Dirichlet series
| L(s) = 1 | + 15·2-s − 20·3-s − 153·4-s − 300·6-s − 720·7-s − 2.11e3·8-s − 9.51e3·9-s − 944·11-s + 3.06e3·12-s − 9.43e3·13-s − 1.08e4·14-s + 1.87e4·16-s − 1.11e4·17-s − 1.42e5·18-s − 8.23e4·19-s + 1.44e4·21-s − 1.41e4·22-s − 3.31e4·23-s + 4.23e4·24-s − 1.41e5·26-s + 3.95e5·27-s + 1.10e5·28-s + 2.05e5·29-s + 2.36e4·31-s − 2.75e4·32-s + 1.88e4·33-s − 1.67e5·34-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.427·3-s − 1.19·4-s − 0.567·6-s − 0.793·7-s − 1.46·8-s − 4.35·9-s − 0.213·11-s + 0.511·12-s − 1.19·13-s − 1.05·14-s + 1.14·16-s − 0.552·17-s − 5.76·18-s − 2.75·19-s + 0.339·21-s − 0.283·22-s − 0.567·23-s + 0.624·24-s − 1.57·26-s + 3.86·27-s + 0.948·28-s + 1.56·29-s + 0.142·31-s − 0.148·32-s + 0.0914·33-s − 0.732·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{24} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(5^{24} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.13921\times 10^{26}\) |
| Root analytic conductor: | \(12.1812\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 5^{24} \cdot 19^{12} ,\ ( \ : [7/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(13.41016736\) |
| \(L(\frac12)\) | \(\approx\) | \(13.41016736\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 19 | \( ( 1 + p^{3} T )^{12} \) | |
| good | 2 | \( 1 - 15 T + 189 p T^{2} - 2925 p T^{3} + 95077 T^{4} - 1212465 T^{5} + 8497539 p T^{6} - 24084505 p^{3} T^{7} + 145676143 p^{4} T^{8} - 1456837525 p^{4} T^{9} + 8614477119 p^{5} T^{10} - 5518878545 p^{9} T^{11} + 123510062423 p^{8} T^{12} - 5518878545 p^{16} T^{13} + 8614477119 p^{19} T^{14} - 1456837525 p^{25} T^{15} + 145676143 p^{32} T^{16} - 24084505 p^{38} T^{17} + 8497539 p^{43} T^{18} - 1212465 p^{49} T^{19} + 95077 p^{56} T^{20} - 2925 p^{64} T^{21} + 189 p^{71} T^{22} - 15 p^{77} T^{23} + p^{84} T^{24} \) |
| 3 | \( 1 + 20 T + 9917 T^{2} - 6350 T^{3} + 43197928 T^{4} - 1126256090 T^{5} + 121508053753 T^{6} - 2105919469900 p T^{7} + 38768227686751 p^{2} T^{8} - 627675692659160 p^{3} T^{9} + 4679619288309574 p^{5} T^{10} - 14233634473596140 p^{7} T^{11} + 4095254722940591296 p^{6} T^{12} - 14233634473596140 p^{14} T^{13} + 4679619288309574 p^{19} T^{14} - 627675692659160 p^{24} T^{15} + 38768227686751 p^{30} T^{16} - 2105919469900 p^{36} T^{17} + 121508053753 p^{42} T^{18} - 1126256090 p^{49} T^{19} + 43197928 p^{56} T^{20} - 6350 p^{63} T^{21} + 9917 p^{70} T^{22} + 20 p^{77} T^{23} + p^{84} T^{24} \) | |
