Dirichlet series
| L(s) = 1 | − 2·2-s + 4·3-s + 2·4-s − 8·5-s − 8·6-s − 56·7-s − 42·8-s − 476·9-s + 16·10-s − 500·11-s + 8·12-s + 112·14-s − 32·15-s + 479·16-s + 952·18-s − 1.71e3·19-s − 16·20-s − 224·21-s + 1.00e3·22-s − 168·24-s + 32·25-s − 916·27-s − 112·28-s − 1.62e3·29-s + 64·30-s − 4.06e3·31-s − 1.32e3·32-s + ⋯ |
| L(s) = 1 | − 1/2·2-s + 4/9·3-s + 1/8·4-s − 0.319·5-s − 2/9·6-s − 8/7·7-s − 0.656·8-s − 5.87·9-s + 4/25·10-s − 4.13·11-s + 1/18·12-s + 4/7·14-s − 0.142·15-s + 1.87·16-s + 2.93·18-s − 4.74·19-s − 0.0399·20-s − 0.507·21-s + 2.06·22-s − 0.291·24-s + 0.0511·25-s − 1.25·27-s − 1/7·28-s − 1.93·29-s + 0.0711·30-s − 4.22·31-s − 1.28·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(13^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.52483\times 10^{19}\) |
| Root analytic conductor: | \(4.17965\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 13^{32} ,\ ( \ : [2]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.02678610576\) |
| \(L(\frac12)\) | \(\approx\) | \(0.02678610576\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} + 21 p T^{3} - 315 T^{4} - 67 p^{2} T^{5} + 61 p^{4} T^{6} - 517 p^{4} T^{7} + 29519 p^{2} T^{8} + 1711 p^{6} T^{9} - 663 p^{5} T^{10} - 18443 p^{5} T^{11} - 2706561 p^{4} T^{12} - 263551 p^{6} T^{13} - 43443 p^{11} T^{14} - 969859 p^{9} T^{15} + 141979377 p^{6} T^{16} - 969859 p^{13} T^{17} - 43443 p^{19} T^{18} - 263551 p^{18} T^{19} - 2706561 p^{20} T^{20} - 18443 p^{25} T^{21} - 663 p^{29} T^{22} + 1711 p^{34} T^{23} + 29519 p^{34} T^{24} - 517 p^{40} T^{25} + 61 p^{44} T^{26} - 67 p^{46} T^{27} - 315 p^{48} T^{28} + 21 p^{53} T^{29} + p^{57} T^{30} + p^{61} T^{31} + p^{64} T^{32} \) |
| 3 | \( ( 1 - 2 T + 244 T^{2} - 110 p^{2} T^{3} + 11543 p T^{4} - 18736 p^{2} T^{5} + 42694 p^{4} T^{6} - 240496 p^{4} T^{7} + 1199144 p^{5} T^{8} - 240496 p^{8} T^{9} + 42694 p^{12} T^{10} - 18736 p^{14} T^{11} + 11543 p^{17} T^{12} - 110 p^{22} T^{13} + 244 p^{24} T^{14} - 2 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 5 | \( 1 + 8 T + 32 T^{2} + 12096 T^{3} - 527442 T^{4} - 6817816 T^{5} + 35492224 T^{6} - 3499118776 T^{7} + 214598054801 T^{8} + 1770375811288 T^{9} + 6931925138112 T^{10} + 204666718110056 T^{11} - 91059199232161746 T^{12} + 449348487028373168 T^{13} - 924223306284337248 T^{14} + \)\(13\!\cdots\!88\)\( T^{15} + \)\(53\!\cdots\!36\)\( T^{16} + \)\(13\!\cdots\!88\)\( p^{4} T^{17} - 924223306284337248 p^{8} T^{18} + 449348487028373168 p^{12} T^{19} - 91059199232161746 p^{16} T^{20} + 204666718110056 p^{20} T^{21} + 6931925138112 p^{24} T^{22} + 1770375811288 p^{28} T^{23} + 214598054801 p^{32} T^{24} - 3499118776 p^{36} T^{25} + 35492224 p^{40} T^{26} - 6817816 p^{44} T^{27} - 527442 p^{48} T^{28} + 12096 p^{52} T^{29} + 32 p^{56} T^{30} + 8 p^{60} T^{31} + p^{64} T^{32} \) | |
