Dirichlet series
| L(s) = 1 | − 2.08e14·2-s − 2.82e23·3-s − 1.07e30·4-s − 4.88e34·5-s + 5.88e37·6-s − 5.68e40·7-s + 4.68e44·8-s − 6.39e47·9-s + 1.01e49·10-s + 6.68e50·11-s + 3.04e53·12-s − 5.31e54·13-s + 1.18e55·14-s + 1.38e58·15-s + 4.95e59·16-s + 2.98e61·17-s + 1.33e62·18-s − 2.58e63·19-s + 5.25e64·20-s + 1.60e64·21-s − 1.39e65·22-s − 5.91e65·23-s − 1.32e68·24-s − 4.76e69·25-s + 1.10e69·26-s + 2.09e71·27-s + 6.11e70·28-s + ⋯ |
| L(s) = 1 | − 0.261·2-s − 0.682·3-s − 1.69·4-s − 1.22·5-s + 0.178·6-s − 0.0836·7-s + 0.928·8-s − 3.72·9-s + 0.321·10-s + 0.188·11-s + 1.15·12-s − 0.384·13-s + 0.0218·14-s + 0.839·15-s + 1.23·16-s + 3.69·17-s + 0.972·18-s − 1.30·19-s + 2.08·20-s + 0.0570·21-s − 0.0493·22-s − 0.0232·23-s − 0.633·24-s − 3.01·25-s + 0.100·26-s + 2.93·27-s + 0.141·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(100-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+99/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(1\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.20253\times 10^{14}\) |
| Root analytic conductor: | \(7.87830\) |
| Motivic weight: | \(99\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 1,\ (\ :[99/2]^{8}),\ 1)\) |
Particular Values
| \(L(50)\) | \(\approx\) | \(0.03454262504\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03454262504\) |
| \(L(\frac{101}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| good | 2 | \( 1 + 26005077112815 p^{3} T + \)\(27\!\cdots\!15\)\( p^{12} T^{2} - \)\(55\!\cdots\!45\)\( p^{21} T^{3} + \)\(11\!\cdots\!27\)\( p^{39} T^{4} - \)\(12\!\cdots\!95\)\( p^{55} T^{5} + \)\(94\!\cdots\!15\)\( p^{81} T^{6} + \)\(43\!\cdots\!65\)\( p^{107} T^{7} - \)\(71\!\cdots\!43\)\( p^{139} T^{8} + \)\(43\!\cdots\!65\)\( p^{206} T^{9} + \)\(94\!\cdots\!15\)\( p^{279} T^{10} - \)\(12\!\cdots\!95\)\( p^{352} T^{11} + \)\(11\!\cdots\!27\)\( p^{435} T^{12} - \)\(55\!\cdots\!45\)\( p^{516} T^{13} + \)\(27\!\cdots\!15\)\( p^{606} T^{14} + 26005077112815 p^{696} T^{15} + p^{792} T^{16} \) |
| 3 | \( 1 + \)\(34\!\cdots\!20\)\( p^{4} T + \)\(10\!\cdots\!80\)\( p^{8} T^{2} + \)\(13\!\cdots\!80\)\( p^{17} T^{3} + \)\(45\!\cdots\!48\)\( p^{31} T^{4} + \)\(20\!\cdots\!80\)\( p^{45} T^{5} + \)\(65\!\cdots\!80\)\( p^{63} T^{6} + \)\(13\!\cdots\!20\)\( p^{86} T^{7} + \)\(13\!\cdots\!98\)\( p^{107} T^{8} + \)\(13\!\cdots\!20\)\( p^{185} T^{9} + \)\(65\!\cdots\!80\)\( p^{261} T^{10} + \)\(20\!\cdots\!80\)\( p^{342} T^{11} + \)\(45\!\cdots\!48\)\( p^{427} T^{12} + \)\(13\!\cdots\!80\)\( p^{512} T^{13} + \)\(10\!\cdots\!80\)\( p^{602} T^{14} + \)\(34\!\cdots\!20\)\( p^{697} T^{15} + p^{792} T^{16} \) | |
