Dirichlet series
| L(s) = 1 | − 6·2-s − 432·3-s − 519·4-s − 68·5-s + 2.59e3·6-s − 2.34e3·7-s + 3.71e3·8-s + 9.91e4·9-s + 408·10-s + 898·11-s + 2.24e5·12-s − 8.17e3·13-s + 1.40e4·14-s + 2.93e4·15-s + 1.01e5·16-s − 4.49e4·17-s − 5.94e5·18-s − 4.01e4·19-s + 3.52e4·20-s + 1.01e6·21-s − 5.38e3·22-s − 2.83e3·23-s − 1.60e6·24-s − 4.79e5·25-s + 4.90e4·26-s − 1.60e7·27-s + 1.21e6·28-s + ⋯ |
| L(s) = 1 | − 0.530·2-s − 9.23·3-s − 4.05·4-s − 0.243·5-s + 4.89·6-s − 2.58·7-s + 2.56·8-s + 45.3·9-s + 0.129·10-s + 0.203·11-s + 37.4·12-s − 1.03·13-s + 1.36·14-s + 2.24·15-s + 6.16·16-s − 2.22·17-s − 24.0·18-s − 1.34·19-s + 0.986·20-s + 23.8·21-s − 0.107·22-s − 0.0485·23-s − 23.6·24-s − 6.14·25-s + 0.547·26-s − 157.·27-s + 10.4·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 59^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 59^{16}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(3^{16} \cdot 59^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.63159\times 10^{27}\) |
| Root analytic conductor: | \(7.43586\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(16\) |
| Selberg data: | \((32,\ 3^{16} \cdot 59^{16} ,\ ( \ : [7/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{3} T )^{16} \) |
| 59 | \( ( 1 - p^{3} T )^{16} \) | |
| good | 2 | \( 1 + 3 p T + 555 T^{2} + 2729 T^{3} + 90557 p T^{4} + 801421 T^{5} + 23563035 p T^{6} + 6064111 p^{5} T^{7} + 319507921 p^{5} T^{8} + 1249075169 p^{5} T^{9} + 30022780525 p^{6} T^{10} + 28089249987 p^{8} T^{11} + 78484723671 p^{12} T^{12} + 4464698946637 p^{8} T^{13} + 93818649401511 p^{9} T^{14} + 39580945805855 p^{12} T^{15} + 788182218735117 p^{13} T^{16} + 39580945805855 p^{19} T^{17} + 93818649401511 p^{23} T^{18} + 4464698946637 p^{29} T^{19} + 78484723671 p^{40} T^{20} + 28089249987 p^{43} T^{21} + 30022780525 p^{48} T^{22} + 1249075169 p^{54} T^{23} + 319507921 p^{61} T^{24} + 6064111 p^{68} T^{25} + 23563035 p^{71} T^{26} + 801421 p^{77} T^{27} + 90557 p^{85} T^{28} + 2729 p^{91} T^{29} + 555 p^{98} T^{30} + 3 p^{106} T^{31} + p^{112} T^{32} \) |
| 5 | \( 1 + 68 T + 484439 T^{2} + 32331978 T^{3} + 23151305886 p T^{4} + 1553151288618 p T^{5} + 3706537404251679 p T^{6} + 59466026220969652 p^{2} T^{7} + \)\(23\!\cdots\!29\)\( T^{8} + \)\(51\!\cdots\!44\)\( p T^{9} + \)\(19\!\cdots\!12\)\( p^{3} T^{10} + \)\(29\!\cdots\!64\)\( p^{3} T^{11} + \)\(36\!\cdots\!38\)\( p^{4} T^{12} + \)\(26\!\cdots\!52\)\( p^{6} T^{13} + \)\(24\!\cdots\!54\)\( p^{7} T^{14} + \)\(49\!\cdots\!28\)\( p^{7} T^{15} + \)\(39\!\cdots\!32\)\( p^{8} T^{16} + \)\(49\!\cdots\!28\)\( p^{14} T^{17} + \)\(24\!\cdots\!54\)\( p^{21} T^{18} + \)\(26\!\cdots\!52\)\( p^{27} T^{19} + \)\(36\!\cdots\!38\)\( p^{32} T^{20} + \)\(29\!\cdots\!64\)\( p^{38} T^{21} + \)\(19\!\cdots\!12\)\( p^{45} T^{22} + \)\(51\!\cdots\!44\)\( p^{50} T^{23} + \)\(23\!\cdots\!29\)\( p^{56} T^{24} + 59466026220969652 p^{65} T^{25} + 3706537404251679 p^{71} T^{26} + 1553151288618 p^{78} T^{27} + 23151305886 p^{85} T^{28} + 32331978 p^{91} T^{29} + 484439 p^{98} T^{30} + 68 p^{105} T^{31} + p^{112} T^{32} \) | |