| 7 | \( 1 + 8 p T + 32 p^{2} T^{2} - 128300 T^{3} - 2631698 T^{4} + 254843912 T^{5} + 26628206536 T^{6} - 53777376424 T^{7} - 538919421319 p^{2} T^{8} - 1024746864091444 T^{9} + 93470177610969744 T^{10} + 26885730332508892 p T^{11} + 53702670901391448726 T^{12} + \)\(30\!\cdots\!80\)\( T^{13} + \)\(75\!\cdots\!00\)\( T^{14} + \)\(38\!\cdots\!44\)\( T^{15} - \)\(36\!\cdots\!52\)\( T^{16} + \)\(38\!\cdots\!44\)\( p^{4} T^{17} + \)\(75\!\cdots\!00\)\( p^{8} T^{18} + \)\(30\!\cdots\!80\)\( p^{12} T^{19} + 53702670901391448726 p^{16} T^{20} + 26885730332508892 p^{21} T^{21} + 93470177610969744 p^{24} T^{22} - 1024746864091444 p^{28} T^{23} - 538919421319 p^{34} T^{24} - 53777376424 p^{36} T^{25} + 26628206536 p^{40} T^{26} + 254843912 p^{44} T^{27} - 2631698 p^{48} T^{28} - 128300 p^{52} T^{29} + 32 p^{58} T^{30} + 8 p^{61} T^{31} + p^{64} T^{32} \) | |
| 11 | \( 1 + 500 T + 125000 T^{2} + 23323032 T^{3} + 4070502606 T^{4} + 670241943908 T^{5} + 98290057040512 T^{6} + 12771275182877444 T^{7} + 1557426041928232937 T^{8} + \)\(19\!\cdots\!96\)\( T^{9} + \)\(25\!\cdots\!68\)\( T^{10} + \)\(32\!\cdots\!24\)\( T^{11} + \)\(42\!\cdots\!58\)\( T^{12} + \)\(61\!\cdots\!20\)\( T^{13} + \)\(85\!\cdots\!52\)\( T^{14} + \)\(11\!\cdots\!16\)\( T^{15} + \)\(13\!\cdots\!56\)\( T^{16} + \)\(11\!\cdots\!16\)\( p^{4} T^{17} + \)\(85\!\cdots\!52\)\( p^{8} T^{18} + \)\(61\!\cdots\!20\)\( p^{12} T^{19} + \)\(42\!\cdots\!58\)\( p^{16} T^{20} + \)\(32\!\cdots\!24\)\( p^{20} T^{21} + \)\(25\!\cdots\!68\)\( p^{24} T^{22} + \)\(19\!\cdots\!96\)\( p^{28} T^{23} + 1557426041928232937 p^{32} T^{24} + 12771275182877444 p^{36} T^{25} + 98290057040512 p^{40} T^{26} + 670241943908 p^{44} T^{27} + 4070502606 p^{48} T^{28} + 23323032 p^{52} T^{29} + 125000 p^{56} T^{30} + 500 p^{60} T^{31} + p^{64} T^{32} \) | |
| 17 | \( 1 - 900808 T^{2} + 390368392284 T^{4} - 108276885202120544 T^{6} + \)\(21\!\cdots\!18\)\( T^{8} - \)\(33\!\cdots\!64\)\( T^{10} + \)\(41\!\cdots\!08\)\( T^{12} - \)\(43\!\cdots\!44\)\( T^{14} + \)\(38\!\cdots\!79\)\( T^{16} - \)\(43\!\cdots\!44\)\( p^{8} T^{18} + \)\(41\!\cdots\!08\)\( p^{16} T^{20} - \)\(33\!\cdots\!64\)\( p^{24} T^{22} + \)\(21\!\cdots\!18\)\( p^{32} T^{24} - 108276885202120544 p^{40} T^{26} + 390368392284 p^{48} T^{28} - 900808 p^{56} T^{30} + p^{64} T^{32} \) | |