| 7 | \( 1 + 720 T + 2357087 T^{2} + 2127079040 T^{3} + 2884242066112 T^{4} + 2473490327990240 T^{5} + 50978725822254611 p^{2} T^{6} + \)\(15\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!83\)\( T^{8} + \)\(46\!\cdots\!00\)\( T^{9} + \)\(43\!\cdots\!62\)\( T^{10} - \)\(79\!\cdots\!20\)\( T^{11} + \)\(26\!\cdots\!08\)\( T^{12} - \)\(79\!\cdots\!20\)\( p^{7} T^{13} + \)\(43\!\cdots\!62\)\( p^{14} T^{14} + \)\(46\!\cdots\!00\)\( p^{21} T^{15} + \)\(14\!\cdots\!83\)\( p^{28} T^{16} + \)\(15\!\cdots\!60\)\( p^{35} T^{17} + 50978725822254611 p^{44} T^{18} + 2473490327990240 p^{49} T^{19} + 2884242066112 p^{56} T^{20} + 2127079040 p^{63} T^{21} + 2357087 p^{70} T^{22} + 720 p^{77} T^{23} + p^{84} T^{24} \) | |
| 11 | \( 1 + 944 T + 98264444 T^{2} + 98488059760 T^{3} + 5503625662914530 T^{4} + 5057098141828864464 T^{5} + \)\(21\!\cdots\!48\)\( T^{6} + \)\(17\!\cdots\!44\)\( T^{7} + \)\(68\!\cdots\!23\)\( T^{8} + \)\(47\!\cdots\!60\)\( T^{9} + \)\(17\!\cdots\!44\)\( T^{10} + \)\(10\!\cdots\!44\)\( T^{11} + \)\(36\!\cdots\!68\)\( T^{12} + \)\(10\!\cdots\!44\)\( p^{7} T^{13} + \)\(17\!\cdots\!44\)\( p^{14} T^{14} + \)\(47\!\cdots\!60\)\( p^{21} T^{15} + \)\(68\!\cdots\!23\)\( p^{28} T^{16} + \)\(17\!\cdots\!44\)\( p^{35} T^{17} + \)\(21\!\cdots\!48\)\( p^{42} T^{18} + 5057098141828864464 p^{49} T^{19} + 5503625662914530 p^{56} T^{20} + 98488059760 p^{63} T^{21} + 98264444 p^{70} T^{22} + 944 p^{77} T^{23} + p^{84} T^{24} \) | |
| 13 | \( 1 + 9430 T + 423059267 T^{2} + 4129197841140 T^{3} + 90698492412802520 T^{4} + \)\(87\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!23\)\( T^{6} + \)\(12\!\cdots\!30\)\( T^{7} + \)\(14\!\cdots\!39\)\( T^{8} + \)\(93\!\cdots\!40\)\( p T^{9} + \)\(12\!\cdots\!42\)\( T^{10} + \)\(95\!\cdots\!80\)\( T^{11} + \)\(87\!\cdots\!40\)\( T^{12} + \)\(95\!\cdots\!80\)\( p^{7} T^{13} + \)\(12\!\cdots\!42\)\( p^{14} T^{14} + \)\(93\!\cdots\!40\)\( p^{22} T^{15} + \)\(14\!\cdots\!39\)\( p^{28} T^{16} + \)\(12\!\cdots\!30\)\( p^{35} T^{17} + \)\(13\!\cdots\!23\)\( p^{42} T^{18} + \)\(87\!\cdots\!40\)\( p^{49} T^{19} + 90698492412802520 p^{56} T^{20} + 4129197841140 p^{63} T^{21} + 423059267 p^{70} T^{22} + 9430 p^{77} T^{23} + p^{84} T^{24} \) | |
| 17 | \( 1 + 11190 T + 1084595661 T^{2} - 7262583912310 T^{3} + 409541267391400408 T^{4} - \)\(11\!\cdots\!50\)\( T^{5} + \)\(22\!\cdots\!97\)\( T^{6} - \)\(52\!\cdots\!30\)\( T^{7} + \)\(10\!\cdots\!75\)\( T^{8} - \)\(24\!\cdots\!40\)\( T^{9} + \)\(48\!\cdots\!78\)\( T^{10} - \)\(62\!\cdots\!00\)\( T^{11} + \)\(27\!\cdots\!92\)\( T^{12} - \)\(62\!\cdots\!00\)\( p^{7} T^{13} + \)\(48\!\cdots\!78\)\( p^{14} T^{14} - \)\(24\!\cdots\!40\)\( p^{21} T^{15} + \)\(10\!\cdots\!75\)\( p^{28} T^{16} - \)\(52\!\cdots\!30\)\( p^{35} T^{17} + \)\(22\!\cdots\!97\)\( p^{42} T^{18} - \)\(11\!\cdots\!50\)\( p^{49} T^{19} + 409541267391400408 p^{56} T^{20} - 7262583912310 p^{63} T^{21} + 1084595661 p^{70} T^{22} + 11190 p^{77} T^{23} + p^{84} T^{24} \) | |
