Dirichlet series
| L(s) = 1 | − 3·2-s − 101·3-s − 455·4-s + 226·5-s + 303·6-s − 3.43e3·7-s + 1.59e3·8-s + 289·9-s − 678·10-s + 451·11-s + 4.59e4·12-s − 2.19e4·13-s + 1.02e4·14-s − 2.28e4·15-s + 9.42e4·16-s − 8.65e3·17-s − 867·18-s + 1.01e4·19-s − 1.02e5·20-s + 3.46e5·21-s − 1.35e3·22-s − 5.21e4·23-s − 1.61e5·24-s − 3.41e5·25-s + 6.59e4·26-s + 3.36e5·27-s + 1.56e6·28-s + ⋯ |
| L(s) = 1 | − 0.265·2-s − 2.15·3-s − 3.55·4-s + 0.808·5-s + 0.572·6-s − 3.77·7-s + 1.10·8-s + 0.132·9-s − 0.214·10-s + 0.102·11-s + 7.67·12-s − 2.77·13-s + 1.00·14-s − 1.74·15-s + 5.75·16-s − 0.427·17-s − 0.0350·18-s + 0.338·19-s − 2.87·20-s + 8.16·21-s − 0.0270·22-s − 0.893·23-s − 2.38·24-s − 4.37·25-s + 0.735·26-s + 3.28·27-s + 13.4·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(20\) |
| Conductor: | \(7^{10} \cdot 13^{10}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.44600\times 10^{14}\) |
| Root analytic conductor: | \(5.33170\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((20,\ 7^{10} \cdot 13^{10} ,\ ( \ : [7/2]^{10} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.1934583736\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1934583736\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( ( 1 + p^{3} T )^{10} \) |
| 13 | \( ( 1 + p^{3} T )^{10} \) | |
| good | 2 | \( 1 + 3 T + 29 p^{4} T^{2} + 579 p T^{3} + 14443 p^{3} T^{4} + 54399 p^{2} T^{5} + 2703987 p^{3} T^{6} + 2378529 p^{4} T^{7} + 113234435 p^{5} T^{8} + 12883539 p^{9} T^{9} + 503598959 p^{10} T^{10} + 12883539 p^{16} T^{11} + 113234435 p^{19} T^{12} + 2378529 p^{25} T^{13} + 2703987 p^{31} T^{14} + 54399 p^{37} T^{15} + 14443 p^{45} T^{16} + 579 p^{50} T^{17} + 29 p^{60} T^{18} + 3 p^{63} T^{19} + p^{70} T^{20} \) |
| 3 | \( 1 + 101 T + 3304 p T^{2} + 635773 T^{3} + 38506697 T^{4} + 688799866 p T^{5} + 34260308480 p T^{6} + 174916341298 p^{3} T^{7} + 2703823585646 p^{4} T^{8} + 115790155365380 p^{4} T^{9} + 639775014430736 p^{6} T^{10} + 115790155365380 p^{11} T^{11} + 2703823585646 p^{18} T^{12} + 174916341298 p^{24} T^{13} + 34260308480 p^{29} T^{14} + 688799866 p^{36} T^{15} + 38506697 p^{42} T^{16} + 635773 p^{49} T^{17} + 3304 p^{57} T^{18} + 101 p^{63} T^{19} + p^{70} T^{20} \) | |
| 5 | \( 1 - 226 T + 392568 T^{2} - 105355378 T^{3} + 85136045597 T^{4} - 23691063729288 T^{5} + 2532462269301402 p T^{6} - 698699255512506072 p T^{7} + 56498801781909873881 p^{2} T^{8} - \)\(29\!\cdots\!22\)\( p^{3} T^{9} + \)\(19\!\cdots\!18\)\( p^{4} T^{10} - \)\(29\!\cdots\!22\)\( p^{10} T^{11} + 56498801781909873881 p^{16} T^{12} - 698699255512506072 p^{22} T^{13} + 2532462269301402 p^{29} T^{14} - 23691063729288 p^{35} T^{15} + 85136045597 p^{42} T^{16} - 105355378 p^{49} T^{17} + 392568 p^{56} T^{18} - 226 p^{63} T^{19} + p^{70} T^{20} \) | |
