Dirichlet series
| L(s) = 1 | + 91·3-s + 801·5-s + 4.82e3·7-s + 6.47e4·9-s + 4.27e4·11-s + 3.01e5·13-s + 7.28e4·15-s + 1.13e4·17-s + 1.23e6·19-s + 4.38e5·21-s − 9.50e4·23-s + 8.68e6·25-s + 6.50e6·27-s + 1.20e7·29-s + 1.04e7·31-s + 3.88e6·33-s + 3.86e6·35-s + 1.84e7·37-s + 2.74e7·39-s − 9.44e6·41-s − 1.55e7·43-s + 5.18e7·45-s − 6.48e6·47-s − 3.69e7·49-s + 1.03e6·51-s + 1.53e7·53-s + 3.42e7·55-s + ⋯ |
| L(s) = 1 | + 0.648·3-s + 0.573·5-s + 0.758·7-s + 3.29·9-s + 0.879·11-s + 2.92·13-s + 0.371·15-s + 0.0329·17-s + 2.17·19-s + 0.492·21-s − 0.0708·23-s + 4.44·25-s + 2.35·27-s + 3.15·29-s + 2.03·31-s + 0.570·33-s + 0.434·35-s + 1.62·37-s + 1.89·39-s − 0.522·41-s − 0.695·43-s + 1.88·45-s − 0.193·47-s − 0.915·49-s + 0.0213·51-s + 0.266·53-s + 0.504·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{72} \cdot 7^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.00032\times 10^{31}\) |
| Root analytic conductor: | \(7.59499\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{72} \cdot 7^{18} ,\ ( \ : [9/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(23.72576881\) |
| \(L(\frac12)\) | \(\approx\) | \(23.72576881\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 4820 T + 60175839 T^{2} + 60825170528 p T^{3} - 4660841400996 p^{3} T^{4} + 2347563489157552 p^{5} T^{5} + 7885754443147628 p^{8} T^{6} + 230375366381432992 p^{11} T^{7} + 15178090949264336896 p^{14} T^{8} - 13763238141217003520 p^{17} T^{9} + 15178090949264336896 p^{23} T^{10} + 230375366381432992 p^{29} T^{11} + 7885754443147628 p^{35} T^{12} + 2347563489157552 p^{41} T^{13} - 4660841400996 p^{48} T^{14} + 60825170528 p^{55} T^{15} + 60175839 p^{63} T^{16} - 4820 p^{72} T^{17} + p^{81} T^{18} \) | |
| good | 3 | \( 1 - 91 T - 18830 p T^{2} + 4525535 T^{3} + 1541734915 T^{4} - 58709266 p^{7} T^{5} - 2743055284700 p^{2} T^{6} + 35321987054270 p^{4} T^{7} + 2266222475380703 p^{4} T^{8} - 80795945773589431 p^{6} T^{9} + 2472219562270472350 p^{6} T^{10} + 73466664912084676393 p^{9} T^{11} - \)\(14\!\cdots\!21\)\( p^{8} T^{12} - \)\(64\!\cdots\!42\)\( p^{10} T^{13} + \)\(51\!\cdots\!08\)\( p^{10} T^{14} + \)\(13\!\cdots\!98\)\( p^{12} T^{15} - \)\(18\!\cdots\!86\)\( p^{12} T^{16} - \)\(39\!\cdots\!80\)\( p^{15} T^{17} + \)\(61\!\cdots\!72\)\( p^{18} T^{18} - \)\(39\!\cdots\!80\)\( p^{24} T^{19} - \)\(18\!\cdots\!86\)\( p^{30} T^{20} + \)\(13\!