| 23 | \( 1 + 33140 T + 14261802087 T^{2} - 104809922073320 T^{3} + 87187883427576824216 T^{4} - \)\(19\!\cdots\!20\)\( p T^{5} + \)\(43\!\cdots\!35\)\( T^{6} - \)\(34\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!45\)\( p T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!26\)\( T^{10} - \)\(58\!\cdots\!20\)\( T^{11} + \)\(47\!\cdots\!96\)\( T^{12} - \)\(58\!\cdots\!20\)\( p^{7} T^{13} + \)\(12\!\cdots\!26\)\( p^{14} T^{14} - \)\(15\!\cdots\!00\)\( p^{21} T^{15} + \)\(10\!\cdots\!45\)\( p^{29} T^{16} - \)\(34\!\cdots\!20\)\( p^{35} T^{17} + \)\(43\!\cdots\!35\)\( p^{42} T^{18} - \)\(19\!\cdots\!20\)\( p^{50} T^{19} + 87187883427576824216 p^{56} T^{20} - 104809922073320 p^{63} T^{21} + 14261802087 p^{70} T^{22} + 33140 p^{77} T^{23} + p^{84} T^{24} \) | |
| 29 | \( 1 - 205284 T + 103668561907 T^{2} - 653208161191828 p T^{3} + \)\(61\!\cdots\!00\)\( T^{4} - \)\(99\!\cdots\!56\)\( T^{5} + \)\(24\!\cdots\!19\)\( T^{6} - \)\(36\!\cdots\!72\)\( T^{7} + \)\(75\!\cdots\!95\)\( T^{8} - \)\(99\!\cdots\!92\)\( T^{9} + \)\(18\!\cdots\!42\)\( T^{10} - \)\(21\!\cdots\!20\)\( T^{11} + \)\(34\!\cdots\!72\)\( T^{12} - \)\(21\!\cdots\!20\)\( p^{7} T^{13} + \)\(18\!\cdots\!42\)\( p^{14} T^{14} - \)\(99\!\cdots\!92\)\( p^{21} T^{15} + \)\(75\!\cdots\!95\)\( p^{28} T^{16} - \)\(36\!\cdots\!72\)\( p^{35} T^{17} + \)\(24\!\cdots\!19\)\( p^{42} T^{18} - \)\(99\!\cdots\!56\)\( p^{49} T^{19} + \)\(61\!\cdots\!00\)\( p^{56} T^{20} - 653208161191828 p^{64} T^{21} + 103668561907 p^{70} T^{22} - 205284 p^{77} T^{23} + p^{84} T^{24} \) | |
| 31 | \( 1 - 23648 T + 199578121880 T^{2} - 188227222579024 T^{3} + \)\(18\!\cdots\!86\)\( T^{4} + \)\(34\!\cdots\!04\)\( T^{5} + \)\(11\!\cdots\!96\)\( T^{6} + \)\(35\!\cdots\!40\)\( T^{7} + \)\(52\!\cdots\!95\)\( T^{8} + \)\(17\!\cdots\!72\)\( T^{9} + \)\(19\!\cdots\!16\)\( T^{10} + \)\(61\!\cdots\!40\)\( T^{11} + \)\(58\!\cdots\!04\)\( T^{12} + \)\(61\!\cdots\!40\)\( p^{7} T^{13} + \)\(19\!\cdots\!16\)\( p^{14} T^{14} + \)\(17\!\cdots\!72\)\( p^{21} T^{15} + \)\(52\!\cdots\!95\)\( p^{28} T^{16} + \)\(35\!\cdots\!40\)\( p^{35} T^{17} + \)\(11\!\cdots\!96\)\( p^{42} T^{18} + \)\(34\!\cdots\!04\)\( p^{49} T^{19} + \)\(18\!\cdots\!86\)\( p^{56} T^{20} - 188227222579024 p^{63} T^{21} + 199578121880 p^{70} T^{22} - 23648 p^{77} T^{23} + p^{84} T^{24} \) | |