| 5 | \( 1 + \)\(97\!\cdots\!12\)\( p T + \)\(22\!\cdots\!64\)\( p^{5} T^{2} + \)\(30\!\cdots\!28\)\( p^{10} T^{3} + \)\(71\!\cdots\!04\)\( p^{18} T^{4} + \)\(13\!\cdots\!28\)\( p^{27} T^{5} + \)\(94\!\cdots\!24\)\( p^{37} T^{6} + \)\(24\!\cdots\!32\)\( p^{50} T^{7} + \)\(23\!\cdots\!66\)\( p^{64} T^{8} + \)\(24\!\cdots\!32\)\( p^{149} T^{9} + \)\(94\!\cdots\!24\)\( p^{235} T^{10} + \)\(13\!\cdots\!28\)\( p^{324} T^{11} + \)\(71\!\cdots\!04\)\( p^{414} T^{12} + \)\(30\!\cdots\!28\)\( p^{505} T^{13} + \)\(22\!\cdots\!64\)\( p^{599} T^{14} + \)\(97\!\cdots\!12\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
| 7 | \( 1 + \)\(81\!\cdots\!00\)\( p T + \)\(11\!\cdots\!00\)\( p^{5} T^{2} + \)\(90\!\cdots\!00\)\( p^{10} T^{3} + \)\(57\!\cdots\!96\)\( p^{16} T^{4} - \)\(21\!\cdots\!00\)\( p^{23} T^{5} + \)\(84\!\cdots\!00\)\( p^{31} T^{6} - \)\(91\!\cdots\!00\)\( p^{40} T^{7} + \)\(39\!\cdots\!94\)\( p^{50} T^{8} - \)\(91\!\cdots\!00\)\( p^{139} T^{9} + \)\(84\!\cdots\!00\)\( p^{229} T^{10} - \)\(21\!\cdots\!00\)\( p^{320} T^{11} + \)\(57\!\cdots\!96\)\( p^{412} T^{12} + \)\(90\!\cdots\!00\)\( p^{505} T^{13} + \)\(11\!\cdots\!00\)\( p^{599} T^{14} + \)\(81\!\cdots\!00\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
| 11 | \( 1 - \)\(60\!\cdots\!76\)\( p T + \)\(23\!\cdots\!20\)\( p^{3} T^{2} - \)\(20\!\cdots\!60\)\( p^{6} T^{3} + \)\(22\!\cdots\!20\)\( p^{10} T^{4} - \)\(45\!\cdots\!48\)\( p^{14} T^{5} + \)\(14\!\cdots\!48\)\( p^{19} T^{6} - \)\(29\!\cdots\!40\)\( p^{25} T^{7} + \)\(48\!\cdots\!70\)\( p^{32} T^{8} - \)\(29\!\cdots\!40\)\( p^{124} T^{9} + \)\(14\!\cdots\!48\)\( p^{217} T^{10} - \)\(45\!\cdots\!48\)\( p^{311} T^{11} + \)\(22\!\cdots\!20\)\( p^{406} T^{12} - \)\(20\!\cdots\!60\)\( p^{501} T^{13} + \)\(23\!\cdots\!20\)\( p^{597} T^{14} - \)\(60\!\cdots\!76\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
| 13 | \( 1 + \)\(40\!\cdots\!80\)\( p T + \)\(31\!\cdots\!40\)\( p^{2} T^{2} + \)\(25\!\cdots\!40\)\( p^{3} T^{3} + \)\(45\!\cdots\!24\)\( p^{6} T^{4} + \)\(15\!\cdots\!20\)\( p^{10} T^{5} + \)\(12\!\cdots\!60\)\( p^{15} T^{6} + \)\(23\!\cdots\!20\)\( p^{21} T^{7} + \)\(89\!\cdots\!26\)\( p^{28} T^{8} + \)\(23\!\cdots\!20\)\( p^{120} T^{9} + \)\(12\!\cdots\!60\)\( p^{213} T^{10} + \)\(15\!\cdots\!20\)\( p^{307} T^{11} + \)\(45\!\cdots\!24\)\( p^{402} T^{12} + \)\(25\!\cdots\!40\)\( p^{498} T^{13} + \)\(31\!\cdots\!40\)\( p^{596} T^{14} + \)\(40\!\cdots\!80\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