| 19 | \( 1 + 1712 T + 1465472 T^{2} + 970886788 T^{3} + 599154831898 T^{4} + 332939374660184 T^{5} + 163258157163487624 T^{6} + 74753624049830398400 T^{7} + \)\(32\!\cdots\!49\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(48\!\cdots\!48\)\( T^{10} + \)\(17\!\cdots\!72\)\( T^{11} + \)\(58\!\cdots\!02\)\( T^{12} + \)\(19\!\cdots\!28\)\( T^{13} + \)\(17\!\cdots\!68\)\( p^{2} T^{14} + \)\(10\!\cdots\!04\)\( p T^{15} + \)\(70\!\cdots\!60\)\( T^{16} + \)\(10\!\cdots\!04\)\( p^{5} T^{17} + \)\(17\!\cdots\!68\)\( p^{10} T^{18} + \)\(19\!\cdots\!28\)\( p^{12} T^{19} + \)\(58\!\cdots\!02\)\( p^{16} T^{20} + \)\(17\!\cdots\!72\)\( p^{20} T^{21} + \)\(48\!\cdots\!48\)\( p^{24} T^{22} + \)\(12\!\cdots\!00\)\( p^{28} T^{23} + \)\(32\!\cdots\!49\)\( p^{32} T^{24} + 74753624049830398400 p^{36} T^{25} + 163258157163487624 p^{40} T^{26} + 332939374660184 p^{44} T^{27} + 599154831898 p^{48} T^{28} + 970886788 p^{52} T^{29} + 1465472 p^{56} T^{30} + 1712 p^{60} T^{31} + p^{64} T^{32} \) | |
| 23 | \( 1 - 2055052 T^{2} + 2170526769570 T^{4} - 1566575052785000768 T^{6} + \)\(87\!\cdots\!21\)\( T^{8} - \)\(39\!\cdots\!36\)\( T^{10} + \)\(15\!\cdots\!58\)\( T^{12} - \)\(53\!\cdots\!84\)\( T^{14} + \)\(15\!\cdots\!84\)\( T^{16} - \)\(53\!\cdots\!84\)\( p^{8} T^{18} + \)\(15\!\cdots\!58\)\( p^{16} T^{20} - \)\(39\!\cdots\!36\)\( p^{24} T^{22} + \)\(87\!\cdots\!21\)\( p^{32} T^{24} - 1566575052785000768 p^{40} T^{26} + 2170526769570 p^{48} T^{28} - 2055052 p^{56} T^{30} + p^{64} T^{32} \) | |
| 29 | \( ( 1 + 814 T + 4407208 T^{2} + 2838563372 T^{3} + 8935236067050 T^{4} + 4772689957292634 T^{5} + 11201444806298853104 T^{6} + \)\(50\!\cdots\!78\)\( T^{7} + \)\(95\!\cdots\!91\)\( T^{8} + \)\(50\!\cdots\!78\)\( p^{4} T^{9} + 11201444806298853104 p^{8} T^{10} + 4772689957292634 p^{12} T^{11} + 8935236067050 p^{16} T^{12} + 2838563372 p^{20} T^{13} + 4407208 p^{24} T^{14} + 814 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 31 | \( 1 + 4060 T + 8241800 T^{2} + 11888153764 T^{3} + 12102195192780 T^{4} + 6844355067784460 T^{5} - 1291690806395612552 T^{6} - \)\(91\!\cdots\!16\)\( T^{7} - \)\(40\!\cdots\!44\)\( p T^{8} - \)\(80\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!36\)\( T^{10} + \)\(10\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!04\)\( T^{12} + \)\(11\!\cdots\!48\)\( T^{13} + \)\(36\!\cdots\!28\)\( T^{14} - \)\(41\!\cdots\!28\)\( T^{15} - \)\(74\!\cdots\!42\)\( T^{16} - \)\(41\!\cdots\!28\)\( p^{4} T^{17} + \)\(36\!\cdots\!28\)\( p^{8} T^{18} + \)\(11\!\cdots\!48\)\( p^{12} T^{19} + \)\(15\!\cdots\!04\)\( p^{16} T^{20} + \)\(10\!\cdots\!52\)\( p^{20} T^{21} + \)\(12\!\cdots\!36\)\( p^{24} T^{22} - \)\(80\!\cdots\!00\)\( p^{28} T^{23} - \)\(40\!\cdots\!44\)\( p^{33} T^{24} - \)\(91\!\cdots\!16\)\( p^{36} T^{25} - 1291690806395612552 p^{40} T^{26} + 6844355067784460 p^{44} T^{27} + 12102195192780 p^{48} T^{28} + 11888153764 p^{52} T^{29} + 8241800 p^{56} T^{30} + 4060 p^{60} T^{31} + p^{64} T^{32} \) | |