\cdots\!98\)\( p^{39} T^{21} + \)\(51\!\cdots\!08\)\( p^{46} T^{22} - \)\(64\!\cdots\!42\)\( p^{55} T^{23} - \)\(14\!\cdots\!21\)\( p^{62} T^{24} + 73466664912084676393 p^{72} T^{25} + 2472219562270472350 p^{78} T^{26} - 80795945773589431 p^{87} T^{27} + 2266222475380703 p^{94} T^{28} + 35321987054270 p^{103} T^{29} - 2743055284700 p^{110} T^{30} - 58709266 p^{124} T^{31} + 1541734915 p^{126} T^{32} + 4525535 p^{135} T^{33} - 18830 p^{145} T^{34} - 91 p^{153} T^{35} + p^{162} T^{36} \) |
| 5 | \( 1 - 801 T - 8044032 T^{2} + 3975414583 T^{3} + 1308770212959 p^{2} T^{4} - 10229326152997218 T^{5} - 16631070344312752076 p T^{6} + \)\(36\!\cdots\!98\)\( T^{7} + \)\(13\!\cdots\!03\)\( T^{8} - \)\(17\!\cdots\!61\)\( T^{9} - \)\(13\!\cdots\!04\)\( T^{10} + \)\(11\!\cdots\!03\)\( p T^{11} + \)\(66\!\cdots\!31\)\( p^{2} T^{12} - \)\(10\!\cdots\!86\)\( p^{3} T^{13} + \)\(17\!\cdots\!72\)\( p^{5} T^{14} + \)\(55\!\cdots\!06\)\( p^{5} T^{15} - \)\(15\!\cdots\!22\)\( p^{6} T^{16} - \)\(28\!\cdots\!44\)\( p^{8} T^{17} + \)\(15\!\cdots\!56\)\( p^{8} T^{18} - \)\(28\!\cdots\!44\)\( p^{17} T^{19} - \)\(15\!\cdots\!22\)\( p^{24} T^{20} + \)\(55\!\cdots\!06\)\( p^{32} T^{21} + \)\(17\!\cdots\!72\)\( p^{41} T^{22} - \)\(10\!\cdots\!86\)\( p^{48} T^{23} + \)\(66\!\cdots\!31\)\( p^{56} T^{24} + \)\(11\!\cdots\!03\)\( p^{64} T^{25} - \)\(13\!\cdots\!04\)\( p^{72} T^{26} - \)\(17\!\cdots\!61\)\( p^{81} T^{27} + \)\(13\!\cdots\!03\)\( p^{90} T^{28} + \)\(36\!\cdots\!98\)\( p^{99} T^{29} - 16631070344312752076 p^{109} T^{30} - 10229326152997218 p^{117} T^{31} + 1308770212959 p^{128} T^{32} + 3975414583 p^{135} T^{33} - 8044032 p^{144} T^{34} - 801 p^{153} T^{35} + p^{162} T^{36} \) | |
| 11 | \( 1 - 42719 T - 5974525818 T^{2} + 484004075865579 T^{3} + 8190747288791330915 T^{4} - \)\(22\!\cdots\!86\)\( T^{5} + \)\(71\!\cdots\!96\)\( T^{6} + \)\(51\!\cdots\!74\)\( T^{7} - \)\(46\!\cdots\!17\)\( T^{8} + \)\(10\!\cdots\!57\)\( T^{9} + \)\(13\!\cdots\!74\)\( T^{10} - \)\(52\!\cdots\!61\)\( T^{11} - \)\(18\!\cdots\!93\)\( T^{12} + \)\(21\!\cdots\!18\)\( T^{13} - \)\(24\!\cdots\!96\)\( T^{14} - \)\(42\!\cdots\!58\)\( p T^{15} + \)\(20\!\cdots\!30\)\( T^{16} + \)\(45\!\cdots\!52\)\( T^{17} - \)\(64\!\cdots\!32\)\( T^{18} + \)\(45\!\cdots\!52\)\( p^{9} T^{19} + \)\(20\!\cdots\!30\)\( p^{18} T^{20} - \)\(42\!