| 17 | \( 1 - \)\(17\!\cdots\!20\)\( p T + \)\(19\!\cdots\!80\)\( p^{2} T^{2} - \)\(16\!\cdots\!40\)\( p^{3} T^{3} + \)\(10\!\cdots\!16\)\( p^{4} T^{4} - \)\(19\!\cdots\!60\)\( p^{7} T^{5} + \)\(17\!\cdots\!80\)\( p^{11} T^{6} - \)\(15\!\cdots\!80\)\( p^{15} T^{7} + \)\(81\!\cdots\!86\)\( p^{20} T^{8} - \)\(15\!\cdots\!80\)\( p^{114} T^{9} + \)\(17\!\cdots\!80\)\( p^{209} T^{10} - \)\(19\!\cdots\!60\)\( p^{304} T^{11} + \)\(10\!\cdots\!16\)\( p^{400} T^{12} - \)\(16\!\cdots\!40\)\( p^{498} T^{13} + \)\(19\!\cdots\!80\)\( p^{596} T^{14} - \)\(17\!\cdots\!20\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
| 19 | \( 1 + \)\(13\!\cdots\!80\)\( p T + \)\(50\!\cdots\!12\)\( p^{2} T^{2} + \)\(36\!\cdots\!60\)\( p^{4} T^{3} + \)\(20\!\cdots\!32\)\( p^{7} T^{4} + \)\(69\!\cdots\!20\)\( p^{10} T^{5} + \)\(14\!\cdots\!04\)\( p^{14} T^{6} + \)\(23\!\cdots\!00\)\( p^{18} T^{7} + \)\(40\!\cdots\!70\)\( p^{22} T^{8} + \)\(23\!\cdots\!00\)\( p^{117} T^{9} + \)\(14\!\cdots\!04\)\( p^{212} T^{10} + \)\(69\!\cdots\!20\)\( p^{307} T^{11} + \)\(20\!\cdots\!32\)\( p^{403} T^{12} + \)\(36\!\cdots\!60\)\( p^{499} T^{13} + \)\(50\!\cdots\!12\)\( p^{596} T^{14} + \)\(13\!\cdots\!80\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
| 23 | \( 1 + \)\(59\!\cdots\!60\)\( T + \)\(13\!\cdots\!80\)\( p T^{2} + \)\(19\!\cdots\!60\)\( p^{3} T^{3} + \)\(67\!\cdots\!32\)\( p^{5} T^{4} + \)\(76\!\cdots\!20\)\( p^{8} T^{5} + \)\(42\!\cdots\!40\)\( p^{11} T^{6} + \)\(26\!\cdots\!20\)\( p^{15} T^{7} + \)\(39\!\cdots\!18\)\( p^{19} T^{8} + \)\(26\!\cdots\!20\)\( p^{114} T^{9} + \)\(42\!\cdots\!40\)\( p^{209} T^{10} + \)\(76\!\cdots\!20\)\( p^{305} T^{11} + \)\(67\!\cdots\!32\)\( p^{401} T^{12} + \)\(19\!\cdots\!60\)\( p^{498} T^{13} + \)\(13\!\cdots\!80\)\( p^{595} T^{14} + \)\(59\!\cdots\!60\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 29 | \( 1 + \)\(43\!\cdots\!80\)\( T + \)\(10\!\cdots\!88\)\( p T^{2} + \)\(39\!\cdots\!40\)\( p^{2} T^{3} + \)\(59\!\cdots\!68\)\( p^{4} T^{4} + \)\(12\!\cdots\!80\)\( p^{6} T^{5} + \)\(72\!\cdots\!64\)\( p^{8} T^{6} + \)\(63\!\cdots\!00\)\( p^{11} T^{7} + \)\(79\!\cdots\!70\)\( p^{14} T^{8} + \)\(63\!\cdots\!00\)\( p^{110} T^{9} + \)\(72\!\cdots\!64\)\( p^{206} T^{10} + \)\(12\!\cdots\!80\)\( p^{303} T^{11} + \)\(59\!\cdots\!68\)\( p^{400} T^{12} + \)\(39\!\cdots\!40\)\( p^{497} T^{13} + \)\(10\!\cdots\!88\)\( p^{595} T^{14} + \)\(43\!\cdots\!80\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 31 | \( 1 + \)\(13\!\cdots\!04\)\( p T + \)\(21\!\cdots\!20\)\( p^{2} T^{2} + \)\(15\!\cdots\!40\)\( p^{3} T^{3} + \)\(72\!\cdots\!20\)\( p^{5} T^{4} + \)\(80\!\cdots\!12\)\( p^{7} T^{5} + \)\(16\!\cdots\!88\)\( p^{10} T^{6} + \)\(32\!\cdots\!60\)\( p^{13} T^{7} + \)\(96\!\cdots\!70\)\( p^{16} T^{8} + \)\(32\!\cdots\!60\)\( p^{112} T^{9} + \)\(16\!\cdots\!88\)\( p^{208} T^{10} + \)\(80\!\cdots\!12\)\( p^{304} T^{11} + \)\(72\!\cdots\!20\)\( p^{401} T^{12} + \)\(15\!\cdots\!40\)\( p^{498} T^{13} + \)\(21\!\cdots\!20\)\( p^{596} T^{14} + \)\(13\!\cdots\!04\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