| 11 | \( 1 - 41 p T + 56496956 T^{2} - 223746252591 T^{3} + 1977629085663425 T^{4} - 11976961523774559062 T^{5} + \)\(67\!\cdots\!40\)\( T^{6} - \)\(35\!\cdots\!18\)\( T^{7} + \)\(19\!\cdots\!78\)\( T^{8} - \)\(82\!\cdots\!24\)\( T^{9} + \)\(43\!\cdots\!56\)\( T^{10} - \)\(82\!\cdots\!24\)\( p^{7} T^{11} + \)\(19\!\cdots\!78\)\( p^{14} T^{12} - \)\(35\!\cdots\!18\)\( p^{21} T^{13} + \)\(67\!\cdots\!40\)\( p^{28} T^{14} - 11976961523774559062 p^{35} T^{15} + 1977629085663425 p^{42} T^{16} - 223746252591 p^{49} T^{17} + 56496956 p^{56} T^{18} - 41 p^{64} T^{19} + p^{70} T^{20} \) | |
| 17 | \( 1 + 8654 T + 1656675752 T^{2} + 17675786187738 T^{3} + 1479368154241447553 T^{4} + \)\(18\!\cdots\!36\)\( T^{5} + \)\(98\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!48\)\( T^{7} + \)\(54\!\cdots\!90\)\( T^{8} + \)\(62\!\cdots\!72\)\( T^{9} + \)\(25\!\cdots\!40\)\( T^{10} + \)\(62\!\cdots\!72\)\( p^{7} T^{11} + \)\(54\!\cdots\!90\)\( p^{14} T^{12} + \)\(12\!\cdots\!48\)\( p^{21} T^{13} + \)\(98\!\cdots\!60\)\( p^{28} T^{14} + \)\(18\!\cdots\!36\)\( p^{35} T^{15} + 1479368154241447553 p^{42} T^{16} + 17675786187738 p^{49} T^{17} + 1656675752 p^{56} T^{18} + 8654 p^{63} T^{19} + p^{70} T^{20} \) | |
| 19 | \( 1 - 10130 T + 6159013852 T^{2} - 3029604524602 p T^{3} + 18382102388302585909 T^{4} - \)\(15\!\cdots\!56\)\( T^{5} + \)\(35\!\cdots\!02\)\( T^{6} - \)\(26\!\cdots\!40\)\( T^{7} + \)\(48\!\cdots\!05\)\( T^{8} - \)\(16\!\cdots\!78\)\( p T^{9} + \)\(49\!\cdots\!26\)\( T^{10} - \)\(16\!\cdots\!78\)\( p^{8} T^{11} + \)\(48\!\cdots\!05\)\( p^{14} T^{12} - \)\(26\!\cdots\!40\)\( p^{21} T^{13} + \)\(35\!\cdots\!02\)\( p^{28} T^{14} - \)\(15\!\cdots\!56\)\( p^{35} T^{15} + 18382102388302585909 p^{42} T^{16} - 3029604524602 p^{50} T^{17} + 6159013852 p^{56} T^{18} - 10130 p^{63} T^{19} + p^{70} T^{20} \) | |
| 23 | \( 1 + 52155 T + 7450114434 T^{2} - 204342962406759 T^{3} + 16493396243029070963 T^{4} - \)\(14\!\cdots\!50\)\( T^{5} + \)\(21\!\cdots\!04\)\( T^{6} - \)\(42\!\cdots\!86\)\( T^{7} + \)\(91\!\cdots\!57\)\( T^{8} - \)\(29\!\cdots\!07\)\( T^{9} + \)\(18\!\cdots\!02\)\( T^{10} - \)\(29\!\cdots\!07\)\( p^{7} T^{11} + \)\(91\!\cdots\!57\)\( p^{14} T^{12} - \)\(42\!\cdots\!86\)\( p^{21} T^{13} + \)\(21\!\cdots\!04\)\( p^{28} T^{14} - \)\(14\!\cdots\!50\)\( p^{35} T^{15} + 16493396243029070963 p^{42} T^{16} - 204342962406759 p^{49} T^{17} + 7450114434 p^{56} T^{18} + 52155 p^{63} T^{19} + p^{70} T^{20} \) | |