| 7 | \( 1 + 2343 T + 8386396 T^{2} + 340924638 p^{2} T^{3} + 36305360326168 T^{4} + 61555776166460035 T^{5} + \)\(10\!\cdots\!22\)\( T^{6} + \)\(15\!\cdots\!89\)\( T^{7} + \)\(22\!\cdots\!20\)\( T^{8} + \)\(28\!\cdots\!54\)\( T^{9} + \)\(52\!\cdots\!94\)\( p T^{10} + \)\(42\!\cdots\!75\)\( T^{11} + \)\(48\!\cdots\!01\)\( T^{12} + \)\(51\!\cdots\!20\)\( T^{13} + \)\(52\!\cdots\!84\)\( T^{14} + \)\(72\!\cdots\!50\)\( p T^{15} + \)\(47\!\cdots\!48\)\( T^{16} + \)\(72\!\cdots\!50\)\( p^{8} T^{17} + \)\(52\!\cdots\!84\)\( p^{14} T^{18} + \)\(51\!\cdots\!20\)\( p^{21} T^{19} + \)\(48\!\cdots\!01\)\( p^{28} T^{20} + \)\(42\!\cdots\!75\)\( p^{35} T^{21} + \)\(52\!\cdots\!94\)\( p^{43} T^{22} + \)\(28\!\cdots\!54\)\( p^{49} T^{23} + \)\(22\!\cdots\!20\)\( p^{56} T^{24} + \)\(15\!\cdots\!89\)\( p^{63} T^{25} + \)\(10\!\cdots\!22\)\( p^{70} T^{26} + 61555776166460035 p^{77} T^{27} + 36305360326168 p^{84} T^{28} + 340924638 p^{93} T^{29} + 8386396 p^{98} T^{30} + 2343 p^{105} T^{31} + p^{112} T^{32} \) | |
| 11 | \( 1 - 898 T + 163741540 T^{2} - 42234715948 T^{3} + 12957347483852297 T^{4} + 3797236701767068896 T^{5} + \)\(66\!\cdots\!08\)\( T^{6} + \)\(48\!\cdots\!06\)\( T^{7} + \)\(25\!\cdots\!97\)\( T^{8} + \)\(25\!\cdots\!00\)\( T^{9} + \)\(77\!\cdots\!64\)\( T^{10} + \)\(84\!\cdots\!72\)\( T^{11} + \)\(20\!\cdots\!06\)\( T^{12} + \)\(20\!\cdots\!12\)\( T^{13} + \)\(46\!\cdots\!24\)\( T^{14} + \)\(41\!\cdots\!24\)\( T^{15} + \)\(94\!\cdots\!38\)\( T^{16} + \)\(41\!\cdots\!24\)\( p^{7} T^{17} + \)\(46\!\cdots\!24\)\( p^{14} T^{18} + \)\(20\!\cdots\!12\)\( p^{21} T^{19} + \)\(20\!\cdots\!06\)\( p^{28} T^{20} + \)\(84\!\cdots\!72\)\( p^{35} T^{21} + \)\(77\!\cdots\!64\)\( p^{42} T^{22} + \)\(25\!\cdots\!00\)\( p^{49} T^{23} + \)\(25\!\cdots\!97\)\( p^{56} T^{24} + \)\(48\!\cdots\!06\)\( p^{63} T^{25} + \)\(66\!\cdots\!08\)\( p^{70} T^{26} + 3797236701767068896 p^{77} T^{27} + 12957347483852297 p^{84} T^{28} - 42234715948 p^{91} T^{29} + 163741540 p^{98} T^{30} - 898 p^{105} T^{31} + p^{112} T^{32} \) | |
| 13 | \( 1 + 8172 T + 511935818 T^{2} + 3986968146014 T^{3} + 135894539819802669 T^{4} + 79319665280264615160 p T^{5} + \)\(24\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!42\)\( T^{7} + \)\(26\!\cdots\!33\)\( p T^{8} + \)\(19\!\cdots\!66\)\( p T^{9} + \)\(38\!\cdots\!62\)\( T^{10} + \)\(26\!\cdots\!94\)\( T^{11} + \)\(35\!\cdots\!34\)\( T^{12} + \)\(24\!\cdots\!14\)\( T^{13} + \)\(28\!\cdots\!94\)\( T^{14} + \)\(17\!\cdots\!86\)\( T^{15} + \)\(19\!\cdots\!62\)\( T^{16} + \)\(17\!\cdots\!86\)\( p^{7} T^{17} + \)\(28\!\cdots\!94\)\( p^{14} T^{18} + \)\(24\!\cdots\!14\)\( p^{21} T^{19} + \)\(35\!\cdots\!34\)\( p^{28} T^{20} + \)\(26\!\cdots\!94\)\( p^{35} T^{21} + \)\(38\!\cdots\!62\)\( p^{42} T^{22} + \)\(19\!\cdots\!66\)\( p^{50} T^{23} + \)\(26\!\cdots\!33\)\( p^{57} T^{24} + \)\(18\!\cdots\!42\)\( p^{63} T^{25} + \)\(24\!\cdots\!00\)\( p^{70} T^{26} + 79319665280264615160 p^{78} T^{27} + 135894539819802669 p^{84} T^{28} + 3986968146014 p^{91} T^{29} + 511935818 p^{98} T^{30} + 8172 p^{105} T^{31} + p^{112} T^{32} \) | |
| 17 | \( 1 + 44985 T + 5494681630 T^{2} + 205790956761078 T^{3} + 14099821456028369852 T^{4} + \)\(45\!\cdots\!21\)\( T^{5} + \)\(22\!\cdots\!54\)\( T^{6} + \)\(38\!\cdots\!13\)\( p T^{7} + \)\(26\!\cdots\!36\)\( T^{8} + \)\(66\!\cdots\!74\)\( T^{9} + \)\(22\!\cdots\!92\)\( T^{10} + \)\(52\!\cdots\!47\)\( T^{11} + \)\(15\!\cdots\!65\)\( T^{12} + \)\(32\!\cdots\!92\)\( T^{13} + \)\(87\!\cdots\!76\)\( T^{14} + \)\(16\!\cdots\!14\)\( T^{15} + \)\(39\!\cdots\!08\)\( T^{16} + \)\(16\!\cdots\!14\)\( p^{7} T^{17} + \)\(87\!\cdots\!76\)\( p^{14} T^{18} + \)\(32\!\cdots\!92\)\( p^{21} T^{19} + \)\(15\!\cdots\!65\)\( p^{28} T^{20} + \)\(52\!\cdots\!47\)\( p^{35} T^{21} + \)\(22\!\cdots\!92\)\( p^{42} T^{22} + \)\(66\!\cdots\!74\)\( p^{49} T^{23} + \)\(26\!\cdots\!36\)\( p^{56} T^{24} + \)\(38\!\cdots\!13\)\( p^{64} T^{25} + \)\(22\!\cdots\!54\)\( p^{70} T^{26} + \)\(45\!\cdots\!21\)\( p^{77} T^{27} + 14099821456028369852 p^{84} T^{28} + 205790956761078 p^{91} T^{29} + 5494681630 p^{98} T^{30} + 44985 p^{105} T^{31} + p^{112} T^{32} \) | |