| 37 | \( 1 - \)\(70\!\cdots\!20\)\( T + \)\(10\!\cdots\!60\)\( T^{2} + \)\(98\!\cdots\!20\)\( p T^{3} + \)\(40\!\cdots\!64\)\( p^{2} T^{4} + \)\(12\!\cdots\!20\)\( p^{3} T^{5} + \)\(25\!\cdots\!60\)\( p^{5} T^{6} + \)\(27\!\cdots\!60\)\( p^{7} T^{7} + \)\(28\!\cdots\!98\)\( p^{9} T^{8} + \)\(27\!\cdots\!60\)\( p^{106} T^{9} + \)\(25\!\cdots\!60\)\( p^{203} T^{10} + \)\(12\!\cdots\!20\)\( p^{300} T^{11} + \)\(40\!\cdots\!64\)\( p^{398} T^{12} + \)\(98\!\cdots\!20\)\( p^{496} T^{13} + \)\(10\!\cdots\!60\)\( p^{594} T^{14} - \)\(70\!\cdots\!20\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 41 | \( 1 - \)\(16\!\cdots\!96\)\( T + \)\(62\!\cdots\!20\)\( p T^{2} - \)\(35\!\cdots\!60\)\( p^{2} T^{3} + \)\(45\!\cdots\!20\)\( p^{3} T^{4} - \)\(29\!\cdots\!88\)\( p^{4} T^{5} + \)\(50\!\cdots\!28\)\( p^{6} T^{6} - \)\(82\!\cdots\!40\)\( p^{8} T^{7} + \)\(96\!\cdots\!70\)\( p^{10} T^{8} - \)\(82\!\cdots\!40\)\( p^{107} T^{9} + \)\(50\!\cdots\!28\)\( p^{204} T^{10} - \)\(29\!\cdots\!88\)\( p^{301} T^{11} + \)\(45\!\cdots\!20\)\( p^{399} T^{12} - \)\(35\!\cdots\!60\)\( p^{497} T^{13} + \)\(62\!\cdots\!20\)\( p^{595} T^{14} - \)\(16\!\cdots\!96\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 43 | \( 1 + \)\(11\!\cdots\!00\)\( T + \)\(48\!\cdots\!00\)\( p T^{2} + \)\(27\!\cdots\!00\)\( p^{2} T^{3} + \)\(29\!\cdots\!28\)\( p^{3} T^{4} + \)\(44\!\cdots\!00\)\( p^{5} T^{5} + \)\(68\!\cdots\!00\)\( p^{7} T^{6} + \)\(94\!\cdots\!00\)\( p^{9} T^{7} + \)\(12\!\cdots\!58\)\( p^{11} T^{8} + \)\(94\!\cdots\!00\)\( p^{108} T^{9} + \)\(68\!\cdots\!00\)\( p^{205} T^{10} + \)\(44\!\cdots\!00\)\( p^{302} T^{11} + \)\(29\!\cdots\!28\)\( p^{399} T^{12} + \)\(27\!\cdots\!00\)\( p^{497} T^{13} + \)\(48\!\cdots\!00\)\( p^{595} T^{14} + \)\(11\!\cdots\!00\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 47 | \( 1 - \)\(61\!\cdots\!80\)\( p T + \)\(69\!\cdots\!20\)\( p^{2} T^{2} - \)\(17\!\cdots\!60\)\( p^{3} T^{3} + \)\(24\!\cdots\!76\)\( p^{4} T^{4} - \)\(65\!\cdots\!80\)\( p^{6} T^{5} + \)\(27\!\cdots\!60\)\( p^{8} T^{6} - \)\(47\!\cdots\!40\)\( p^{10} T^{7} + \)\(22\!\cdots\!86\)\( p^{12} T^{8} - \)\(47\!\cdots\!40\)\( p^{109} T^{9} + \)\(27\!\cdots\!60\)\( p^{206} T^{10} - \)\(65\!\cdots\!80\)\( p^{303} T^{11} + \)\(24\!\cdots\!76\)\( p^{400} T^{12} - \)\(17\!\cdots\!60\)\( p^{498} T^{13} + \)\(69\!\cdots\!20\)\( p^{596} T^{14} - \)\(61\!\cdots\!80\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