| 29 | \( 1 - 520154 T + 205070784894 T^{2} - 55099585704416946 T^{3} + \)\(12\!\cdots\!67\)\( T^{4} - \)\(24\!\cdots\!12\)\( T^{5} + \)\(41\!\cdots\!32\)\( T^{6} - \)\(62\!\cdots\!48\)\( T^{7} + \)\(90\!\cdots\!17\)\( T^{8} - \)\(11\!\cdots\!86\)\( T^{9} + \)\(16\!\cdots\!50\)\( T^{10} - \)\(11\!\cdots\!86\)\( p^{7} T^{11} + \)\(90\!\cdots\!17\)\( p^{14} T^{12} - \)\(62\!\cdots\!48\)\( p^{21} T^{13} + \)\(41\!\cdots\!32\)\( p^{28} T^{14} - \)\(24\!\cdots\!12\)\( p^{35} T^{15} + \)\(12\!\cdots\!67\)\( p^{42} T^{16} - 55099585704416946 p^{49} T^{17} + 205070784894 p^{56} T^{18} - 520154 p^{63} T^{19} + p^{70} T^{20} \) | |
| 31 | \( 1 - 692605 T + 391098579576 T^{2} - 155861930403143553 T^{3} + \)\(53\!\cdots\!85\)\( T^{4} - \)\(15\!\cdots\!26\)\( T^{5} + \)\(13\!\cdots\!38\)\( p T^{6} - \)\(93\!\cdots\!84\)\( T^{7} + \)\(19\!\cdots\!25\)\( T^{8} - \)\(37\!\cdots\!49\)\( T^{9} + \)\(64\!\cdots\!94\)\( T^{10} - \)\(37\!\cdots\!49\)\( p^{7} T^{11} + \)\(19\!\cdots\!25\)\( p^{14} T^{12} - \)\(93\!\cdots\!84\)\( p^{21} T^{13} + \)\(13\!\cdots\!38\)\( p^{29} T^{14} - \)\(15\!\cdots\!26\)\( p^{35} T^{15} + \)\(53\!\cdots\!85\)\( p^{42} T^{16} - 155861930403143553 p^{49} T^{17} + 391098579576 p^{56} T^{18} - 692605 p^{63} T^{19} + p^{70} T^{20} \) | |
| 37 | \( 1 + 20511 T + 441867149942 T^{2} + 30576846679905641 T^{3} + \)\(98\!\cdots\!05\)\( T^{4} + \)\(10\!\cdots\!26\)\( T^{5} + \)\(15\!\cdots\!68\)\( T^{6} + \)\(18\!\cdots\!50\)\( T^{7} + \)\(20\!\cdots\!66\)\( T^{8} + \)\(20\!\cdots\!20\)\( T^{9} + \)\(21\!\cdots\!24\)\( T^{10} + \)\(20\!\cdots\!20\)\( p^{7} T^{11} + \)\(20\!\cdots\!66\)\( p^{14} T^{12} + \)\(18\!\cdots\!50\)\( p^{21} T^{13} + \)\(15\!\cdots\!68\)\( p^{28} T^{14} + \)\(10\!\cdots\!26\)\( p^{35} T^{15} + \)\(98\!\cdots\!05\)\( p^{42} T^{16} + 30576846679905641 p^{49} T^{17} + 441867149942 p^{56} T^{18} + 20511 p^{63} T^{19} + p^{70} T^{20} \) | |
| 41 | \( 1 - 355049 T + 1127216244432 T^{2} - 331025580866177921 T^{3} + \)\(66\!\cdots\!37\)\( T^{4} - \)\(18\!\cdots\!32\)\( T^{5} + \)\(26\!\cdots\!08\)\( T^{6} - \)\(65\!\cdots\!36\)\( T^{7} + \)\(76\!\cdots\!34\)\( T^{8} - \)\(17\!\cdots\!74\)\( T^{9} + \)\(16\!\cdots\!64\)\( T^{10} - \)\(17\!\cdots\!74\)\( p^{7} T^{11} + \)\(76\!\cdots\!34\)\( p^{14} T^{12} - \)\(65\!\cdots\!36\)\( p^{21} T^{13} + \)\(26\!\cdots\!08\)\( p^{28} T^{14} - \)\(18\!\cdots\!32\)\( p^{35} T^{15} + \)\(66\!\cdots\!37\)\( p^{42} T^{16} - 331025580866177921 p^{49} T^{17} + 1127216244432 p^{56} T^{18} - 355049 p^{63} T^{19} + p^{70} T^{20} \) | |