| 19 | \( 1 + 40137 T + 6809402349 T^{2} + 225411665068823 T^{3} + 21791992588224206783 T^{4} + \)\(60\!\cdots\!90\)\( T^{5} + \)\(23\!\cdots\!10\)\( p T^{6} + \)\(10\!\cdots\!70\)\( T^{7} + \)\(67\!\cdots\!63\)\( T^{8} + \)\(13\!\cdots\!21\)\( T^{9} + \)\(86\!\cdots\!99\)\( T^{10} + \)\(16\!\cdots\!81\)\( T^{11} + \)\(10\!\cdots\!91\)\( T^{12} + \)\(18\!\cdots\!18\)\( T^{13} + \)\(10\!\cdots\!70\)\( T^{14} + \)\(18\!\cdots\!20\)\( T^{15} + \)\(10\!\cdots\!88\)\( T^{16} + \)\(18\!\cdots\!20\)\( p^{7} T^{17} + \)\(10\!\cdots\!70\)\( p^{14} T^{18} + \)\(18\!\cdots\!18\)\( p^{21} T^{19} + \)\(10\!\cdots\!91\)\( p^{28} T^{20} + \)\(16\!\cdots\!81\)\( p^{35} T^{21} + \)\(86\!\cdots\!99\)\( p^{42} T^{22} + \)\(13\!\cdots\!21\)\( p^{49} T^{23} + \)\(67\!\cdots\!63\)\( p^{56} T^{24} + \)\(10\!\cdots\!70\)\( p^{63} T^{25} + \)\(23\!\cdots\!10\)\( p^{71} T^{26} + \)\(60\!\cdots\!90\)\( p^{77} T^{27} + 21791992588224206783 p^{84} T^{28} + 225411665068823 p^{91} T^{29} + 6809402349 p^{98} T^{30} + 40137 p^{105} T^{31} + p^{112} T^{32} \) | |
| 23 | \( 1 + 2833 T + 20814787185 T^{2} - 648124018861281 T^{3} + \)\(21\!\cdots\!99\)\( T^{4} - \)\(12\!\cdots\!26\)\( T^{5} + \)\(17\!\cdots\!58\)\( T^{6} - \)\(12\!\cdots\!02\)\( T^{7} + \)\(12\!\cdots\!27\)\( T^{8} - \)\(92\!\cdots\!11\)\( T^{9} + \)\(72\!\cdots\!23\)\( T^{10} - \)\(52\!\cdots\!11\)\( T^{11} + \)\(15\!\cdots\!25\)\( p T^{12} - \)\(24\!\cdots\!90\)\( T^{13} + \)\(15\!\cdots\!54\)\( T^{14} - \)\(97\!\cdots\!88\)\( T^{15} + \)\(56\!\cdots\!48\)\( T^{16} - \)\(97\!\cdots\!88\)\( p^{7} T^{17} + \)\(15\!\cdots\!54\)\( p^{14} T^{18} - \)\(24\!\cdots\!90\)\( p^{21} T^{19} + \)\(15\!\cdots\!25\)\( p^{29} T^{20} - \)\(52\!\cdots\!11\)\( p^{35} T^{21} + \)\(72\!\cdots\!23\)\( p^{42} T^{22} - \)\(92\!\cdots\!11\)\( p^{49} T^{23} + \)\(12\!\cdots\!27\)\( p^{56} T^{24} - \)\(12\!\cdots\!02\)\( p^{63} T^{25} + \)\(17\!\cdots\!58\)\( p^{70} T^{26} - \)\(12\!\cdots\!26\)\( p^{77} T^{27} + \)\(21\!\cdots\!99\)\( p^{84} T^{28} - 648124018861281 p^{91} T^{29} + 20814787185 p^{98} T^{30} + 2833 p^{105} T^{31} + p^{112} T^{32} \) | |
| 29 | \( 1 - 144375 T + 144322094863 T^{2} - 16887377443378959 T^{3} + \)\(99\!\cdots\!65\)\( T^{4} - \)\(10\!\cdots\!38\)\( T^{5} + \)\(44\!\cdots\!34\)\( T^{6} - \)\(43\!\cdots\!22\)\( T^{7} + \)\(15\!\cdots\!29\)\( T^{8} - \)\(14\!\cdots\!47\)\( T^{9} + \)\(40\!\cdots\!45\)\( T^{10} - \)\(38\!\cdots\!37\)\( T^{11} + \)\(91\!\cdots\!07\)\( T^{12} - \)\(89\!\cdots\!82\)\( T^{13} + \)\(18\!\cdots\!54\)\( T^{14} - \)\(17\!\cdots\!60\)\( T^{15} + \)\(32\!\cdots\!44\)\( T^{16} - \)\(17\!\cdots\!60\)\( p^{7} T^{17} + \)\(18\!\cdots\!54\)\( p^{14} T^{18} - \)\(89\!\cdots\!82\)\( p^{21} T^{19} + \)\(91\!\cdots\!07\)\( p^{28} T^{20} - \)\(38\!\cdots\!37\)\( p^{35} T^{21} + \)\(40\!\cdots\!45\)\( p^{42} T^{22} - \)\(14\!\cdots\!47\)\( p^{49} T^{23} + \)\(15\!\cdots\!29\)\( p^{56} T^{24} - \)\(43\!\cdots\!22\)\( p^{63} T^{25} + \)\(44\!\cdots\!34\)\( p^{70} T^{26} - \)\(10\!\cdots\!38\)\( p^{77} T^{27} + \)\(99\!\cdots\!65\)\( p^{84} T^{28} - 16887377443378959 p^{91} T^{29} + 144322094863 p^{98} T^{30} - 144375 p^{105} T^{31} + p^{112} T^{32} \) | |