| 37 | \( 1 + 10468 T + 54789512 T^{2} + 198419905432 T^{3} + 586867972468884 T^{4} + 1547505974482038836 T^{5} + \)\(10\!\cdots\!96\)\( p T^{6} + \)\(82\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!22\)\( T^{8} + \)\(32\!\cdots\!92\)\( T^{9} + \)\(60\!\cdots\!72\)\( T^{10} + \)\(10\!\cdots\!12\)\( T^{11} + \)\(17\!\cdots\!76\)\( T^{12} + \)\(27\!\cdots\!24\)\( T^{13} + \)\(42\!\cdots\!24\)\( T^{14} + \)\(61\!\cdots\!16\)\( T^{15} + \)\(85\!\cdots\!35\)\( T^{16} + \)\(61\!\cdots\!16\)\( p^{4} T^{17} + \)\(42\!\cdots\!24\)\( p^{8} T^{18} + \)\(27\!\cdots\!24\)\( p^{12} T^{19} + \)\(17\!\cdots\!76\)\( p^{16} T^{20} + \)\(10\!\cdots\!12\)\( p^{20} T^{21} + \)\(60\!\cdots\!72\)\( p^{24} T^{22} + \)\(32\!\cdots\!92\)\( p^{28} T^{23} + \)\(16\!\cdots\!22\)\( p^{32} T^{24} + \)\(82\!\cdots\!00\)\( p^{36} T^{25} + \)\(10\!\cdots\!96\)\( p^{41} T^{26} + 1547505974482038836 p^{44} T^{27} + 586867972468884 p^{48} T^{28} + 198419905432 p^{52} T^{29} + 54789512 p^{56} T^{30} + 10468 p^{60} T^{31} + p^{64} T^{32} \) | |
| 41 | \( 1 + 3440 T + 5916800 T^{2} + 5159810592 T^{3} + 15051867372708 T^{4} + 38175029731524752 T^{5} + 55575036078264147712 T^{6} - \)\(29\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!70\)\( T^{8} + \)\(40\!\cdots\!72\)\( T^{9} + \)\(10\!\cdots\!56\)\( T^{10} + \)\(57\!\cdots\!32\)\( T^{11} + \)\(76\!\cdots\!00\)\( T^{12} + \)\(82\!\cdots\!84\)\( p T^{13} + \)\(94\!\cdots\!60\)\( T^{14} + \)\(23\!\cdots\!80\)\( T^{15} - \)\(71\!\cdots\!49\)\( T^{16} + \)\(23\!\cdots\!80\)\( p^{4} T^{17} + \)\(94\!\cdots\!60\)\( p^{8} T^{18} + \)\(82\!\cdots\!84\)\( p^{13} T^{19} + \)\(76\!\cdots\!00\)\( p^{16} T^{20} + \)\(57\!\cdots\!32\)\( p^{20} T^{21} + \)\(10\!\cdots\!56\)\( p^{24} T^{22} + \)\(40\!\cdots\!72\)\( p^{28} T^{23} + \)\(10\!\cdots\!70\)\( p^{32} T^{24} - \)\(29\!\cdots\!40\)\( p^{36} T^{25} + 55575036078264147712 p^{40} T^{26} + 38175029731524752 p^{44} T^{27} + 15051867372708 p^{48} T^{28} + 5159810592 p^{52} T^{29} + 5916800 p^{56} T^{30} + 3440 p^{60} T^{31} + p^{64} T^{32} \) | |
| 43 | \( 1 - 24901612 T^{2} + 309983664307950 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!97\)\( T^{8} - \)\(83\!\cdots\!72\)\( T^{10} + \)\(36\!\cdots\!10\)\( T^{12} - \)\(14\!\cdots\!76\)\( T^{14} + \)\(49\!\cdots\!40\)\( T^{16} - \)\(14\!\cdots\!76\)\( p^{8} T^{18} + \)\(36\!\cdots\!10\)\( p^{16} T^{20} - \)\(83\!\cdots\!72\)\( p^{24} T^{22} + \)\(16\!\cdots\!97\)\( p^{32} T^{24} - \)\(25\!\cdots\!00\)\( p^{40} T^{26} + 309983664307950 p^{48} T^{28} - 24901612 p^{56} T^{30} + p^{64} T^{32} \) | |
