(11\!\cdots\!40\)\( T^{4} + \)\(22\!\cdots\!01\)\( T^{5} + \)\(33\!\cdots\!86\)\( T^{6} + \)\(21\!\cdots\!59\)\( T^{7} + \)\(96\!\cdots\!08\)\( T^{8} - \)\(51\!\cdots\!75\)\( T^{9} - \)\(12\!\cdots\!74\)\( T^{10} - \)\(51\!\cdots\!05\)\( T^{11} - \)\(15\!\cdots\!70\)\( T^{12} - \)\(51\!\cdots\!05\)\( p^{8} T^{13} - \)\(12\!\cdots\!74\)\( p^{16} T^{14} - \)\(51\!\cdots\!75\)\( p^{24} T^{15} + \)\(96\!\cdots\!08\)\( p^{32} T^{16} + \)\(21\!\cdots\!59\)\( p^{40} T^{17} + \)\(33\!\cdots\!86\)\( p^{48} T^{18} + \)\(22\!\cdots\!01\)\( p^{56} T^{19} + \)\(11\!\cdots\!40\)\( p^{64} T^{20} + \)\(15\!\cdots\!27\)\( p^{72} T^{21} + 219421003168794 p^{80} T^{22} + 7511901 p^{88} T^{23} + p^{96} T^{24} \) | |
| 61 | \( 1 + 37215576 T + 801351511399062 T^{2} + \)\(12\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!93\)\( T^{4} + \)\(13\!\cdots\!60\)\( p T^{5} - \)\(12\!\cdots\!50\)\( T^{6} - \)\(38\!\cdots\!92\)\( T^{7} - \)\(59\!\cdots\!10\)\( T^{8} - \)\(64\!\cdots\!36\)\( T^{9} - \)\(41\!\cdots\!10\)\( T^{10} + \)\(10\!\cdots\!04\)\( T^{11} + \)\(74\!\cdots\!73\)\( T^{12} + \)\(10\!\cdots\!04\)\( p^{8} T^{13} - \)\(41\!\cdots\!10\)\( p^{16} T^{14} - \)\(64\!\cdots\!36\)\( p^{24} T^{15} - \)\(59\!\cdots\!10\)\( p^{32} T^{16} - \)\(38\!\cdots\!92\)\( p^{40} T^{17} - \)\(12\!\cdots\!50\)\( p^{48} T^{18} + \)\(13\!\cdots\!60\)\( p^{57} T^{19} + \)\(13\!\cdots\!93\)\( p^{64} T^{20} + \)\(12\!\cdots\!20\)\( p^{72} T^{21} + 801351511399062 p^{80} T^{22} + 37215576 p^{88} T^{23} + p^{96} T^{24} \) | |
| 67 | \( 1 + 36824553 T + 235041725822940 T^{2} + \)\(20\!\cdots\!73\)\( T^{3} + \)\(97\!\cdots\!62\)\( T^{4} + \)\(96\!\cdots\!49\)\( T^{5} + \)\(10\!\cdots\!34\)\( T^{6} + \)\(80\!\cdots\!19\)\( T^{7} + \)\(10\!\cdots\!94\)\( T^{8} - \)\(43\!\cdots\!05\)\( T^{9} + \)\(18\!\cdots\!76\)\( T^{10} + \)\(28\!\cdots\!67\)\( T^{11} - \)\(88\!\cdots\!26\)\( T^{12} + \)\(28\!\cdots\!67\)\( p^{8} T^{13} + \)\(18\!\cdots\!76\)\( p^{16} T^{14} - \)\(43\!\cdots\!05\)\( p^{24} T^{15} + \)\(10\!\cdots\!94\)\( p^{32} T^{16} + \)\(80\!\cdots\!19\)\( p^{40} T^{17} + \)\(10\!\cdots\!34\)\( p^{48} T^{18} + \)\(96\!\cdots\!49\)\( p^{56} T^{19} + \)\(97\!\cdots\!62\)\( p^{64} T^{20} + \)\(20\!\cdots\!73\)\( p^{72} T^{21} + 235041725822940 p^{80} T^{22} + 36824553 p^{88} T^{23} + p^{96} T^{24} \) | |
| 71 | \( ( 1 - 15005778 T + 1741360781651802 T^{2} - \)\(18\!\cdots\!26\)\( T^{3} + \)\(15\!\cdots\!35\)\( T^{4} - \)\(67\!\cdots\!92\)\( T^{5} + \)\(99\!\cdots\!28\)\( T^{6} - \)\(67\!\cdots\!92\)\( p^{8} T^{7} + \)\(15\!\cdots\!35\)\( p^{16} T^{8} - \)\(18\!\cdots\!26\)\( p^{24} T^{9} + 1741360781651802 p^{32} T^{10} - 15005778 p^{40} T^{11} + p^{48} T^{12} )^{2} \) | |