| 31 | \( 1 + 141759 T + 311746829431 T^{2} + 38304726706398301 T^{3} + \)\(14\!\cdots\!43\)\( p T^{4} + \)\(48\!\cdots\!90\)\( T^{5} + \)\(44\!\cdots\!86\)\( T^{6} + \)\(38\!\cdots\!98\)\( T^{7} + \)\(29\!\cdots\!53\)\( T^{8} + \)\(21\!\cdots\!79\)\( T^{9} + \)\(15\!\cdots\!33\)\( T^{10} + \)\(86\!\cdots\!39\)\( T^{11} + \)\(64\!\cdots\!39\)\( T^{12} + \)\(28\!\cdots\!66\)\( T^{13} + \)\(22\!\cdots\!42\)\( T^{14} + \)\(82\!\cdots\!24\)\( T^{15} + \)\(66\!\cdots\!32\)\( T^{16} + \)\(82\!\cdots\!24\)\( p^{7} T^{17} + \)\(22\!\cdots\!42\)\( p^{14} T^{18} + \)\(28\!\cdots\!66\)\( p^{21} T^{19} + \)\(64\!\cdots\!39\)\( p^{28} T^{20} + \)\(86\!\cdots\!39\)\( p^{35} T^{21} + \)\(15\!\cdots\!33\)\( p^{42} T^{22} + \)\(21\!\cdots\!79\)\( p^{49} T^{23} + \)\(29\!\cdots\!53\)\( p^{56} T^{24} + \)\(38\!\cdots\!98\)\( p^{63} T^{25} + \)\(44\!\cdots\!86\)\( p^{70} T^{26} + \)\(48\!\cdots\!90\)\( p^{77} T^{27} + \)\(14\!\cdots\!43\)\( p^{85} T^{28} + 38304726706398301 p^{91} T^{29} + 311746829431 p^{98} T^{30} + 141759 p^{105} T^{31} + p^{112} T^{32} \) | |
| 37 | \( 1 + 297971 T + 17316545788 p T^{2} + 174031755215644656 T^{3} + \)\(22\!\cdots\!82\)\( T^{4} + \)\(59\!\cdots\!53\)\( T^{5} + \)\(56\!\cdots\!16\)\( T^{6} + \)\(14\!\cdots\!09\)\( T^{7} + \)\(11\!\cdots\!34\)\( T^{8} + \)\(27\!\cdots\!96\)\( T^{9} + \)\(18\!\cdots\!42\)\( T^{10} + \)\(42\!\cdots\!61\)\( T^{11} + \)\(24\!\cdots\!29\)\( T^{12} + \)\(55\!\cdots\!08\)\( T^{13} + \)\(29\!\cdots\!76\)\( T^{14} + \)\(61\!\cdots\!82\)\( T^{15} + \)\(29\!\cdots\!52\)\( T^{16} + \)\(61\!\cdots\!82\)\( p^{7} T^{17} + \)\(29\!\cdots\!76\)\( p^{14} T^{18} + \)\(55\!\cdots\!08\)\( p^{21} T^{19} + \)\(24\!\cdots\!29\)\( p^{28} T^{20} + \)\(42\!\cdots\!61\)\( p^{35} T^{21} + \)\(18\!\cdots\!42\)\( p^{42} T^{22} + \)\(27\!\cdots\!96\)\( p^{49} T^{23} + \)\(11\!\cdots\!34\)\( p^{56} T^{24} + \)\(14\!\cdots\!09\)\( p^{63} T^{25} + \)\(56\!\cdots\!16\)\( p^{70} T^{26} + \)\(59\!\cdots\!53\)\( p^{77} T^{27} + \)\(22\!\cdots\!82\)\( p^{84} T^{28} + 174031755215644656 p^{91} T^{29} + 17316545788 p^{99} T^{30} + 297971 p^{105} T^{31} + p^{112} T^{32} \) | |
| 41 | \( 1 - 659077 T + 1837498595458 T^{2} - 1066858548141989670 T^{3} + \)\(16\!\cdots\!60\)\( T^{4} - \)\(89\!\cdots\!05\)\( T^{5} + \)\(10\!\cdots\!34\)\( T^{6} - \)\(50\!\cdots\!97\)\( T^{7} + \)\(46\!\cdots\!44\)\( T^{8} - \)\(21\!\cdots\!50\)\( T^{9} + \)\(17\!\cdots\!84\)\( T^{10} - \)\(74\!\cdots\!71\)\( T^{11} + \)\(51\!\cdots\!81\)\( T^{12} - \)\(20\!\cdots\!84\)\( T^{13} + \)\(12\!\cdots\!48\)\( T^{14} - \)\(48\!\cdots\!94\)\( T^{15} + \)\(27\!\cdots\!76\)\( T^{16} - \)\(48\!\cdots\!94\)\( p^{7} T^{17} + \)\(12\!\cdots\!48\)\( p^{14} T^{18} - \)\(20\!\cdots\!84\)\( p^{21} T^{19} + \)\(51\!\cdots\!81\)\( p^{28} T^{20} - \)\(74\!\cdots\!71\)\( p^{35} T^{21} + \)\(17\!\cdots\!84\)\( p^{42} T^{22} - \)\(21\!\cdots\!50\)\( p^{49} T^{23} + \)\(46\!\cdots\!44\)\( p^{56} T^{24} - \)\(50\!\cdots\!97\)\( p^{63} T^{25} + \)\(10\!\cdots\!34\)\( p^{70} T^{26} - \)\(89\!\cdots\!05\)\( p^{77} T^{27} + \)\(16\!\cdots\!60\)\( p^{84} T^{28} - 1066858548141989670 p^{91} T^{29} + 1837498595458 p^{98} T^{30} - 659077 p^{105} T^{31} + p^{112} T^{32} \) | |