| 37 | \( 1 - 513730 T + 651306228064 T^{2} - 238598337634255990 T^{3} + \)\(19\!\cdots\!46\)\( T^{4} - \)\(55\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!32\)\( p T^{6} - \)\(91\!\cdots\!50\)\( T^{7} + \)\(55\!\cdots\!19\)\( T^{8} - \)\(12\!\cdots\!60\)\( T^{9} + \)\(69\!\cdots\!84\)\( T^{10} - \)\(14\!\cdots\!60\)\( T^{11} + \)\(72\!\cdots\!48\)\( T^{12} - \)\(14\!\cdots\!60\)\( p^{7} T^{13} + \)\(69\!\cdots\!84\)\( p^{14} T^{14} - \)\(12\!\cdots\!60\)\( p^{21} T^{15} + \)\(55\!\cdots\!19\)\( p^{28} T^{16} - \)\(91\!\cdots\!50\)\( p^{35} T^{17} + \)\(10\!\cdots\!32\)\( p^{43} T^{18} - \)\(55\!\cdots\!30\)\( p^{49} T^{19} + \)\(19\!\cdots\!46\)\( p^{56} T^{20} - 238598337634255990 p^{63} T^{21} + 651306228064 p^{70} T^{22} - 513730 p^{77} T^{23} + p^{84} T^{24} \) | |
| 41 | \( 1 - 1745968 T + 2634189997684 T^{2} - 2588894315721040752 T^{3} + \)\(23\!\cdots\!70\)\( T^{4} - \)\(16\!\cdots\!36\)\( T^{5} + \)\(11\!\cdots\!24\)\( T^{6} - \)\(68\!\cdots\!72\)\( T^{7} + \)\(39\!\cdots\!75\)\( T^{8} - \)\(20\!\cdots\!96\)\( T^{9} + \)\(10\!\cdots\!80\)\( T^{10} - \)\(48\!\cdots\!80\)\( T^{11} + \)\(22\!\cdots\!40\)\( T^{12} - \)\(48\!\cdots\!80\)\( p^{7} T^{13} + \)\(10\!\cdots\!80\)\( p^{14} T^{14} - \)\(20\!\cdots\!96\)\( p^{21} T^{15} + \)\(39\!\cdots\!75\)\( p^{28} T^{16} - \)\(68\!\cdots\!72\)\( p^{35} T^{17} + \)\(11\!\cdots\!24\)\( p^{42} T^{18} - \)\(16\!\cdots\!36\)\( p^{49} T^{19} + \)\(23\!\cdots\!70\)\( p^{56} T^{20} - 2588894315721040752 p^{63} T^{21} + 2634189997684 p^{70} T^{22} - 1745968 p^{77} T^{23} + p^{84} T^{24} \) | |
| 43 | \( 1 + 387120 T + 1350717726452 T^{2} + 649528234897494480 T^{3} + \)\(95\!\cdots\!06\)\( T^{4} + \)\(45\!\cdots\!40\)\( T^{5} + \)\(45\!\cdots\!44\)\( T^{6} + \)\(18\!\cdots\!00\)\( T^{7} + \)\(15\!\cdots\!63\)\( T^{8} + \)\(53\!\cdots\!20\)\( T^{9} + \)\(44\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!80\)\( T^{11} + \)\(11\!\cdots\!60\)\( T^{12} + \)\(12\!\cdots\!80\)\( p^{7} T^{13} + \)\(44\!\cdots\!24\)\( p^{14} T^{14} + \)\(53\!\cdots\!20\)\( p^{21} T^{15} + \)\(15\!\cdots\!63\)\( p^{28} T^{16} + \)\(18\!\cdots\!00\)\( p^{35} T^{17} + \)\(45\!\cdots\!44\)\( p^{42} T^{18} + \)\(45\!\cdots\!40\)\( p^{49} T^{19} + \)\(95\!\cdots\!06\)\( p^{56} T^{20} + 649528234897494480 p^{63} T^{21} + 1350717726452 p^{70} T^{22} + 387120 p^{77} T^{23} + p^{84} T^{24} \) | |