\cdots\!58\)\( p^{28} T^{21} - \)\(24\!\cdots\!96\)\( p^{36} T^{22} + \)\(21\!\cdots\!18\)\( p^{45} T^{23} - \)\(18\!\cdots\!93\)\( p^{54} T^{24} - \)\(52\!\cdots\!61\)\( p^{63} T^{25} + \)\(13\!\cdots\!74\)\( p^{72} T^{26} + \)\(10\!\cdots\!57\)\( p^{81} T^{27} - \)\(46\!\cdots\!17\)\( p^{90} T^{28} + \)\(51\!\cdots\!74\)\( p^{99} T^{29} + \)\(71\!\cdots\!96\)\( p^{108} T^{30} - \)\(22\!\cdots\!86\)\( p^{117} T^{31} + 8190747288791330915 p^{126} T^{32} + 484004075865579 p^{135} T^{33} - 5974525818 p^{144} T^{34} - 42719 p^{153} T^{35} + p^{162} T^{36} \) | |
| 13 | \( ( 1 - 150810 T + 59764083333 T^{2} - 661250334845232 p T^{3} + \)\(19\!\cdots\!20\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{5} + \)\(39\!\cdots\!12\)\( T^{6} - \)\(44\!\cdots\!16\)\( T^{7} + \)\(57\!\cdots\!78\)\( T^{8} - \)\(55\!\cdots\!60\)\( T^{9} + \)\(57\!\cdots\!78\)\( p^{9} T^{10} - \)\(44\!\cdots\!16\)\( p^{18} T^{11} + \)\(39\!\cdots\!12\)\( p^{27} T^{12} - \)\(24\!\cdots\!20\)\( p^{36} T^{13} + \)\(19\!\cdots\!20\)\( p^{45} T^{14} - 661250334845232 p^{55} T^{15} + 59764083333 p^{63} T^{16} - 150810 p^{72} T^{17} + p^{81} T^{18} )^{2} \) | |
| 17 | \( 1 - 11357 T - 201097922868 T^{2} - 1599967542909505 p T^{3} + \)\(88\!\cdots\!99\)\( T^{4} + \)\(28\!\cdots\!42\)\( T^{5} + \)\(33\!\cdots\!40\)\( T^{6} + \)\(40\!\cdots\!78\)\( T^{7} - \)\(52\!\cdots\!89\)\( T^{8} - \)\(12\!\cdots\!41\)\( T^{9} + \)\(11\!\cdots\!36\)\( T^{10} + \)\(77\!\cdots\!51\)\( T^{11} + \)\(66\!\cdots\!51\)\( T^{12} + \)\(67\!\cdots\!14\)\( T^{13} - \)\(76\!\cdots\!64\)\( T^{14} - \)\(24\!\cdots\!02\)\( T^{15} - \)\(16\!\cdots\!34\)\( T^{16} + \)\(15\!\cdots\!36\)\( T^{17} + \)\(10\!\cdots\!28\)\( T^{18} + \)\(15\!\cdots\!36\)\( p^{9} T^{19} - \)\(16\!\cdots\!34\)\( p^{18} T^{20} - \)\(24\!\cdots\!02\)\( p^{27} T^{21} - \)\(76\!\cdots\!64\)\( p^{36} T^{22} + \)\(67\!\cdots\!14\)\( p^{45} T^{23} + \)\(66\!\cdots\!51\)\( p^{54} T^{24} + \)\(77\!\cdots\!51\)\( p^{63} T^{25} + \)\(11\!\cdots\!36\)\( p^{72} T^{26} - \)\(12\!\cdots\!41\)\( p^{81} T^{27} - \)\(52\!\cdots\!89\)\( p^{90} T^{28} + \)\(40\!\cdots\!78\)\( p^{99} T^{29} + \)\(33\!\cdots\!40\)\( p^{108} T^{30} + \)\(28\!\cdots\!42\)\( p^{117} T^{31} + \)\(88\!\cdots\!99\)\( p^{126} T^{32} - 1599967542909505 p^{136} T^{33} - 201097922868 p^{144} T^{34} - 11357 p^{153} T^{35} + p^{162} T^{36} \) | |