| 73 | \( 1 - 95080185 T + 7788143704773942 T^{2} - \)\(45\!\cdots\!95\)\( T^{3} + \)\(23\!\cdots\!24\)\( T^{4} - \)\(10\!\cdots\!97\)\( T^{5} + \)\(44\!\cdots\!82\)\( T^{6} - \)\(16\!\cdots\!31\)\( T^{7} + \)\(60\!\cdots\!68\)\( T^{8} - \)\(20\!\cdots\!37\)\( T^{9} + \)\(66\!\cdots\!10\)\( T^{10} - \)\(20\!\cdots\!11\)\( T^{11} + \)\(58\!\cdots\!98\)\( T^{12} - \)\(20\!\cdots\!11\)\( p^{8} T^{13} + \)\(66\!\cdots\!10\)\( p^{16} T^{14} - \)\(20\!\cdots\!37\)\( p^{24} T^{15} + \)\(60\!\cdots\!68\)\( p^{32} T^{16} - \)\(16\!\cdots\!31\)\( p^{40} T^{17} + \)\(44\!\cdots\!82\)\( p^{48} T^{18} - \)\(10\!\cdots\!97\)\( p^{56} T^{19} + \)\(23\!\cdots\!24\)\( p^{64} T^{20} - \)\(45\!\cdots\!95\)\( p^{72} T^{21} + 7788143704773942 p^{80} T^{22} - 95080185 p^{88} T^{23} + p^{96} T^{24} \) | |
| 79 | \( 1 - 8514456 T - 5984367856004523 T^{2} + \)\(11\!\cdots\!72\)\( p T^{3} + \)\(17\!\cdots\!63\)\( T^{4} - \)\(37\!\cdots\!36\)\( p T^{5} - \)\(38\!\cdots\!80\)\( T^{6} + \)\(40\!\cdots\!48\)\( T^{7} + \)\(75\!\cdots\!69\)\( T^{8} - \)\(16\!\cdots\!28\)\( T^{9} - \)\(14\!\cdots\!53\)\( T^{10} - \)\(47\!\cdots\!80\)\( T^{11} + \)\(25\!\cdots\!46\)\( T^{12} - \)\(47\!\cdots\!80\)\( p^{8} T^{13} - \)\(14\!\cdots\!53\)\( p^{16} T^{14} - \)\(16\!\cdots\!28\)\( p^{24} T^{15} + \)\(75\!\cdots\!69\)\( p^{32} T^{16} + \)\(40\!\cdots\!48\)\( p^{40} T^{17} - \)\(38\!\cdots\!80\)\( p^{48} T^{18} - \)\(37\!\cdots\!36\)\( p^{57} T^{19} + \)\(17\!\cdots\!63\)\( p^{64} T^{20} + \)\(11\!\cdots\!72\)\( p^{73} T^{21} - 5984367856004523 p^{80} T^{22} - 8514456 p^{88} T^{23} + p^{96} T^{24} \) | |
| 83 | \( 1 - 9924703546116687 T^{2} + \)\(57\!\cdots\!79\)\( T^{4} - \)\(26\!\cdots\!48\)\( T^{6} + \)\(92\!\cdots\!29\)\( T^{8} - \)\(26\!\cdots\!13\)\( T^{10} + \)\(66\!\cdots\!42\)\( T^{12} - \)\(26\!\cdots\!13\)\( p^{16} T^{14} + \)\(92\!\cdots\!29\)\( p^{32} T^{16} - \)\(26\!\cdots\!48\)\( p^{48} T^{18} + \)\(57\!\cdots\!79\)\( p^{64} T^{20} - 9924703546116687 p^{80} T^{22} + p^{96} T^{24} \) | |
| 89 | \( 1 + 83038554 T + 15954309489433614 T^{2} + \)\(11\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!17\)\( T^{4} + \)\(65\!\cdots\!84\)\( T^{5} + \)\(51\!\cdots\!46\)\( T^{6} + \)\(20\!\cdots\!10\)\( T^{7} + \)\(16\!\cdots\!82\)\( T^{8} + \)\(24\!\cdots\!74\)\( T^{9} + \)\(29\!\cdots\!54\)\( T^{10} - \)\(77\!\cdots\!52\)\( T^{11} + \)\(38\!\cdots\!37\)\( T^{12} - \)\(77\!\cdots\!52\)\( p^{8} T^{13} + \)\(29\!\cdots\!54\)\( p^{16} T^{14} + \)\(24\!\cdots\!74\)\( p^{24} T^{15} + \)\(16\!\cdots\!82\)\( p^{32} T^{16} + \)\(20\!\cdots\!10\)\( p^{40} T^{17} + \)\(51\!\cdots\!46\)\( p^{48} T^{18} + \)\(65\!\cdots\!84\)\( p^{56} T^{19} + \)\(11\!\cdots\!17\)\( p^{64} T^{20} + \)\(11\!\cdots\!68\)\( p^{72} T^{21} + 15954309489433614 p^{80} T^{22} + 83038554 p^{88} T^{23} + p^{96} T^{24} \) | |