| 47 | \( 1 + 1484 T + 1101128 T^{2} - 8571170172 T^{3} + 67218351180012 T^{4} + 197396893555353260 T^{5} + \)\(25\!\cdots\!96\)\( T^{6} - \)\(27\!\cdots\!20\)\( T^{7} + \)\(18\!\cdots\!48\)\( T^{8} + \)\(96\!\cdots\!72\)\( T^{9} + \)\(16\!\cdots\!92\)\( T^{10} + \)\(79\!\cdots\!80\)\( T^{11} + \)\(29\!\cdots\!72\)\( T^{12} + \)\(27\!\cdots\!40\)\( T^{13} + \)\(63\!\cdots\!92\)\( T^{14} + \)\(86\!\cdots\!68\)\( T^{15} + \)\(34\!\cdots\!58\)\( T^{16} + \)\(86\!\cdots\!68\)\( p^{4} T^{17} + \)\(63\!\cdots\!92\)\( p^{8} T^{18} + \)\(27\!\cdots\!40\)\( p^{12} T^{19} + \)\(29\!\cdots\!72\)\( p^{16} T^{20} + \)\(79\!\cdots\!80\)\( p^{20} T^{21} + \)\(16\!\cdots\!92\)\( p^{24} T^{22} + \)\(96\!\cdots\!72\)\( p^{28} T^{23} + \)\(18\!\cdots\!48\)\( p^{32} T^{24} - \)\(27\!\cdots\!20\)\( p^{36} T^{25} + \)\(25\!\cdots\!96\)\( p^{40} T^{26} + 197396893555353260 p^{44} T^{27} + 67218351180012 p^{48} T^{28} - 8571170172 p^{52} T^{29} + 1101128 p^{56} T^{30} + 1484 p^{60} T^{31} + p^{64} T^{32} \) | |
| 53 | \( ( 1 - 3602 T + 54510142 T^{2} - 169091095000 T^{3} + 1347824194996053 T^{4} - 3593979789946993116 T^{5} + \)\(19\!\cdots\!86\)\( T^{6} - \)\(44\!\cdots\!26\)\( T^{7} + \)\(19\!\cdots\!96\)\( T^{8} - \)\(44\!\cdots\!26\)\( p^{4} T^{9} + \)\(19\!\cdots\!86\)\( p^{8} T^{10} - 3593979789946993116 p^{12} T^{11} + 1347824194996053 p^{16} T^{12} - 169091095000 p^{20} T^{13} + 54510142 p^{24} T^{14} - 3602 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 59 | \( 1 - 4840 T + 11712800 T^{2} - 18813732156 T^{3} + 358871166925950 T^{4} - 1617204569870149288 T^{5} + \)\(38\!\cdots\!88\)\( T^{6} + \)\(26\!\cdots\!04\)\( T^{7} + \)\(24\!\cdots\!57\)\( T^{8} - \)\(29\!\cdots\!24\)\( T^{9} + \)\(10\!\cdots\!08\)\( T^{10} + \)\(27\!\cdots\!20\)\( T^{11} - \)\(71\!\cdots\!38\)\( T^{12} - \)\(47\!\cdots\!56\)\( T^{13} + \)\(25\!\cdots\!24\)\( T^{14} - \)\(33\!\cdots\!20\)\( T^{15} - \)\(17\!\cdots\!24\)\( T^{16} - \)\(33\!\cdots\!20\)\( p^{4} T^{17} + \)\(25\!\cdots\!24\)\( p^{8} T^{18} - \)\(47\!\cdots\!56\)\( p^{12} T^{19} - \)\(71\!\cdots\!38\)\( p^{16} T^{20} + \)\(27\!\cdots\!20\)\( p^{20} T^{21} + \)\(10\!\cdots\!08\)\( p^{24} T^{22} - \)\(29\!\cdots\!24\)\( p^{28} T^{23} + \)\(24\!\cdots\!57\)\( p^{32} T^{24} + \)\(26\!\cdots\!04\)\( p^{36} T^{25} + \)\(38\!\cdots\!88\)\( p^{40} T^{26} - 1617204569870149288 p^{44} T^{27} + 358871166925950 p^{48} T^{28} - 18813732156 p^{52} T^{29} + 11712800 p^{56} T^{30} - 4840 p^{60} T^{31} + p^{64} T^{32} \) | |
| 61 | \( ( 1 - 162 T + 50329136 T^{2} - 34462042204 T^{3} + 1451032679422714 T^{4} - 948669137342177966 T^{5} + \)\(29\!\cdots\!40\)\( T^{6} - \)\(17\!\cdots\!78\)\( T^{7} + \)\(46\!\cdots\!19\)\( T^{8} - \)\(17\!\cdots\!78\)\( p^{4} T^{9} + \)\(29\!\cdots\!40\)\( p^{8} T^{10} - 948669137342177966 p^{12} T^{11} + 1451032679422714 p^{16} T^{12} - 34462042204 p^{20} T^{13} + 50329136 p^{24} T^{14} - 162 p^{28} T^{15} + p^{32} T^{16} )^{2} \) | |