| 19 | \( 1 - 1237749 T - 1095582739242 T^{2} + 1881824767941403825 T^{3} + \)\(63\!\cdots\!87\)\( T^{4} - \)\(15\!\cdots\!66\)\( T^{5} - \)\(25\!\cdots\!56\)\( T^{6} + \)\(95\!\cdots\!50\)\( T^{7} + \)\(64\!\cdots\!19\)\( T^{8} - \)\(44\!\cdots\!17\)\( T^{9} - \)\(68\!\cdots\!22\)\( T^{10} + \)\(17\!\cdots\!81\)\( T^{11} - \)\(48\!\cdots\!53\)\( T^{12} - \)\(55\!\cdots\!42\)\( T^{13} + \)\(42\!\cdots\!00\)\( T^{14} + \)\(13\!\cdots\!66\)\( T^{15} - \)\(20\!\cdots\!02\)\( T^{16} - \)\(16\!\cdots\!16\)\( T^{17} + \)\(77\!\cdots\!56\)\( T^{18} - \)\(16\!\cdots\!16\)\( p^{9} T^{19} - \)\(20\!\cdots\!02\)\( p^{18} T^{20} + \)\(13\!\cdots\!66\)\( p^{27} T^{21} + \)\(42\!\cdots\!00\)\( p^{36} T^{22} - \)\(55\!\cdots\!42\)\( p^{45} T^{23} - \)\(48\!\cdots\!53\)\( p^{54} T^{24} + \)\(17\!\cdots\!81\)\( p^{63} T^{25} - \)\(68\!\cdots\!22\)\( p^{72} T^{26} - \)\(44\!\cdots\!17\)\( p^{81} T^{27} + \)\(64\!\cdots\!19\)\( p^{90} T^{28} + \)\(95\!\cdots\!50\)\( p^{99} T^{29} - \)\(25\!\cdots\!56\)\( p^{108} T^{30} - \)\(15\!\cdots\!66\)\( p^{117} T^{31} + \)\(63\!\cdots\!87\)\( p^{126} T^{32} + 1881824767941403825 p^{135} T^{33} - 1095582739242 p^{144} T^{34} - 1237749 p^{153} T^{35} + p^{162} T^{36} \) | |
| 23 | \( 1 + 95019 T - 9289652045214 T^{2} + 3447857591468862445 T^{3} + \)\(41\!\cdots\!23\)\( T^{4} - \)\(31\!\cdots\!74\)\( T^{5} - \)\(11\!\cdots\!24\)\( T^{6} + \)\(11\!\cdots\!78\)\( T^{7} + \)\(26\!\cdots\!67\)\( T^{8} - \)\(24\!\cdots\!41\)\( T^{9} - \)\(60\!\cdots\!42\)\( T^{10} + \)\(36\!\cdots\!81\)\( T^{11} + \)\(14\!\cdots\!63\)\( T^{12} - \)\(51\!\cdots\!22\)\( T^{13} - \)\(34\!\cdots\!88\)\( T^{14} + \)\(80\!\cdots\!90\)\( T^{15} + \)\(66\!\cdots\!34\)\( T^{16} - \)\(65\!\cdots\!28\)\( T^{17} - \)\(12\!\cdots\!16\)\( T^{18} - \)\(65\!\cdots\!28\)\( p^{9} T^{19} + \)\(66\!\cdots\!34\)\( p^{18} T^{20} + \)\(80\!\cdots\!90\)\( p^{27} T^{21} - \)\(34\!\cdots\!88\)\( p^{36} T^{22} - \)\(51\!\cdots\!22\)\( p^{45} T^{23} + \)\(14\!\cdots\!63\)\( p^{54} T^{24} + \)\(36\!\cdots\!81\)\( p^{63} T^{25} - \)\(60\!\cdots\!42\)\( p^{72} T^{26} - \)\(24\!\cdots\!41\)\( p^{81} T^{27} + \)\(26\!\cdots\!67\)\( p^{90} T^{28} + \)\(11\!\cdots\!78\)\( p^{99} T^{29} - \)\(11\!\cdots\!24\)\( p^{108} T^{30} - \)\(31\!\cdots\!74\)\( p^{117} T^{31} + \)\(41\!\cdots\!23\)\( p^{126} T^{32} + 3447857591468862445 p^{135} T^{33} - 9289652045214 p^{144} T^{34} + 95019 p^{153} T^{35} + p^{162} T^{36} \) | |