| 43 | \( 1 + 1431608 T + 2523178123406 T^{2} + 2698404381912396320 T^{3} + \)\(31\!\cdots\!79\)\( T^{4} + \)\(28\!\cdots\!56\)\( T^{5} + \)\(26\!\cdots\!02\)\( T^{6} + \)\(20\!\cdots\!36\)\( T^{7} + \)\(16\!\cdots\!25\)\( T^{8} + \)\(11\!\cdots\!96\)\( T^{9} + \)\(82\!\cdots\!60\)\( T^{10} + \)\(53\!\cdots\!04\)\( T^{11} + \)\(34\!\cdots\!10\)\( T^{12} + \)\(20\!\cdots\!72\)\( T^{13} + \)\(11\!\cdots\!72\)\( T^{14} + \)\(65\!\cdots\!24\)\( T^{15} + \)\(35\!\cdots\!30\)\( T^{16} + \)\(65\!\cdots\!24\)\( p^{7} T^{17} + \)\(11\!\cdots\!72\)\( p^{14} T^{18} + \)\(20\!\cdots\!72\)\( p^{21} T^{19} + \)\(34\!\cdots\!10\)\( p^{28} T^{20} + \)\(53\!\cdots\!04\)\( p^{35} T^{21} + \)\(82\!\cdots\!60\)\( p^{42} T^{22} + \)\(11\!\cdots\!96\)\( p^{49} T^{23} + \)\(16\!\cdots\!25\)\( p^{56} T^{24} + \)\(20\!\cdots\!36\)\( p^{63} T^{25} + \)\(26\!\cdots\!02\)\( p^{70} T^{26} + \)\(28\!\cdots\!56\)\( p^{77} T^{27} + \)\(31\!\cdots\!79\)\( p^{84} T^{28} + 2698404381912396320 p^{91} T^{29} + 2523178123406 p^{98} T^{30} + 1431608 p^{105} T^{31} + p^{112} T^{32} \) | |
| 47 | \( 1 + 1574073 T + 3535103246343 T^{2} + 4432622288871442935 T^{3} + \)\(66\!\cdots\!87\)\( T^{4} + \)\(72\!\cdots\!62\)\( T^{5} + \)\(87\!\cdots\!78\)\( T^{6} + \)\(85\!\cdots\!10\)\( T^{7} + \)\(88\!\cdots\!95\)\( T^{8} + \)\(79\!\cdots\!05\)\( T^{9} + \)\(73\!\cdots\!73\)\( T^{10} + \)\(60\!\cdots\!01\)\( T^{11} + \)\(51\!\cdots\!71\)\( T^{12} + \)\(39\!\cdots\!70\)\( T^{13} + \)\(31\!\cdots\!06\)\( T^{14} + \)\(22\!\cdots\!08\)\( T^{15} + \)\(16\!\cdots\!68\)\( T^{16} + \)\(22\!\cdots\!08\)\( p^{7} T^{17} + \)\(31\!\cdots\!06\)\( p^{14} T^{18} + \)\(39\!\cdots\!70\)\( p^{21} T^{19} + \)\(51\!\cdots\!71\)\( p^{28} T^{20} + \)\(60\!\cdots\!01\)\( p^{35} T^{21} + \)\(73\!\cdots\!73\)\( p^{42} T^{22} + \)\(79\!\cdots\!05\)\( p^{49} T^{23} + \)\(88\!\cdots\!95\)\( p^{56} T^{24} + \)\(85\!\cdots\!10\)\( p^{63} T^{25} + \)\(87\!\cdots\!78\)\( p^{70} T^{26} + \)\(72\!\cdots\!62\)\( p^{77} T^{27} + \)\(66\!\cdots\!87\)\( p^{84} T^{28} + 4432622288871442935 p^{91} T^{29} + 3535103246343 p^{98} T^{30} + 1574073 p^{105} T^{31} + p^{112} T^{32} \) | |
| 53 | \( 1 - 587736 T + 10691953380791 T^{2} - 4079852343577778034 T^{3} + \)\(55\!\cdots\!98\)\( T^{4} - \)\(21\!\cdots\!46\)\( p T^{5} + \)\(19\!\cdots\!63\)\( T^{6} - \)\(78\!\cdots\!56\)\( T^{7} + \)\(48\!\cdots\!85\)\( T^{8} + \)\(45\!\cdots\!76\)\( T^{9} + \)\(99\!\cdots\!36\)\( T^{10} + \)\(19\!\cdots\!56\)\( T^{11} + \)\(17\!\cdots\!18\)\( T^{12} + \)\(43\!\cdots\!68\)\( T^{13} + \)\(46\!\cdots\!02\)\( p T^{14} + \)\(68\!\cdots\!84\)\( T^{15} + \)\(31\!\cdots\!36\)\( T^{16} + \)\(68\!\cdots\!84\)\( p^{7} T^{17} + \)\(46\!\cdots\!02\)\( p^{15} T^{18} + \)\(43\!\cdots\!68\)\( p^{21} T^{19} + \)\(17\!\cdots\!18\)\( p^{28} T^{20} + \)\(19\!\cdots\!56\)\( p^{35} T^{21} + \)\(99\!\cdots\!36\)\( p^{42} T^{22} + \)\(45\!\cdots\!76\)\( p^{49} T^{23} + \)\(48\!\cdots\!85\)\( p^{56} T^{24} - \)\(78\!\cdots\!56\)\( p^{63} T^{25} + \)\(19\!\cdots\!63\)\( p^{70} T^{26} - \)\(21\!\cdots\!46\)\( p^{78} T^{27} + \)\(55\!\cdots\!98\)\( p^{84} T^{28} - 4079852343577778034 p^{91} T^{29} + 10691953380791 p^{98} T^{30} - 587736 p^{105} T^{31} + p^{112} T^{32} \) | |