| 43 | \( 1 - 1256772 T + 2739342286754 T^{2} - 2499916103337403152 T^{3} + \)\(31\!\cdots\!91\)\( T^{4} - \)\(22\!\cdots\!44\)\( T^{5} + \)\(20\!\cdots\!20\)\( T^{6} - \)\(11\!\cdots\!16\)\( T^{7} + \)\(88\!\cdots\!77\)\( T^{8} - \)\(44\!\cdots\!68\)\( T^{9} + \)\(27\!\cdots\!38\)\( T^{10} - \)\(44\!\cdots\!68\)\( p^{7} T^{11} + \)\(88\!\cdots\!77\)\( p^{14} T^{12} - \)\(11\!\cdots\!16\)\( p^{21} T^{13} + \)\(20\!\cdots\!20\)\( p^{28} T^{14} - \)\(22\!\cdots\!44\)\( p^{35} T^{15} + \)\(31\!\cdots\!91\)\( p^{42} T^{16} - 2499916103337403152 p^{49} T^{17} + 2739342286754 p^{56} T^{18} - 1256772 p^{63} T^{19} + p^{70} T^{20} \) | |
| 47 | \( 1 - 1260721 T + 3983650038820 T^{2} - 3551581950269901593 T^{3} + \)\(64\!\cdots\!33\)\( T^{4} - \)\(42\!\cdots\!02\)\( T^{5} + \)\(60\!\cdots\!14\)\( T^{6} - \)\(29\!\cdots\!48\)\( T^{7} + \)\(38\!\cdots\!89\)\( T^{8} - \)\(14\!\cdots\!45\)\( T^{9} + \)\(20\!\cdots\!02\)\( T^{10} - \)\(14\!\cdots\!45\)\( p^{7} T^{11} + \)\(38\!\cdots\!89\)\( p^{14} T^{12} - \)\(29\!\cdots\!48\)\( p^{21} T^{13} + \)\(60\!\cdots\!14\)\( p^{28} T^{14} - \)\(42\!\cdots\!02\)\( p^{35} T^{15} + \)\(64\!\cdots\!33\)\( p^{42} T^{16} - 3551581950269901593 p^{49} T^{17} + 3983650038820 p^{56} T^{18} - 1260721 p^{63} T^{19} + p^{70} T^{20} \) | |
| 53 | \( 1 - 928854 T + 7937938581694 T^{2} - 5394581156781635010 T^{3} + \)\(28\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!96\)\( T^{5} + \)\(59\!\cdots\!00\)\( T^{6} - \)\(12\!\cdots\!20\)\( T^{7} + \)\(89\!\cdots\!85\)\( T^{8} - \)\(55\!\cdots\!54\)\( T^{9} + \)\(11\!\cdots\!74\)\( T^{10} - \)\(55\!\cdots\!54\)\( p^{7} T^{11} + \)\(89\!\cdots\!85\)\( p^{14} T^{12} - \)\(12\!\cdots\!20\)\( p^{21} T^{13} + \)\(59\!\cdots\!00\)\( p^{28} T^{14} - \)\(12\!\cdots\!96\)\( p^{35} T^{15} + \)\(28\!\cdots\!23\)\( p^{42} T^{16} - 5394581156781635010 p^{49} T^{17} + 7937938581694 p^{56} T^{18} - 928854 p^{63} T^{19} + p^{70} T^{20} \) | |