| 29 | \( ( 1 - 6005058 T + 82840462383765 T^{2} - \)\(45\!\cdots\!28\)\( T^{3} + \)\(35\!\cdots\!48\)\( T^{4} - \)\(17\!\cdots\!64\)\( T^{5} + \)\(98\!\cdots\!60\)\( T^{6} - \)\(14\!\cdots\!92\)\( p T^{7} + \)\(19\!\cdots\!62\)\( T^{8} - \)\(70\!\cdots\!32\)\( T^{9} + \)\(19\!\cdots\!62\)\( p^{9} T^{10} - \)\(14\!\cdots\!92\)\( p^{19} T^{11} + \)\(98\!\cdots\!60\)\( p^{27} T^{12} - \)\(17\!\cdots\!64\)\( p^{36} T^{13} + \)\(35\!\cdots\!48\)\( p^{45} T^{14} - \)\(45\!\cdots\!28\)\( p^{54} T^{15} + 82840462383765 p^{63} T^{16} - 6005058 p^{72} T^{17} + p^{81} T^{18} )^{2} \) | |
| 31 | \( 1 - 10454399 T - 55693073134414 T^{2} + \)\(68\!\cdots\!91\)\( T^{3} + \)\(26\!\cdots\!71\)\( T^{4} - \)\(20\!\cdots\!14\)\( T^{5} - \)\(14\!\cdots\!88\)\( T^{6} + \)\(54\!\cdots\!34\)\( T^{7} + \)\(46\!\cdots\!91\)\( T^{8} - \)\(11\!\cdots\!11\)\( T^{9} - \)\(12\!\cdots\!50\)\( T^{10} + \)\(20\!\cdots\!87\)\( T^{11} + \)\(40\!\cdots\!63\)\( T^{12} - \)\(39\!\cdots\!82\)\( T^{13} - \)\(97\!\cdots\!88\)\( T^{14} - \)\(54\!\cdots\!26\)\( T^{15} + \)\(21\!\cdots\!30\)\( T^{16} + \)\(24\!\cdots\!24\)\( T^{17} - \)\(58\!\cdots\!84\)\( T^{18} + \)\(24\!\cdots\!24\)\( p^{9} T^{19} + \)\(21\!\cdots\!30\)\( p^{18} T^{20} - \)\(54\!\cdots\!26\)\( p^{27} T^{21} - \)\(97\!\cdots\!88\)\( p^{36} T^{22} - \)\(39\!\cdots\!82\)\( p^{45} T^{23} + \)\(40\!\cdots\!63\)\( p^{54} T^{24} + \)\(20\!\cdots\!87\)\( p^{63} T^{25} - \)\(12\!\cdots\!50\)\( p^{72} T^{26} - \)\(11\!\cdots\!11\)\( p^{81} T^{27} + \)\(46\!\cdots\!91\)\( p^{90} T^{28} + \)\(54\!\cdots\!34\)\( p^{99} T^{29} - \)\(14\!\cdots\!88\)\( p^{108} T^{30} - \)\(20\!\cdots\!14\)\( p^{117} T^{31} + \)\(26\!\cdots\!71\)\( p^{126} T^{32} + \)\(68\!\cdots\!91\)\( p^{135} T^{33} - 55693073134414 p^{144} T^{34} - 10454399 p^{153} T^{35} + p^{162} T^{36} \) | |
| 37 | \( 1 - 18487597 T - 302035180905176 T^{2} + \)\(62\!\cdots\!15\)\( T^{3} + \)\(47\!\cdots\!07\)\( T^{4} - \)\(97\!\cdots\!54\)\( T^{5} - \)\(44\!\cdots\!68\)\( T^{6} + \)\(77\!\cdots\!22\)\( T^{7} + \)\(78\!\cdots\!07\)\( T^{8} - \)\(12\!\cdots\!13\)\( p T^{9} + \)\(71\!\cdots\!96\)\( T^{10} + \)\(81\!\cdots\!95\)\( T^{11} - \)\(11\!\cdots\!69\)\( T^{12} - \)\(26\!\cdots\!42\)\( T^{13} + \)\(14\!\cdots\!40\)\( T^{14} + \)\(43\!