| 61 | \( 1 + 6117131 T + 47369369697139 T^{2} + \)\(20\!\cdots\!75\)\( T^{3} + \)\(95\!\cdots\!09\)\( T^{4} + \)\(33\!\cdots\!50\)\( T^{5} + \)\(11\!\cdots\!62\)\( T^{6} + \)\(34\!\cdots\!50\)\( T^{7} + \)\(10\!\cdots\!13\)\( T^{8} + \)\(25\!\cdots\!15\)\( T^{9} + \)\(65\!\cdots\!69\)\( T^{10} + \)\(14\!\cdots\!65\)\( T^{11} + \)\(33\!\cdots\!91\)\( T^{12} + \)\(68\!\cdots\!26\)\( T^{13} + \)\(13\!\cdots\!10\)\( T^{14} + \)\(25\!\cdots\!80\)\( T^{15} + \)\(48\!\cdots\!56\)\( T^{16} + \)\(25\!\cdots\!80\)\( p^{7} T^{17} + \)\(13\!\cdots\!10\)\( p^{14} T^{18} + \)\(68\!\cdots\!26\)\( p^{21} T^{19} + \)\(33\!\cdots\!91\)\( p^{28} T^{20} + \)\(14\!\cdots\!65\)\( p^{35} T^{21} + \)\(65\!\cdots\!69\)\( p^{42} T^{22} + \)\(25\!\cdots\!15\)\( p^{49} T^{23} + \)\(10\!\cdots\!13\)\( p^{56} T^{24} + \)\(34\!\cdots\!50\)\( p^{63} T^{25} + \)\(11\!\cdots\!62\)\( p^{70} T^{26} + \)\(33\!\cdots\!50\)\( p^{77} T^{27} + \)\(95\!\cdots\!09\)\( p^{84} T^{28} + \)\(20\!\cdots\!75\)\( p^{91} T^{29} + 47369369697139 p^{98} T^{30} + 6117131 p^{105} T^{31} + p^{112} T^{32} \) | |
| 67 | \( 1 + 16518710 T + 196069542561581 T^{2} + \)\(16\!\cdots\!34\)\( T^{3} + \)\(12\!\cdots\!16\)\( T^{4} + \)\(74\!\cdots\!54\)\( T^{5} + \)\(40\!\cdots\!41\)\( T^{6} + \)\(19\!\cdots\!50\)\( T^{7} + \)\(86\!\cdots\!89\)\( T^{8} + \)\(34\!\cdots\!08\)\( T^{9} + \)\(13\!\cdots\!52\)\( T^{10} + \)\(45\!\cdots\!88\)\( T^{11} + \)\(14\!\cdots\!34\)\( T^{12} + \)\(44\!\cdots\!56\)\( T^{13} + \)\(12\!\cdots\!42\)\( T^{14} + \)\(34\!\cdots\!44\)\( T^{15} + \)\(87\!\cdots\!28\)\( T^{16} + \)\(34\!\cdots\!44\)\( p^{7} T^{17} + \)\(12\!\cdots\!42\)\( p^{14} T^{18} + \)\(44\!\cdots\!56\)\( p^{21} T^{19} + \)\(14\!\cdots\!34\)\( p^{28} T^{20} + \)\(45\!\cdots\!88\)\( p^{35} T^{21} + \)\(13\!\cdots\!52\)\( p^{42} T^{22} + \)\(34\!\cdots\!08\)\( p^{49} T^{23} + \)\(86\!\cdots\!89\)\( p^{56} T^{24} + \)\(19\!\cdots\!50\)\( p^{63} T^{25} + \)\(40\!\cdots\!41\)\( p^{70} T^{26} + \)\(74\!\cdots\!54\)\( p^{77} T^{27} + \)\(12\!\cdots\!16\)\( p^{84} T^{28} + \)\(16\!\cdots\!34\)\( p^{91} T^{29} + 196069542561581 p^{98} T^{30} + 16518710 p^{105} T^{31} + p^{112} T^{32} \) | |
| 71 | \( 1 + 10882582 T + 86794164883412 T^{2} + \)\(58\!\cdots\!10\)\( T^{3} + \)\(35\!\cdots\!71\)\( T^{4} + \)\(19\!\cdots\!48\)\( T^{5} + \)\(96\!\cdots\!06\)\( T^{6} + \)\(45\!\cdots\!92\)\( T^{7} + \)\(19\!\cdots\!21\)\( T^{8} + \)\(83\!\cdots\!06\)\( T^{9} + \)\(33\!\cdots\!62\)\( T^{10} + \)\(12\!\cdots\!22\)\( T^{11} + \)\(64\!\cdots\!90\)\( p T^{12} + \)\(15\!\cdots\!30\)\( T^{13} + \)\(52\!\cdots\!22\)\( T^{14} + \)\(16\!\cdots\!10\)\( T^{15} + \)\(52\!\cdots\!10\)\( T^{16} + \)\(16\!\cdots\!10\)\( p^{7} T^{17} + \)\(52\!\cdots\!22\)\( p^{14} T^{18} + \)\(15\!\cdots\!30\)\( p^{21} T^{19} + \)\(64\!\cdots\!90\)\( p^{29} T^{20} + \)\(12\!\cdots\!22\)\( p^{35} T^{21} + \)\(33\!\cdots\!62\)\( p^{42} T^{22} + \)\(83\!\cdots\!06\)\( p^{49} T^{23} + \)\(19\!\cdots\!21\)\( p^{56} T^{24} + \)\(45\!\cdots\!92\)\( p^{63} T^{25} + \)\(96\!\cdots\!06\)\( p^{70} T^{26} + \)\(19\!\cdots\!48\)\( p^{77} T^{27} + \)\(35\!\cdots\!71\)\( p^{84} T^{28} + \)\(58\!\cdots\!10\)\( p^{91} T^{29} + 86794164883412 p^{98} T^{30} + 10882582 p^{105} T^{31} + p^{112} T^{32} \) | |