| 67 | \( 1 + 2216 T + 2455328 T^{2} + 13868719420 T^{3} + 191560793302474 T^{4} + 898372074234738224 T^{5} + \)\(16\!\cdots\!12\)\( T^{6} + \)\(21\!\cdots\!32\)\( T^{7} + \)\(17\!\cdots\!37\)\( T^{8} + \)\(67\!\cdots\!20\)\( T^{9} + \)\(98\!\cdots\!24\)\( T^{10} - \)\(10\!\cdots\!32\)\( T^{11} - \)\(41\!\cdots\!98\)\( T^{12} - \)\(34\!\cdots\!20\)\( T^{13} - \)\(81\!\cdots\!44\)\( T^{14} - \)\(58\!\cdots\!80\)\( T^{15} - \)\(12\!\cdots\!12\)\( T^{16} - \)\(58\!\cdots\!80\)\( p^{4} T^{17} - \)\(81\!\cdots\!44\)\( p^{8} T^{18} - \)\(34\!\cdots\!20\)\( p^{12} T^{19} - \)\(41\!\cdots\!98\)\( p^{16} T^{20} - \)\(10\!\cdots\!32\)\( p^{20} T^{21} + \)\(98\!\cdots\!24\)\( p^{24} T^{22} + \)\(67\!\cdots\!20\)\( p^{28} T^{23} + \)\(17\!\cdots\!37\)\( p^{32} T^{24} + \)\(21\!\cdots\!32\)\( p^{36} T^{25} + \)\(16\!\cdots\!12\)\( p^{40} T^{26} + 898372074234738224 p^{44} T^{27} + 191560793302474 p^{48} T^{28} + 13868719420 p^{52} T^{29} + 2455328 p^{56} T^{30} + 2216 p^{60} T^{31} + p^{64} T^{32} \) | |
| 71 | \( 1 - 1240 T + 768800 T^{2} + 2105529156 T^{3} + 1084959599367594 T^{4} - 2190189464483365936 T^{5} + \)\(18\!\cdots\!08\)\( T^{6} - \)\(27\!\cdots\!56\)\( T^{7} + \)\(12\!\cdots\!45\)\( T^{8} - \)\(23\!\cdots\!64\)\( T^{9} + \)\(49\!\cdots\!20\)\( T^{10} - \)\(14\!\cdots\!52\)\( T^{11} - \)\(45\!\cdots\!94\)\( T^{12} + \)\(13\!\cdots\!36\)\( T^{13} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(34\!\cdots\!44\)\( p T^{15} - \)\(82\!\cdots\!68\)\( p^{2} T^{16} + \)\(34\!\cdots\!44\)\( p^{5} T^{17} - \)\(12\!\cdots\!20\)\( p^{8} T^{18} + \)\(13\!\cdots\!36\)\( p^{12} T^{19} - \)\(45\!\cdots\!94\)\( p^{16} T^{20} - \)\(14\!\cdots\!52\)\( p^{20} T^{21} + \)\(49\!\cdots\!20\)\( p^{24} T^{22} - \)\(23\!\cdots\!64\)\( p^{28} T^{23} + \)\(12\!\cdots\!45\)\( p^{32} T^{24} - \)\(27\!\cdots\!56\)\( p^{36} T^{25} + \)\(18\!\cdots\!08\)\( p^{40} T^{26} - 2190189464483365936 p^{44} T^{27} + 1084959599367594 p^{48} T^{28} + 2105529156 p^{52} T^{29} + 768800 p^{56} T^{30} - 1240 p^{60} T^{31} + p^{64} T^{32} \) | |
| 73 | \( 1 - 15448 T + 119320352 T^{2} - 673174693736 T^{3} + 2488100657363446 T^{4} - 439794287857885936 T^{5} - \)\(63\!\cdots\!16\)\( T^{6} + \)\(62\!\cdots\!20\)\( T^{7} - \)\(47\!\cdots\!31\)\( T^{8} + \)\(27\!\cdots\!12\)\( T^{9} - \)\(13\!\cdots\!00\)\( T^{10} + \)\(44\!\cdots\!16\)\( T^{11} - \)\(23\!\cdots\!62\)\( T^{12} - \)\(97\!\cdots\!12\)\( T^{13} + \)\(86\!\cdots\!96\)\( T^{14} - \)\(55\!\cdots\!96\)\( T^{15} + \)\(31\!\cdots\!12\)\( T^{16} - \)\(55\!\cdots\!96\)\( p^{4} T^{17} + \)\(86\!\cdots\!96\)\( p^{8} T^{18} - \)\(97\!\cdots\!12\)\( p^{12} T^{19} - \)\(23\!\cdots\!62\)\( p^{16} T^{20} + \)\(44\!\cdots\!16\)\( p^{20} T^{21} - \)\(13\!\cdots\!00\)\( p^{24} T^{22} + \)\(27\!\cdots\!12\)\( p^{28} T^{23} - \)\(47\!\cdots\!31\)\( p^{32} T^{24} + \)\(62\!\cdots\!20\)\( p^{36} T^{25} - \)\(63\!\cdots\!16\)\( p^{40} T^{26} - 439794287857885936 p^{44} T^{27} + 2488100657363446 p^{48} T^{28} - 673174693736 p^{52} T^{29} + 119320352 p^{56} T^{30} - 15448 p^{60} T^{31} + p^{64} T^{32} \) | |