| 59 | \( 1 - 3144446 T + 18138495692978 T^{2} - 43856503817449364322 T^{3} + \)\(14\!\cdots\!57\)\( T^{4} - \)\(29\!\cdots\!24\)\( T^{5} + \)\(75\!\cdots\!28\)\( T^{6} - \)\(13\!\cdots\!12\)\( T^{7} + \)\(27\!\cdots\!38\)\( T^{8} - \)\(44\!\cdots\!48\)\( T^{9} + \)\(79\!\cdots\!76\)\( T^{10} - \)\(44\!\cdots\!48\)\( p^{7} T^{11} + \)\(27\!\cdots\!38\)\( p^{14} T^{12} - \)\(13\!\cdots\!12\)\( p^{21} T^{13} + \)\(75\!\cdots\!28\)\( p^{28} T^{14} - \)\(29\!\cdots\!24\)\( p^{35} T^{15} + \)\(14\!\cdots\!57\)\( p^{42} T^{16} - 43856503817449364322 p^{49} T^{17} + 18138495692978 p^{56} T^{18} - 3144446 p^{63} T^{19} + p^{70} T^{20} \) | |
| 61 | \( 1 - 6322923 T + 38154622663492 T^{2} - \)\(15\!\cdots\!47\)\( T^{3} + \)\(56\!\cdots\!93\)\( T^{4} - \)\(16\!\cdots\!24\)\( T^{5} + \)\(46\!\cdots\!28\)\( T^{6} - \)\(11\!\cdots\!60\)\( T^{7} + \)\(24\!\cdots\!42\)\( T^{8} - \)\(49\!\cdots\!62\)\( T^{9} + \)\(92\!\cdots\!12\)\( T^{10} - \)\(49\!\cdots\!62\)\( p^{7} T^{11} + \)\(24\!\cdots\!42\)\( p^{14} T^{12} - \)\(11\!\cdots\!60\)\( p^{21} T^{13} + \)\(46\!\cdots\!28\)\( p^{28} T^{14} - \)\(16\!\cdots\!24\)\( p^{35} T^{15} + \)\(56\!\cdots\!93\)\( p^{42} T^{16} - \)\(15\!\cdots\!47\)\( p^{49} T^{17} + 38154622663492 p^{56} T^{18} - 6322923 p^{63} T^{19} + p^{70} T^{20} \) | |
| 67 | \( 1 - 3944507 T + 46794987394610 T^{2} - \)\(16\!\cdots\!77\)\( T^{3} + \)\(10\!\cdots\!25\)\( T^{4} - \)\(31\!\cdots\!40\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} - \)\(38\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!26\)\( T^{8} - \)\(32\!\cdots\!90\)\( T^{9} + \)\(10\!\cdots\!92\)\( T^{10} - \)\(32\!\cdots\!90\)\( p^{7} T^{11} + \)\(14\!\cdots\!26\)\( p^{14} T^{12} - \)\(38\!\cdots\!60\)\( p^{21} T^{13} + \)\(14\!\cdots\!00\)\( p^{28} T^{14} - \)\(31\!\cdots\!40\)\( p^{35} T^{15} + \)\(10\!\cdots\!25\)\( p^{42} T^{16} - \)\(16\!\cdots\!77\)\( p^{49} T^{17} + 46794987394610 p^{56} T^{18} - 3944507 p^{63} T^{19} + p^{70} T^{20} \) | |
| 71 | \( 1 - 6032248 T + 38699894216792 T^{2} - \)\(14\!\cdots\!52\)\( T^{3} + \)\(67\!\cdots\!81\)\( T^{4} - \)\(24\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!68\)\( T^{6} - \)\(36\!\cdots\!24\)\( T^{7} + \)\(12\!\cdots\!18\)\( T^{8} - \)\(38\!\cdots\!76\)\( T^{9} + \)\(12\!\cdots\!64\)\( T^{10} - \)\(38\!\cdots\!76\)\( p^{7} T^{11} + \)\(12\!\cdots\!18\)\( p^{14} T^{12} - \)\(36\!\cdots\!24\)\( p^{21} T^{13} + \)\(10\!\cdots\!68\)\( p^{28} T^{14} - \)\(24\!\cdots\!44\)\( p^{35} T^{15} + \)\(67\!\cdots\!81\)\( p^{42} T^{16} - \)\(14\!\cdots\!52\)\( p^{49} T^{17} + 38699894216792 p^{56} T^{18} - 6032248 p^{63} T^{19} + p^{70} T^{20} \) | |