| 53 | \( 1 + \)\(30\!\cdots\!20\)\( T + \)\(30\!\cdots\!80\)\( T^{2} + \)\(72\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!56\)\( T^{4} + \)\(15\!\cdots\!80\)\( p T^{5} + \)\(13\!\cdots\!40\)\( p^{2} T^{6} + \)\(75\!\cdots\!80\)\( p^{4} T^{7} + \)\(27\!\cdots\!46\)\( p^{4} T^{8} + \)\(75\!\cdots\!80\)\( p^{103} T^{9} + \)\(13\!\cdots\!40\)\( p^{200} T^{10} + \)\(15\!\cdots\!80\)\( p^{298} T^{11} + \)\(43\!\cdots\!56\)\( p^{396} T^{12} + \)\(72\!\cdots\!40\)\( p^{495} T^{13} + \)\(30\!\cdots\!80\)\( p^{594} T^{14} + \)\(30\!\cdots\!20\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 59 | \( 1 - \)\(14\!\cdots\!40\)\( T + \)\(21\!\cdots\!12\)\( T^{2} - \)\(19\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!32\)\( p T^{4} - \)\(32\!\cdots\!40\)\( p^{2} T^{5} + \)\(34\!\cdots\!16\)\( p^{3} T^{6} - \)\(31\!\cdots\!00\)\( p^{4} T^{7} + \)\(25\!\cdots\!30\)\( p^{5} T^{8} - \)\(31\!\cdots\!00\)\( p^{103} T^{9} + \)\(34\!\cdots\!16\)\( p^{201} T^{10} - \)\(32\!\cdots\!40\)\( p^{299} T^{11} + \)\(28\!\cdots\!32\)\( p^{397} T^{12} - \)\(19\!\cdots\!20\)\( p^{495} T^{13} + \)\(21\!\cdots\!12\)\( p^{594} T^{14} - \)\(14\!\cdots\!40\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 61 | \( 1 - \)\(38\!\cdots\!36\)\( T + \)\(30\!\cdots\!20\)\( T^{2} - \)\(96\!\cdots\!60\)\( T^{3} + \)\(63\!\cdots\!20\)\( p T^{4} - \)\(29\!\cdots\!08\)\( p^{2} T^{5} + \)\(12\!\cdots\!28\)\( p^{3} T^{6} - \)\(58\!\cdots\!40\)\( p^{4} T^{7} + \)\(20\!\cdots\!70\)\( p^{5} T^{8} - \)\(58\!\cdots\!40\)\( p^{103} T^{9} + \)\(12\!\cdots\!28\)\( p^{201} T^{10} - \)\(29\!\cdots\!08\)\( p^{299} T^{11} + \)\(63\!\cdots\!20\)\( p^{397} T^{12} - \)\(96\!\cdots\!60\)\( p^{495} T^{13} + \)\(30\!\cdots\!20\)\( p^{594} T^{14} - \)\(38\!\cdots\!36\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 67 | \( 1 - \)\(11\!\cdots\!40\)\( T + \)\(81\!\cdots\!20\)\( T^{2} - \)\(59\!\cdots\!60\)\( p T^{3} + \)\(34\!\cdots\!24\)\( p^{2} T^{4} - \)\(17\!\cdots\!60\)\( p^{3} T^{5} + \)\(76\!\cdots\!40\)\( p^{4} T^{6} - \)\(31\!\cdots\!20\)\( p^{5} T^{7} + \)\(11\!\cdots\!94\)\( p^{6} T^{8} - \)\(31\!\cdots\!20\)\( p^{104} T^{9} + \)\(76\!\cdots\!40\)\( p^{202} T^{10} - \)\(17\!\cdots\!60\)\( p^{300} T^{11} + \)\(34\!\cdots\!24\)\( p^{398} T^{12} - \)\(59\!\cdots\!60\)\( p^{496} T^{13} + \)\(81\!\cdots\!20\)\( p^{594} T^{14} - \)\(11\!\cdots\!40\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 71 | \( 1 - \)\(52\!\cdots\!56\)\( T + \)\(57\!\cdots\!20\)\( T^{2} - \)\(25\!\cdots\!60\)\( p T^{3} + \)\(32\!\cdots\!20\)\( p^{2} T^{4} - \)\(11\!