| 79 | \( ( 1 + 108 p T + 181789616 T^{2} + 1264697206412 T^{3} + 17222120260071052 T^{4} + 98573552246045188900 T^{5} + \)\(10\!\cdots\!52\)\( T^{6} + \)\(51\!\cdots\!68\)\( T^{7} + \)\(46\!\cdots\!78\)\( T^{8} + \)\(51\!\cdots\!68\)\( p^{4} T^{9} + \)\(10\!\cdots\!52\)\( p^{8} T^{10} + 98573552246045188900 p^{12} T^{11} + 17222120260071052 p^{16} T^{12} + 1264697206412 p^{20} T^{13} + 181789616 p^{24} T^{14} + 108 p^{29} T^{15} + p^{32} T^{16} )^{2} \) | |
| 83 | \( 1 + 12788 T + 81766472 T^{2} + 219288483636 T^{3} + 2030465335614048 T^{4} + 45830993951572452908 T^{5} + \)\(44\!\cdots\!96\)\( T^{6} + \)\(25\!\cdots\!88\)\( T^{7} + \)\(16\!\cdots\!16\)\( T^{8} + \)\(16\!\cdots\!32\)\( T^{9} + \)\(14\!\cdots\!56\)\( T^{10} + \)\(91\!\cdots\!60\)\( T^{11} + \)\(55\!\cdots\!64\)\( T^{12} + \)\(46\!\cdots\!64\)\( T^{13} + \)\(38\!\cdots\!88\)\( T^{14} + \)\(25\!\cdots\!16\)\( T^{15} + \)\(16\!\cdots\!58\)\( T^{16} + \)\(25\!\cdots\!16\)\( p^{4} T^{17} + \)\(38\!\cdots\!88\)\( p^{8} T^{18} + \)\(46\!\cdots\!64\)\( p^{12} T^{19} + \)\(55\!\cdots\!64\)\( p^{16} T^{20} + \)\(91\!\cdots\!60\)\( p^{20} T^{21} + \)\(14\!\cdots\!56\)\( p^{24} T^{22} + \)\(16\!\cdots\!32\)\( p^{28} T^{23} + \)\(16\!\cdots\!16\)\( p^{32} T^{24} + \)\(25\!\cdots\!88\)\( p^{36} T^{25} + \)\(44\!\cdots\!96\)\( p^{40} T^{26} + 45830993951572452908 p^{44} T^{27} + 2030465335614048 p^{48} T^{28} + 219288483636 p^{52} T^{29} + 81766472 p^{56} T^{30} + 12788 p^{60} T^{31} + p^{64} T^{32} \) | |
| 89 | \( 1 - 37624 T + 707782688 T^{2} - 9177330980724 T^{3} + 88357900910727246 T^{4} - \)\(58\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!20\)\( T^{6} + \)\(24\!\cdots\!92\)\( T^{7} - \)\(53\!\cdots\!27\)\( T^{8} + \)\(67\!\cdots\!68\)\( T^{9} - \)\(63\!\cdots\!84\)\( T^{10} + \)\(44\!\cdots\!76\)\( T^{11} - \)\(19\!\cdots\!22\)\( T^{12} - \)\(37\!\cdots\!24\)\( T^{13} + \)\(18\!\cdots\!64\)\( T^{14} - \)\(23\!\cdots\!92\)\( T^{15} + \)\(20\!\cdots\!28\)\( T^{16} - \)\(23\!\cdots\!92\)\( p^{4} T^{17} + \)\(18\!\cdots\!64\)\( p^{8} T^{18} - \)\(37\!\cdots\!24\)\( p^{12} T^{19} - \)\(19\!\cdots\!22\)\( p^{16} T^{20} + \)\(44\!\cdots\!76\)\( p^{20} T^{21} - \)\(63\!\cdots\!84\)\( p^{24} T^{22} + \)\(67\!\cdots\!68\)\( p^{28} T^{23} - \)\(53\!\cdots\!27\)\( p^{32} T^{24} + \)\(24\!\cdots\!92\)\( p^{36} T^{25} + \)\(15\!\cdots\!20\)\( p^{40} T^{26} - \)\(58\!\cdots\!20\)\( p^{44} T^{27} + 88357900910727246 p^{48} T^{28} - 9177330980724 p^{52} T^{29} + 707782688 p^{56} T^{30} - 37624 p^{60} T^{31} + p^{64} T^{32} \) | |