| 47 | \( 1 + 363480 T + 3178279603400 T^{2} + 1605012653538918200 T^{3} + \)\(50\!\cdots\!98\)\( T^{4} + \)\(32\!\cdots\!60\)\( T^{5} + \)\(53\!\cdots\!96\)\( T^{6} + \)\(41\!\cdots\!00\)\( T^{7} + \)\(43\!\cdots\!87\)\( T^{8} + \)\(37\!\cdots\!00\)\( T^{9} + \)\(60\!\cdots\!40\)\( p T^{10} + \)\(24\!\cdots\!00\)\( T^{11} + \)\(15\!\cdots\!28\)\( T^{12} + \)\(24\!\cdots\!00\)\( p^{7} T^{13} + \)\(60\!\cdots\!40\)\( p^{15} T^{14} + \)\(37\!\cdots\!00\)\( p^{21} T^{15} + \)\(43\!\cdots\!87\)\( p^{28} T^{16} + \)\(41\!\cdots\!00\)\( p^{35} T^{17} + \)\(53\!\cdots\!96\)\( p^{42} T^{18} + \)\(32\!\cdots\!60\)\( p^{49} T^{19} + \)\(50\!\cdots\!98\)\( p^{56} T^{20} + 1605012653538918200 p^{63} T^{21} + 3178279603400 p^{70} T^{22} + 363480 p^{77} T^{23} + p^{84} T^{24} \) | |
| 53 | \( 1 + 12850 T + 8417157465699 T^{2} - 157275430152955580 T^{3} + \)\(36\!\cdots\!96\)\( T^{4} - \)\(11\!\cdots\!80\)\( T^{5} + \)\(10\!\cdots\!31\)\( T^{6} - \)\(40\!\cdots\!10\)\( T^{7} + \)\(23\!\cdots\!99\)\( T^{8} - \)\(89\!\cdots\!20\)\( T^{9} + \)\(38\!\cdots\!18\)\( T^{10} - \)\(14\!\cdots\!00\)\( T^{11} + \)\(50\!\cdots\!68\)\( T^{12} - \)\(14\!\cdots\!00\)\( p^{7} T^{13} + \)\(38\!\cdots\!18\)\( p^{14} T^{14} - \)\(89\!\cdots\!20\)\( p^{21} T^{15} + \)\(23\!\cdots\!99\)\( p^{28} T^{16} - \)\(40\!\cdots\!10\)\( p^{35} T^{17} + \)\(10\!\cdots\!31\)\( p^{42} T^{18} - \)\(11\!\cdots\!80\)\( p^{49} T^{19} + \)\(36\!\cdots\!96\)\( p^{56} T^{20} - 157275430152955580 p^{63} T^{21} + 8417157465699 p^{70} T^{22} + 12850 p^{77} T^{23} + p^{84} T^{24} \) | |
| 59 | \( 1 - 3619470 T + 20896480282093 T^{2} - 54617521372664567366 T^{3} + \)\(19\!\cdots\!08\)\( T^{4} - \)\(39\!\cdots\!18\)\( T^{5} + \)\(18\!\cdots\!07\)\( p T^{6} - \)\(19\!\cdots\!38\)\( T^{7} + \)\(45\!\cdots\!59\)\( T^{8} - \)\(73\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!94\)\( T^{10} - \)\(22\!\cdots\!20\)\( T^{11} + \)\(41\!\cdots\!24\)\( T^{12} - \)\(22\!\cdots\!20\)\( p^{7} T^{13} + \)\(15\!\cdots\!94\)\( p^{14} T^{14} - \)\(73\!\cdots\!80\)\( p^{21} T^{15} + \)\(45\!\cdots\!59\)\( p^{28} T^{16} - \)\(19\!\cdots\!38\)\( p^{35} T^{17} + \)\(18\!\cdots\!07\)\( p^{43} T^{18} - \)\(39\!\cdots\!18\)\( p^{49} T^{19} + \)\(19\!\cdots\!08\)\( p^{56} T^{20} - 54617521372664567366 p^{63} T^{21} + 20896480282093 p^{70} T^{22} - 3619470 p^{77} T^{23} + p^{84} T^{24} \) | |
| 61 | \( 1 - 574072 T + 23384124289160 T^{2} - 21251529336575012424 T^{3} + \)\(26\!\cdots\!58\)\( T^{4} - \)\(30\!\cdots\!32\)\( T^{5} + \)\(20\!\cdots\!16\)\( T^{6} - \)\(25\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!31\)\( T^{8} - \)\(14\!\cdots\!36\)\( T^{9} + \)\(52\!\cdots\!88\)\( T^{10} - \)\(59\!\cdots\!12\)\( T^{11} + \)\(18\!\cdots\!52\)\( T^{12} - \)\(59\!\cdots\!12\)\( p^{7} T^{13} + \)\(52\!\cdots\!88\)\( p^{14} T^{14} - \)\(14\!\cdots\!36\)\( p^{21} T^{15} + \)\(11\!\cdots\!31\)\( p^{28} T^{16} - \)\(25\!\cdots\!04\)\( p^{35} T^{17} + \)\(20\!\cdots\!16\)\( p^{42} T^{18} - \)\(30\!\cdots\!32\)\( p^{49} T^{19} + \)\(26\!\cdots\!58\)\( p^{56} T^{20} - 21251529336575012424 p^{63} T^{21} + 23384124289160 p^{70} T^{22} - 574072 p^{77} T^{23} + p^{84} T^{24} \) | |