| 97 | \( 1 - 76055551527339963 T^{2} + \)\(27\!\cdots\!15\)\( T^{4} - \)\(60\!\cdots\!12\)\( T^{6} + \)\(93\!\cdots\!37\)\( T^{8} - \)\(10\!\cdots\!61\)\( T^{10} + \)\(96\!\cdots\!06\)\( T^{12} - \)\(10\!\cdots\!61\)\( p^{16} T^{14} + \)\(93\!\cdots\!37\)\( p^{32} T^{16} - \)\(60\!\cdots\!12\)\( p^{48} T^{18} + \)\(27\!\cdots\!15\)\( p^{64} T^{20} - 76055551527339963 p^{80} T^{22} + p^{96} T^{24} \) | |
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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
−2.81886658554225651002083547352, −2.58694429590534010097574615249, −2.38091674890544825921570039176, −2.28988854546036275527212403355, −2.24449742402392849546168402491, −2.20267261059337287145318131528, −2.18149289677340000646789145246, −1.95674817507699207403755588209, −1.78937475522357218527163249118, −1.65157842255586435948164413370, −1.49113976833009532383962044013, −1.43522021556499164068909441824, −1.30283695604044385994111740694, −1.25623677442934416420383848760, −1.25377842846062564738723059038, −1.19119324440511837976364113882, −1.09664468314336194474678021504, −0.793496739176099781290516034171, −0.73785087815471620380136321541, −0.71658942743176952702658810784, −0.37378393798165328516909272659, −0.37243038467135028281951216894, −0.33790150604070740452926394619, −0.26247816492094654381160481443, −0.04705126685352096521971605245, 0.04705126685352096521971605245, 0.26247816492094654381160481443, 0.33790150604070740452926394619, 0.37243038467135028281951216894, 0.37378393798165328516909272659, 0.71658942743176952702658810784, 0.73785087815471620380136321541, 0.793496739176099781290516034171, 1.09664468314336194474678021504, 1.19119324440511837976364113882, 1.25377842846062564738723059038, 1.25623677442934416420383848760, 1.30283695604044385994111740694, 1.43522021556499164068909441824, 1.49113976833009532383962044013, 1.65157842255586435948164413370, 1.78937475522357218527163249118, 1.95674817507699207403755588209, 2.18149289677340000646789145246, 2.20267261059337287145318131528, 2.24449742402392849546168402491, 2.28988854546036275527212403355, 2.38091674890544825921570039176, 2.58694429590534010097574615249, 2.81886658554225651002083547352
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.