\cdots\!88\)\( p^{3} T^{5} + \)\(14\!\cdots\!88\)\( p^{4} T^{6} - \)\(37\!\cdots\!40\)\( p^{5} T^{7} + \)\(51\!\cdots\!70\)\( p^{6} T^{8} - \)\(37\!\cdots\!40\)\( p^{104} T^{9} + \)\(14\!\cdots\!88\)\( p^{202} T^{10} - \)\(11\!\cdots\!88\)\( p^{300} T^{11} + \)\(32\!\cdots\!20\)\( p^{398} T^{12} - \)\(25\!\cdots\!60\)\( p^{496} T^{13} + \)\(57\!\cdots\!20\)\( p^{594} T^{14} - \)\(52\!\cdots\!56\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 73 | \( 1 - \)\(50\!\cdots\!40\)\( T + \)\(23\!\cdots\!40\)\( T^{2} - \)\(81\!\cdots\!60\)\( p T^{3} + \)\(26\!\cdots\!44\)\( p^{2} T^{4} - \)\(50\!\cdots\!40\)\( p^{3} T^{5} + \)\(98\!\cdots\!80\)\( p^{4} T^{6} - \)\(55\!\cdots\!20\)\( p^{5} T^{7} + \)\(15\!\cdots\!94\)\( p^{6} T^{8} - \)\(55\!\cdots\!20\)\( p^{104} T^{9} + \)\(98\!\cdots\!80\)\( p^{202} T^{10} - \)\(50\!\cdots\!40\)\( p^{300} T^{11} + \)\(26\!\cdots\!44\)\( p^{398} T^{12} - \)\(81\!\cdots\!60\)\( p^{496} T^{13} + \)\(23\!\cdots\!40\)\( p^{594} T^{14} - \)\(50\!\cdots\!40\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 79 | \( 1 + \)\(30\!\cdots\!80\)\( T + \)\(74\!\cdots\!88\)\( p T^{2} + \)\(13\!\cdots\!40\)\( p^{2} T^{3} + \)\(19\!\cdots\!72\)\( p^{3} T^{4} + \)\(27\!\cdots\!80\)\( p^{4} T^{5} + \)\(36\!\cdots\!96\)\( p^{5} T^{6} + \)\(44\!\cdots\!00\)\( p^{6} T^{7} + \)\(51\!\cdots\!30\)\( p^{7} T^{8} + \)\(44\!\cdots\!00\)\( p^{105} T^{9} + \)\(36\!\cdots\!96\)\( p^{203} T^{10} + \)\(27\!\cdots\!80\)\( p^{301} T^{11} + \)\(19\!\cdots\!72\)\( p^{399} T^{12} + \)\(13\!\cdots\!40\)\( p^{497} T^{13} + \)\(74\!\cdots\!88\)\( p^{595} T^{14} + \)\(30\!\cdots\!80\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 83 | \( 1 - \)\(22\!\cdots\!20\)\( T + \)\(50\!\cdots\!40\)\( p T^{2} - \)\(50\!\cdots\!60\)\( p^{2} T^{3} + \)\(16\!\cdots\!28\)\( p^{3} T^{4} - \)\(10\!\cdots\!40\)\( p^{4} T^{5} + \)\(34\!\cdots\!80\)\( p^{5} T^{6} + \)\(13\!\cdots\!80\)\( p^{6} T^{7} + \)\(55\!\cdots\!18\)\( p^{7} T^{8} + \)\(13\!\cdots\!80\)\( p^{105} T^{9} + \)\(34\!\cdots\!80\)\( p^{203} T^{10} - \)\(10\!\cdots\!40\)\( p^{301} T^{11} + \)\(16\!\cdots\!28\)\( p^{399} T^{12} - \)\(50\!\cdots\!60\)\( p^{497} T^{13} + \)\(50\!\cdots\!40\)\( p^{595} T^{14} - \)\(22\!\cdots\!20\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 89 | \( 1 - \)\(13\!\cdots\!60\)\( T + \)\(15\!\cdots\!48\)\( p T^{2} - \)\(12\!\cdots\!80\)\( p^{2} T^{3} + \)\(84\!\cdots\!72\)\( p^{3} T^{4} - \)\(47\!\cdots\!60\)\( p^{4} T^{5} + \)\(23\!\cdots\!76\)\( p^{5} T^{6} - \)\(10\!\cdots\!00\)\( p^{6} T^{7} + \)\(37\!\cdots\!30\)\( p^{7} T^{8} - \)\(10\!\cdots\!00\)\( p^{105} T^{9} + \)\(23\!\cdots\!76\)\( p^{203} T^{10} - \)\(47\!\cdots\!60\)\( p^{301} T^{11} + \)\(84\!\cdots\!72\)\( p^{399} T^{12} - \)\(12\!\cdots\!80\)\( p^{497} T^{13} + \)\(15\!\cdots\!48\)\( p^{595} T^{14} - \)\(13\!\cdots\!60\)\( p^{693} T^{15} + p^{792} T^{16} \) | |