| 67 | \( 1 + 2892800 T + 32875950307397 T^{2} + \)\(10\!\cdots\!10\)\( T^{3} + \)\(62\!\cdots\!52\)\( T^{4} + \)\(20\!\cdots\!50\)\( T^{5} + \)\(85\!\cdots\!05\)\( T^{6} + \)\(26\!\cdots\!00\)\( T^{7} + \)\(90\!\cdots\!03\)\( T^{8} + \)\(25\!\cdots\!60\)\( T^{9} + \)\(76\!\cdots\!46\)\( T^{10} + \)\(19\!\cdots\!80\)\( T^{11} + \)\(51\!\cdots\!28\)\( T^{12} + \)\(19\!\cdots\!80\)\( p^{7} T^{13} + \)\(76\!\cdots\!46\)\( p^{14} T^{14} + \)\(25\!\cdots\!60\)\( p^{21} T^{15} + \)\(90\!\cdots\!03\)\( p^{28} T^{16} + \)\(26\!\cdots\!00\)\( p^{35} T^{17} + \)\(85\!\cdots\!05\)\( p^{42} T^{18} + \)\(20\!\cdots\!50\)\( p^{49} T^{19} + \)\(62\!\cdots\!52\)\( p^{56} T^{20} + \)\(10\!\cdots\!10\)\( p^{63} T^{21} + 32875950307397 p^{70} T^{22} + 2892800 p^{77} T^{23} + p^{84} T^{24} \) | |
| 71 | \( 1 + 1390956 T + 37438997963588 T^{2} + \)\(10\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!18\)\( T^{4} + \)\(27\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(48\!\cdots\!48\)\( T^{7} + \)\(19\!\cdots\!67\)\( T^{8} + \)\(63\!\cdots\!84\)\( T^{9} + \)\(22\!\cdots\!28\)\( T^{10} + \)\(68\!\cdots\!08\)\( T^{11} + \)\(21\!\cdots\!84\)\( T^{12} + \)\(68\!\cdots\!08\)\( p^{7} T^{13} + \)\(22\!\cdots\!28\)\( p^{14} T^{14} + \)\(63\!\cdots\!84\)\( p^{21} T^{15} + \)\(19\!\cdots\!67\)\( p^{28} T^{16} + \)\(48\!\cdots\!48\)\( p^{35} T^{17} + \)\(13\!\cdots\!80\)\( p^{42} T^{18} + \)\(27\!\cdots\!40\)\( p^{49} T^{19} + \)\(82\!\cdots\!18\)\( p^{56} T^{20} + \)\(10\!\cdots\!20\)\( p^{63} T^{21} + 37438997963588 p^{70} T^{22} + 1390956 p^{77} T^{23} + p^{84} T^{24} \) | |
| 73 | \( 1 + 1894210 T + 48671501936029 T^{2} + 54272526371483906310 T^{3} + \)\(14\!\cdots\!56\)\( T^{4} + \)\(13\!\cdots\!50\)\( T^{5} + \)\(31\!\cdots\!81\)\( T^{6} + \)\(23\!\cdots\!70\)\( T^{7} + \)\(55\!\cdots\!99\)\( T^{8} + \)\(34\!\cdots\!40\)\( T^{9} + \)\(79\!\cdots\!78\)\( T^{10} + \)\(43\!\cdots\!80\)\( T^{11} + \)\(95\!\cdots\!48\)\( T^{12} + \)\(43\!\cdots\!80\)\( p^{7} T^{13} + \)\(79\!\cdots\!78\)\( p^{14} T^{14} + \)\(34\!\cdots\!40\)\( p^{21} T^{15} + \)\(55\!\cdots\!99\)\( p^{28} T^{16} + \)\(23\!\cdots\!70\)\( p^{35} T^{17} + \)\(31\!\cdots\!81\)\( p^{42} T^{18} + \)\(13\!\cdots\!50\)\( p^{49} T^{19} + \)\(14\!\cdots\!56\)\( p^{56} T^{20} + 54272526371483906310 p^{63} T^{21} + 48671501936029 p^{70} T^{22} + 1894210 p^{77} T^{23} + p^{84} T^{24} \) | |