| 73 | \( 1 - 1248533 T + 43235184164810 T^{2} + 16191084565043391933 T^{3} + \)\(83\!\cdots\!97\)\( T^{4} + \)\(19\!\cdots\!98\)\( T^{5} + \)\(12\!\cdots\!62\)\( T^{6} + \)\(38\!\cdots\!36\)\( T^{7} + \)\(21\!\cdots\!73\)\( T^{8} + \)\(43\!\cdots\!63\)\( T^{9} + \)\(28\!\cdots\!96\)\( T^{10} + \)\(43\!\cdots\!63\)\( p^{7} T^{11} + \)\(21\!\cdots\!73\)\( p^{14} T^{12} + \)\(38\!\cdots\!36\)\( p^{21} T^{13} + \)\(12\!\cdots\!62\)\( p^{28} T^{14} + \)\(19\!\cdots\!98\)\( p^{35} T^{15} + \)\(83\!\cdots\!97\)\( p^{42} T^{16} + 16191084565043391933 p^{49} T^{17} + 43235184164810 p^{56} T^{18} - 1248533 p^{63} T^{19} + p^{70} T^{20} \) | |
| 79 | \( 1 + 14947605 T + 218265257796582 T^{2} + \)\(21\!\cdots\!47\)\( T^{3} + \)\(19\!\cdots\!19\)\( T^{4} + \)\(14\!\cdots\!30\)\( T^{5} + \)\(97\!\cdots\!48\)\( T^{6} + \)\(57\!\cdots\!02\)\( T^{7} + \)\(32\!\cdots\!57\)\( T^{8} + \)\(20\!\cdots\!81\)\( p T^{9} + \)\(73\!\cdots\!94\)\( T^{10} + \)\(20\!\cdots\!81\)\( p^{8} T^{11} + \)\(32\!\cdots\!57\)\( p^{14} T^{12} + \)\(57\!\cdots\!02\)\( p^{21} T^{13} + \)\(97\!\cdots\!48\)\( p^{28} T^{14} + \)\(14\!\cdots\!30\)\( p^{35} T^{15} + \)\(19\!\cdots\!19\)\( p^{42} T^{16} + \)\(21\!\cdots\!47\)\( p^{49} T^{17} + 218265257796582 p^{56} T^{18} + 14947605 p^{63} T^{19} + p^{70} T^{20} \) | |
| 83 | \( 1 + 14177784 T + 272284176497028 T^{2} + \)\(27\!\cdots\!76\)\( T^{3} + \)\(30\!\cdots\!77\)\( T^{4} + \)\(29\!\cdots\!48\)\( p T^{5} + \)\(20\!\cdots\!86\)\( T^{6} + \)\(13\!\cdots\!60\)\( T^{7} + \)\(92\!\cdots\!29\)\( T^{8} + \)\(51\!\cdots\!24\)\( T^{9} + \)\(29\!\cdots\!34\)\( T^{10} + \)\(51\!\cdots\!24\)\( p^{7} T^{11} + \)\(92\!\cdots\!29\)\( p^{14} T^{12} + \)\(13\!\cdots\!60\)\( p^{21} T^{13} + \)\(20\!\cdots\!86\)\( p^{28} T^{14} + \)\(29\!\cdots\!48\)\( p^{36} T^{15} + \)\(30\!\cdots\!77\)\( p^{42} T^{16} + \)\(27\!\cdots\!76\)\( p^{49} T^{17} + 272284176497028 p^{56} T^{18} + 14177784 p^{63} T^{19} + p^{70} T^{20} \) | |
| 89 | \( 1 - 6734836 T + 115365014875412 T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(21\!\cdots\!05\)\( T^{4} + \)\(76\!\cdots\!68\)\( T^{5} + \)\(11\!\cdots\!30\)\( T^{6} - \)\(70\!\cdots\!04\)\( T^{7} + \)\(13\!\cdots\!13\)\( T^{8} - \)\(26\!\cdots\!84\)\( T^{9} + \)\(56\!\cdots\!38\)\( T^{10} - \)\(26\!\cdots\!84\)\( p^{7} T^{11} + \)\(13\!\cdots\!13\)\( p^{14} T^{12} - \)\(70\!\cdots\!04\)\( p^{21} T^{13} + \)\(11\!\cdots\!30\)\( p^{28} T^{14} + \)\(76\!\cdots\!68\)\( p^{35} T^{15} + \)\(21\!\cdots\!05\)\( p^{42} T^{16} - \)\(22\!\cdots\!20\)\( p^{49} T^{17} + 115365014875412 p^{56} T^{18} - 6734836 p^{63} T^{19} + p^{70} T^{20} \) | |