| 97 | \( 1 - 57056 T + 1627693568 T^{2} - 32338516715900 T^{3} + 525585002175249918 T^{4} - \)\(75\!\cdots\!92\)\( T^{5} + \)\(99\!\cdots\!28\)\( T^{6} - \)\(12\!\cdots\!96\)\( T^{7} + \)\(13\!\cdots\!77\)\( T^{8} - \)\(15\!\cdots\!64\)\( T^{9} + \)\(16\!\cdots\!76\)\( T^{10} - \)\(16\!\cdots\!16\)\( T^{11} + \)\(16\!\cdots\!10\)\( T^{12} - \)\(16\!\cdots\!96\)\( T^{13} + \)\(15\!\cdots\!68\)\( T^{14} - \)\(15\!\cdots\!40\)\( T^{15} + \)\(14\!\cdots\!64\)\( T^{16} - \)\(15\!\cdots\!40\)\( p^{4} T^{17} + \)\(15\!\cdots\!68\)\( p^{8} T^{18} - \)\(16\!\cdots\!96\)\( p^{12} T^{19} + \)\(16\!\cdots\!10\)\( p^{16} T^{20} - \)\(16\!\cdots\!16\)\( p^{20} T^{21} + \)\(16\!\cdots\!76\)\( p^{24} T^{22} - \)\(15\!\cdots\!64\)\( p^{28} T^{23} + \)\(13\!\cdots\!77\)\( p^{32} T^{24} - \)\(12\!\cdots\!96\)\( p^{36} T^{25} + \)\(99\!\cdots\!28\)\( p^{40} T^{26} - \)\(75\!\cdots\!92\)\( p^{44} T^{27} + 525585002175249918 p^{48} T^{28} - 32338516715900 p^{52} T^{29} + 1627693568 p^{56} T^{30} - 57056 p^{60} T^{31} + p^{64} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.82108240708696350173111135888, −2.79526703971144864540146254051, −2.76807708148264439928347702392, −2.61526104781184714932099957529, −2.45387039334492293080470674119, −2.30871076451037160344856366767, −2.24472420633326292840605989939, −2.19898452838592702078525697314, −2.13694744208956588209170342767, −2.12284147420633370927446007518, −2.02886800852990758200194306907, −1.83420991775710350080388742179, −1.77213128742932670471679281495, −1.68744934118681677849763308474, −1.37450243099177892501179524803, −1.24277057282067360483056850211, −1.12137147653460521820460411585, −0.853584963507311698085140849774, −0.66536196477165957923388554697, −0.40117636097651519115394245076, −0.34026075807082038374445608633, −0.21447923459336284150992748856, −0.17770356813806043236678945753, −0.17086544505895526218240294820, −0.15506775434369160441470462788, 0.15506775434369160441470462788, 0.17086544505895526218240294820, 0.17770356813806043236678945753, 0.21447923459336284150992748856, 0.34026075807082038374445608633, 0.40117636097651519115394245076, 0.66536196477165957923388554697, 0.853584963507311698085140849774, 1.12137147653460521820460411585, 1.24277057282067360483056850211, 1.37450243099177892501179524803, 1.68744934118681677849763308474, 1.77213128742932670471679281495, 1.83420991775710350080388742179, 2.02886800852990758200194306907, 2.12284147420633370927446007518, 2.13694744208956588209170342767, 2.19898452838592702078525697314, 2.24472420633326292840605989939, 2.30871076451037160344856366767, 2.45387039334492293080470674119, 2.61526104781184714932099957529, 2.76807708148264439928347702392, 2.79526703971144864540146254051, 2.82108240708696350173111135888
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.