| 79 | \( 1 - 13253424 T + 215881007909472 T^{2} - \)\(19\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!94\)\( T^{4} - \)\(12\!\cdots\!80\)\( T^{5} + \)\(95\!\cdots\!28\)\( T^{6} - \)\(55\!\cdots\!32\)\( T^{7} + \)\(34\!\cdots\!35\)\( T^{8} - \)\(17\!\cdots\!88\)\( T^{9} + \)\(95\!\cdots\!24\)\( T^{10} - \)\(42\!\cdots\!36\)\( T^{11} + \)\(20\!\cdots\!92\)\( T^{12} - \)\(42\!\cdots\!36\)\( p^{7} T^{13} + \)\(95\!\cdots\!24\)\( p^{14} T^{14} - \)\(17\!\cdots\!88\)\( p^{21} T^{15} + \)\(34\!\cdots\!35\)\( p^{28} T^{16} - \)\(55\!\cdots\!32\)\( p^{35} T^{17} + \)\(95\!\cdots\!28\)\( p^{42} T^{18} - \)\(12\!\cdots\!80\)\( p^{49} T^{19} + \)\(18\!\cdots\!94\)\( p^{56} T^{20} - \)\(19\!\cdots\!84\)\( p^{63} T^{21} + 215881007909472 p^{70} T^{22} - 13253424 p^{77} T^{23} + p^{84} T^{24} \) | |
| 83 | \( 1 + 5033940 T + 204584499159820 T^{2} + \)\(98\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!22\)\( T^{4} + \)\(95\!\cdots\!40\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} + \)\(60\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!83\)\( T^{8} + \)\(27\!\cdots\!60\)\( T^{9} + \)\(26\!\cdots\!60\)\( T^{10} + \)\(96\!\cdots\!40\)\( T^{11} + \)\(79\!\cdots\!88\)\( T^{12} + \)\(96\!\cdots\!40\)\( p^{7} T^{13} + \)\(26\!\cdots\!60\)\( p^{14} T^{14} + \)\(27\!\cdots\!60\)\( p^{21} T^{15} + \)\(68\!\cdots\!83\)\( p^{28} T^{16} + \)\(60\!\cdots\!00\)\( p^{35} T^{17} + \)\(13\!\cdots\!80\)\( p^{42} T^{18} + \)\(95\!\cdots\!40\)\( p^{49} T^{19} + \)\(20\!\cdots\!22\)\( p^{56} T^{20} + \)\(98\!\cdots\!40\)\( p^{63} T^{21} + 204584499159820 p^{70} T^{22} + 5033940 p^{77} T^{23} + p^{84} T^{24} \) | |
| 89 | \( 1 - 109392 T + 333054597265460 T^{2} + \)\(17\!\cdots\!60\)\( T^{3} + \)\(55\!\cdots\!58\)\( T^{4} + \)\(52\!\cdots\!12\)\( T^{5} + \)\(61\!\cdots\!60\)\( T^{6} + \)\(73\!\cdots\!56\)\( T^{7} + \)\(49\!\cdots\!79\)\( T^{8} + \)\(64\!\cdots\!96\)\( T^{9} + \)\(30\!\cdots\!24\)\( T^{10} + \)\(39\!\cdots\!40\)\( T^{11} + \)\(15\!\cdots\!36\)\( T^{12} + \)\(39\!\cdots\!40\)\( p^{7} T^{13} + \)\(30\!\cdots\!24\)\( p^{14} T^{14} + \)\(64\!\cdots\!96\)\( p^{21} T^{15} + \)\(49\!\cdots\!79\)\( p^{28} T^{16} + \)\(73\!\cdots\!56\)\( p^{35} T^{17} + \)\(61\!\cdots\!60\)\( p^{42} T^{18} + \)\(52\!\cdots\!12\)\( p^{49} T^{19} + \)\(55\!\cdots\!58\)\( p^{56} T^{20} + \)\(17\!\cdots\!60\)\( p^{63} T^{21} + 333054597265460 p^{70} T^{22} - 109392 p^{77} T^{23} + p^{84} T^{24} \) | |