| 73 | \( 1 + 21097441 T + 337490184288491 T^{2} + \)\(38\!\cdots\!29\)\( T^{3} + \)\(38\!\cdots\!67\)\( T^{4} + \)\(31\!\cdots\!82\)\( T^{5} + \)\(23\!\cdots\!22\)\( T^{6} + \)\(15\!\cdots\!58\)\( T^{7} + \)\(93\!\cdots\!75\)\( T^{8} + \)\(51\!\cdots\!81\)\( T^{9} + \)\(26\!\cdots\!81\)\( T^{10} + \)\(12\!\cdots\!83\)\( T^{11} + \)\(55\!\cdots\!07\)\( T^{12} + \)\(22\!\cdots\!10\)\( T^{13} + \)\(89\!\cdots\!66\)\( T^{14} + \)\(32\!\cdots\!00\)\( T^{15} + \)\(11\!\cdots\!20\)\( T^{16} + \)\(32\!\cdots\!00\)\( p^{7} T^{17} + \)\(89\!\cdots\!66\)\( p^{14} T^{18} + \)\(22\!\cdots\!10\)\( p^{21} T^{19} + \)\(55\!\cdots\!07\)\( p^{28} T^{20} + \)\(12\!\cdots\!83\)\( p^{35} T^{21} + \)\(26\!\cdots\!81\)\( p^{42} T^{22} + \)\(51\!\cdots\!81\)\( p^{49} T^{23} + \)\(93\!\cdots\!75\)\( p^{56} T^{24} + \)\(15\!\cdots\!58\)\( p^{63} T^{25} + \)\(23\!\cdots\!22\)\( p^{70} T^{26} + \)\(31\!\cdots\!82\)\( p^{77} T^{27} + \)\(38\!\cdots\!67\)\( p^{84} T^{28} + \)\(38\!\cdots\!29\)\( p^{91} T^{29} + 337490184288491 p^{98} T^{30} + 21097441 p^{105} T^{31} + p^{112} T^{32} \) | |
| 79 | \( 1 + 3784458 T + 142950984155520 T^{2} + \)\(56\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!77\)\( T^{4} + \)\(45\!\cdots\!92\)\( T^{5} + \)\(55\!\cdots\!72\)\( T^{6} + \)\(24\!\cdots\!74\)\( T^{7} + \)\(22\!\cdots\!77\)\( T^{8} + \)\(10\!\cdots\!96\)\( T^{9} + \)\(74\!\cdots\!80\)\( T^{10} + \)\(33\!\cdots\!40\)\( T^{11} + \)\(20\!\cdots\!66\)\( T^{12} + \)\(89\!\cdots\!28\)\( T^{13} + \)\(50\!\cdots\!60\)\( T^{14} + \)\(20\!\cdots\!72\)\( T^{15} + \)\(10\!\cdots\!78\)\( T^{16} + \)\(20\!\cdots\!72\)\( p^{7} T^{17} + \)\(50\!\cdots\!60\)\( p^{14} T^{18} + \)\(89\!\cdots\!28\)\( p^{21} T^{19} + \)\(20\!\cdots\!66\)\( p^{28} T^{20} + \)\(33\!\cdots\!40\)\( p^{35} T^{21} + \)\(74\!\cdots\!80\)\( p^{42} T^{22} + \)\(10\!\cdots\!96\)\( p^{49} T^{23} + \)\(22\!\cdots\!77\)\( p^{56} T^{24} + \)\(24\!\cdots\!74\)\( p^{63} T^{25} + \)\(55\!\cdots\!72\)\( p^{70} T^{26} + \)\(45\!\cdots\!92\)\( p^{77} T^{27} + \)\(10\!\cdots\!77\)\( p^{84} T^{28} + \)\(56\!\cdots\!12\)\( p^{91} T^{29} + 142950984155520 p^{98} T^{30} + 3784458 p^{105} T^{31} + p^{112} T^{32} \) | |
| 83 | \( 1 + 1951425 T + 177736299568080 T^{2} + \)\(12\!\cdots\!64\)\( T^{3} + \)\(16\!\cdots\!06\)\( T^{4} - \)\(59\!\cdots\!05\)\( T^{5} + \)\(10\!\cdots\!22\)\( T^{6} - \)\(11\!\cdots\!03\)\( T^{7} + \)\(50\!\cdots\!14\)\( T^{8} - \)\(81\!\cdots\!40\)\( T^{9} + \)\(21\!\cdots\!30\)\( T^{10} - \)\(38\!\cdots\!35\)\( T^{11} + \)\(78\!\cdots\!73\)\( T^{12} - \)\(14\!\cdots\!64\)\( T^{13} + \)\(25\!\cdots\!76\)\( T^{14} - \)\(48\!\cdots\!06\)\( T^{15} + \)\(72\!\cdots\!88\)\( T^{16} - \)\(48\!\cdots\!06\)\( p^{7} T^{17} + \)\(25\!\cdots\!76\)\( p^{14} T^{18} - \)\(14\!\cdots\!64\)\( p^{21} T^{19} + \)\(78\!\cdots\!73\)\( p^{28} T^{20} - \)\(38\!\cdots\!35\)\( p^{35} T^{21} + \)\(21\!\cdots\!30\)\( p^{42} T^{22} - \)\(81\!\cdots\!40\)\( p^{49} T^{23} + \)\(50\!\cdots\!14\)\( p^{56} T^{24} - \)\(11\!\cdots\!03\)\( p^{63} T^{25} + \)\(10\!\cdots\!22\)\( p^{70} T^{26} - \)\(59\!\cdots\!05\)\( p^{77} T^{27} + \)\(16\!\cdots\!06\)\( p^{84} T^{28} + \)\(12\!\cdots\!64\)\( p^{91} T^{29} + 177736299568080 p^{98} T^{30} + 1951425 p^{105} T^{31} + p^{112} T^{32} \) | |
| 89 | \( 1 - 10499443 T + 426097160615769 T^{2} - \)\(42\!\cdots\!15\)\( T^{3} + \)\(91\!\cdots\!59\)\( T^{4} - \)\(86\!\cdots\!14\)\( T^{5} + \)\(13\!\cdots\!38\)\( T^{6} - \)\(11\!\cdots\!50\)\( T^{7} + \)\(13\!\cdots\!19\)\( T^{8} - \)\(11\!\cdots\!43\)\( T^{9} + \)\(11\!\cdots\!47\)\( T^{10} - \)\(86\!\cdots\!21\)\( T^{11} + \)\(78\!\cdots\!47\)\( T^{12} - \)\(54\!\cdots\!14\)\( T^{13} + \)\(44\!\cdots\!18\)\( T^{14} - \)\(28\!\cdots\!80\)\( T^{15} + \)\(21\!\cdots\!04\)\( T^{16} - \)\(28\!\cdots\!80\)\( p^{7} T^{17} + \)\(44\!\cdots\!18\)\( p^{14} T^{18} - \)\(54\!\cdots\!14\)\( p^{21} T^{19} + \)\(78\!\cdots\!47\)\( p^{28} T^{20} - \)\(86\!\cdots\!21\)\( p^{35} T^{21} + \)\(11\!\cdots\!47\)\( p^{42} T^{22} - \)\(11\!\cdots\!43\)\( p^{49} T^{23} + \)\(13\!\cdots\!19\)\( p^{56} T^{24} - \)\(11\!\cdots\!50\)\( p^{63} T^{25} + \)\(13\!\cdots\!38\)\( p^{70} T^{26} - \)\(86\!\cdots\!14\)\( p^{77} T^{27} + \)\(91\!\cdots\!59\)\( p^{84} T^{28} - \)\(42\!\cdots\!15\)\( p^{91} T^{29} + 426097160615769 p^{98} T^{30} - 10499443 p^{105} T^{31} + p^{112} T^{32} \) | |