| 97 | \( 1 + 12365397 T + 632105311563950 T^{2} + \)\(71\!\cdots\!59\)\( T^{3} + \)\(18\!\cdots\!29\)\( T^{4} + \)\(19\!\cdots\!18\)\( T^{5} + \)\(34\!\cdots\!90\)\( T^{6} + \)\(33\!\cdots\!40\)\( T^{7} + \)\(45\!\cdots\!13\)\( T^{8} + \)\(38\!\cdots\!25\)\( T^{9} + \)\(42\!\cdots\!84\)\( T^{10} + \)\(38\!\cdots\!25\)\( p^{7} T^{11} + \)\(45\!\cdots\!13\)\( p^{14} T^{12} + \)\(33\!\cdots\!40\)\( p^{21} T^{13} + \)\(34\!\cdots\!90\)\( p^{28} T^{14} + \)\(19\!\cdots\!18\)\( p^{35} T^{15} + \)\(18\!\cdots\!29\)\( p^{42} T^{16} + \)\(71\!\cdots\!59\)\( p^{49} T^{17} + 632105311563950 p^{56} T^{18} + 12365397 p^{63} T^{19} + p^{70} T^{20} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.30617286344580801806645952794, −3.99659246380238225653981364205, −3.97886541521018069206200973171, −3.97322238848742945463085317904, −3.82394802507814568175229959069, −3.33820688189962685076633032097, −3.21270505148724694291320282933, −2.92277440972329670489072586058, −2.90862220437325154832540882806, −2.82700525618293524734093244389, −2.81411005390425253491242693502, −2.42923433546159347910989709052, −2.24683943590729460405821842911, −2.14147292648031274676478418209, −2.13209069277458106650381439216, −1.81125503792889124325968887100, −1.31425733908645736905751220784, −0.878207426088264280139779091879, −0.809377055164273670854553376071, −0.69464762461064572171608488659, −0.68742124438595777730001630183, −0.56898514737161423339516710642, −0.39768839727038919485728101541, −0.28917015925136282964672553835, −0.14246684585066868369796002863, 0.14246684585066868369796002863, 0.28917015925136282964672553835, 0.39768839727038919485728101541, 0.56898514737161423339516710642, 0.68742124438595777730001630183, 0.69464762461064572171608488659, 0.809377055164273670854553376071, 0.878207426088264280139779091879, 1.31425733908645736905751220784, 1.81125503792889124325968887100, 2.13209069277458106650381439216, 2.14147292648031274676478418209, 2.24683943590729460405821842911, 2.42923433546159347910989709052, 2.81411005390425253491242693502, 2.82700525618293524734093244389, 2.90862220437325154832540882806, 2.92277440972329670489072586058, 3.21270505148724694291320282933, 3.33820688189962685076633032097, 3.82394802507814568175229959069, 3.97322238848742945463085317904, 3.97886541521018069206200973171, 3.99659246380238225653981364205, 4.30617286344580801806645952794
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.