| 97 | \( 1 + 36832910 T + 1124877481299544 T^{2} + \)\(24\!\cdots\!70\)\( T^{3} + \)\(45\!\cdots\!30\)\( T^{4} + \)\(71\!\cdots\!10\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} + \)\(12\!\cdots\!90\)\( T^{7} + \)\(14\!\cdots\!59\)\( T^{8} + \)\(15\!\cdots\!60\)\( T^{9} + \)\(15\!\cdots\!36\)\( T^{10} + \)\(14\!\cdots\!60\)\( T^{11} + \)\(13\!\cdots\!60\)\( T^{12} + \)\(14\!\cdots\!60\)\( p^{7} T^{13} + \)\(15\!\cdots\!36\)\( p^{14} T^{14} + \)\(15\!\cdots\!60\)\( p^{21} T^{15} + \)\(14\!\cdots\!59\)\( p^{28} T^{16} + \)\(12\!\cdots\!90\)\( p^{35} T^{17} + \)\(10\!\cdots\!44\)\( p^{42} T^{18} + \)\(71\!\cdots\!10\)\( p^{49} T^{19} + \)\(45\!\cdots\!30\)\( p^{56} T^{20} + \)\(24\!\cdots\!70\)\( p^{63} T^{21} + 1124877481299544 p^{70} T^{22} + 36832910 p^{77} T^{23} + p^{84} T^{24} \) | |
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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
−2.53523051145788428084659898273, −2.43806965112344274721072734806, −2.42503733705146157312759647810, −2.42370685819222651210713165247, −2.11452558856605835879629575361, −2.07603403965853387544880841189, −1.99124146999635859730930490437, −1.84769007638618634373674365543, −1.78598860932785044207296978373, −1.61234450122668295951443669689, −1.48428346723612241693713147212, −1.39227234010884212758432422237, −1.35763709856576601716213659597, −1.10350159336770396093642232752, −1.08650860800412980951888608059, −0.74980645760254077108786636919, −0.72620022661356158977213266649, −0.71525474498294778901062247124, −0.62588136771885147266568567782, −0.43789452351163198113494423548, −0.39248186336629491036017635749, −0.35490597835281038890529257949, −0.32091690964701895361324693344, −0.20481898217784595984995861390, −0.19512278836817961810518280659, 0.19512278836817961810518280659, 0.20481898217784595984995861390, 0.32091690964701895361324693344, 0.35490597835281038890529257949, 0.39248186336629491036017635749, 0.43789452351163198113494423548, 0.62588136771885147266568567782, 0.71525474498294778901062247124, 0.72620022661356158977213266649, 0.74980645760254077108786636919, 1.08650860800412980951888608059, 1.10350159336770396093642232752, 1.35763709856576601716213659597, 1.39227234010884212758432422237, 1.48428346723612241693713147212, 1.61234450122668295951443669689, 1.78598860932785044207296978373, 1.84769007638618634373674365543, 1.99124146999635859730930490437, 2.07603403965853387544880841189, 2.11452558856605835879629575361, 2.42370685819222651210713165247, 2.42503733705146157312759647810, 2.43806965112344274721072734806, 2.53523051145788428084659898273
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.