| 97 | \( 1 + 25158976 T + 9489803532067 p T^{2} + \)\(17\!\cdots\!08\)\( T^{3} + \)\(36\!\cdots\!44\)\( T^{4} + \)\(55\!\cdots\!08\)\( T^{5} + \)\(91\!\cdots\!59\)\( T^{6} + \)\(11\!\cdots\!24\)\( T^{7} + \)\(16\!\cdots\!05\)\( T^{8} + \)\(18\!\cdots\!56\)\( T^{9} + \)\(23\!\cdots\!12\)\( T^{10} + \)\(24\!\cdots\!60\)\( T^{11} + \)\(27\!\cdots\!22\)\( T^{12} + \)\(26\!\cdots\!88\)\( T^{13} + \)\(27\!\cdots\!94\)\( T^{14} + \)\(25\!\cdots\!28\)\( T^{15} + \)\(24\!\cdots\!12\)\( T^{16} + \)\(25\!\cdots\!28\)\( p^{7} T^{17} + \)\(27\!\cdots\!94\)\( p^{14} T^{18} + \)\(26\!\cdots\!88\)\( p^{21} T^{19} + \)\(27\!\cdots\!22\)\( p^{28} T^{20} + \)\(24\!\cdots\!60\)\( p^{35} T^{21} + \)\(23\!\cdots\!12\)\( p^{42} T^{22} + \)\(18\!\cdots\!56\)\( p^{49} T^{23} + \)\(16\!\cdots\!05\)\( p^{56} T^{24} + \)\(11\!\cdots\!24\)\( p^{63} T^{25} + \)\(91\!\cdots\!59\)\( p^{70} T^{26} + \)\(55\!\cdots\!08\)\( p^{77} T^{27} + \)\(36\!\cdots\!44\)\( p^{84} T^{28} + \)\(17\!\cdots\!08\)\( p^{91} T^{29} + 9489803532067 p^{99} T^{30} + 25158976 p^{105} T^{31} + p^{112} T^{32} \) | |
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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
−3.21431499932930812189016655562, −3.18196013146654100597708254876, −2.97535620155047033052839010204, −2.82248641663990520093085642815, −2.63562009327498367068622249221, −2.45414143509716166340650843244, −2.33122787075823935700890824755, −2.24430451568000857971306655278, −2.20026742264310832307499913840, −2.04318667862296378303533185899, −1.93410422473796113134792743784, −1.92961638747251886527349150656, −1.79272410444389339094485643461, −1.79225551904875205780366645203, −1.53578013912109900291578162600, −1.32950575629041538734998936575, −1.30053335634186982595138684331, −1.25890634735935087849180963810, −1.25614495766701843157842624862, −1.21660521409085229530038292975, −1.15737645649049497958418711907, −1.02251000544528177480807705273, −0.990043362036185300131713793310, −0.78777653555696522991821047370, −0.77668908563956621872924272170, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.77668908563956621872924272170, 0.78777653555696522991821047370, 0.990043362036185300131713793310, 1.02251000544528177480807705273, 1.15737645649049497958418711907, 1.21660521409085229530038292975, 1.25614495766701843157842624862, 1.25890634735935087849180963810, 1.30053335634186982595138684331, 1.32950575629041538734998936575, 1.53578013912109900291578162600, 1.79225551904875205780366645203, 1.79272410444389339094485643461, 1.92961638747251886527349150656, 1.93410422473796113134792743784, 2.04318667862296378303533185899, 2.20026742264310832307499913840, 2.24430451568000857971306655278, 2.33122787075823935700890824755, 2.45414143509716166340650843244, 2.63562009327498367068622249221, 2.82248641663990520093085642815, 2.97535620155047033052839010204, 3.18196013146654100597708254876, 3.21431499932930812189016655562
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.