| 97 | \( 1 + \)\(89\!\cdots\!20\)\( p T + \)\(58\!\cdots\!20\)\( p^{2} T^{2} + \)\(27\!\cdots\!40\)\( p^{3} T^{3} + \)\(11\!\cdots\!76\)\( p^{4} T^{4} + \)\(38\!\cdots\!40\)\( p^{5} T^{5} + \)\(11\!\cdots\!40\)\( p^{6} T^{6} + \)\(31\!\cdots\!80\)\( p^{7} T^{7} + \)\(75\!\cdots\!66\)\( p^{8} T^{8} + \)\(31\!\cdots\!80\)\( p^{106} T^{9} + \)\(11\!\cdots\!40\)\( p^{204} T^{10} + \)\(38\!\cdots\!40\)\( p^{302} T^{11} + \)\(11\!\cdots\!76\)\( p^{400} T^{12} + \)\(27\!\cdots\!40\)\( p^{498} T^{13} + \)\(58\!\cdots\!20\)\( p^{596} T^{14} + \)\(89\!\cdots\!20\)\( p^{694} T^{15} + p^{792} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.99422428996055355061406310952, −4.31026581966752052291287874800, −4.23149643749900585852866836873, −3.85412196400214236288439201822, −3.75410456219660588955378331928, −3.73300991387566089608525041043, −3.72735732523573372038775861610, −3.58671410904994897755381279588, −3.12561784033871415900743439535, −3.02403823756730519695357521273, −2.79287991358273355125780156377, −2.68094627333850606771841645426, −2.38698308596546250677309757655, −2.08859696690179104973123944383, −2.07466531707269433382459856444, −1.80804204372360563700184258015, −1.49818656562129496114105808424, −1.47885692592129847625270994374, −0.933902180534301809992712401331, −0.890337043804568232452066276259, −0.63471210310869948932054908978, −0.56956582738471993674943656448, −0.40253771477770699112491134797, −0.26531802737573460571075720343, −0.03599348156476027294440974586, 0.03599348156476027294440974586, 0.26531802737573460571075720343, 0.40253771477770699112491134797, 0.56956582738471993674943656448, 0.63471210310869948932054908978, 0.890337043804568232452066276259, 0.933902180534301809992712401331, 1.47885692592129847625270994374, 1.49818656562129496114105808424, 1.80804204372360563700184258015, 2.07466531707269433382459856444, 2.08859696690179104973123944383, 2.38698308596546250677309757655, 2.68094627333850606771841645426, 2.79287991358273355125780156377, 3.02403823756730519695357521273, 3.12561784033871415900743439535, 3.58671410904994897755381279588, 3.72735732523573372038775861610, 3.73300991387566089608525041043, 3.75410456219660588955378331928, 3.85412196400214236288439201822, 4.23149643749900585852866836873, 4.31026581966752052291287874800, 4.99422428996055355061406310952
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.