\cdots\!78\)\( T^{15} - \)\(59\!\cdots\!70\)\( p T^{16} - \)\(27\!\cdots\!88\)\( T^{17} + \)\(34\!\cdots\!24\)\( T^{18} - \)\(27\!\cdots\!88\)\( p^{9} T^{19} - \)\(59\!\cdots\!70\)\( p^{19} T^{20} + \)\(43\!\cdots\!78\)\( p^{27} T^{21} + \)\(14\!\cdots\!40\)\( p^{36} T^{22} - \)\(26\!\cdots\!42\)\( p^{45} T^{23} - \)\(11\!\cdots\!69\)\( p^{54} T^{24} + \)\(81\!\cdots\!95\)\( p^{63} T^{25} + \)\(71\!\cdots\!96\)\( p^{72} T^{26} - \)\(12\!\cdots\!13\)\( p^{82} T^{27} + \)\(78\!\cdots\!07\)\( p^{90} T^{28} + \)\(77\!\cdots\!22\)\( p^{99} T^{29} - \)\(44\!\cdots\!68\)\( p^{108} T^{30} - \)\(97\!\cdots\!54\)\( p^{117} T^{31} + \)\(47\!\cdots\!07\)\( p^{126} T^{32} + \)\(62\!\cdots\!15\)\( p^{135} T^{33} - 302035180905176 p^{144} T^{34} - 18487597 p^{153} T^{35} + p^{162} T^{36} \) | |
| 41 | \( ( 1 + 4723010 T + 1856722852423329 T^{2} + \)\(27\!\cdots\!36\)\( T^{3} + \)\(16\!\cdots\!64\)\( T^{4} - \)\(23\!\cdots\!72\)\( T^{5} + \)\(93\!\cdots\!52\)\( T^{6} - \)\(32\!\cdots\!12\)\( T^{7} + \)\(38\!\cdots\!30\)\( T^{8} - \)\(15\!\cdots\!28\)\( T^{9} + \)\(38\!\cdots\!30\)\( p^{9} T^{10} - \)\(32\!\cdots\!12\)\( p^{18} T^{11} + \)\(93\!\cdots\!52\)\( p^{27} T^{12} - \)\(23\!\cdots\!72\)\( p^{36} T^{13} + \)\(16\!\cdots\!64\)\( p^{45} T^{14} + \)\(27\!\cdots\!36\)\( p^{54} T^{15} + 1856722852423329 p^{63} T^{16} + 4723010 p^{72} T^{17} + p^{81} T^{18} )^{2} \) | |
| 43 | \( ( 1 + 7794044 T + 2289378250425571 T^{2} - \)\(14\!\cdots\!08\)\( T^{3} + \)\(22\!\cdots\!76\)\( T^{4} - \)\(43\!\cdots\!48\)\( T^{5} + \)\(17\!\cdots\!56\)\( T^{6} - \)\(39\!\cdots\!52\)\( T^{7} + \)\(11\!\cdots\!94\)\( T^{8} - \)\(22\!\cdots\!32\)\( T^{9} + \)\(11\!\cdots\!94\)\( p^{9} T^{10} - \)\(39\!\cdots\!52\)\( p^{18} T^{11} + \)\(17\!\cdots\!56\)\( p^{27} T^{12} - \)\(43\!\cdots\!48\)\( p^{36} T^{13} + \)\(22\!\cdots\!76\)\( p^{45} T^{14} - \)\(14\!\cdots\!08\)\( p^{54} T^{15} + 2289378250425571 p^{63} T^{16} + 7794044 p^{72} T^{17} + p^{81} T^{18} )^{2} \) | |
| 47 | \( 1 + 6482527 T - 3339024543208094 T^{2} + \)\(55\!\cdots\!77\)\( T^{3} + \)\(47\!\cdots\!99\)\( T^{4} - \)\(25\!\cdots\!54\)\( T^{5} - \)\(25\!\cdots\!12\)\( T^{6} + \)\(43\!\cdots\!58\)\( T^{7} - \)\(36\!\cdots\!89\)\( T^{8} - \)\(46\!\cdots\!49\)\( T^{9} + \)\(95\!\cdots\!22\)\( T^{10} + \)\(43\!\cdots\!61\)\( T^{11} - \)\(13\!\cdots\!05\)\( T^{12} - \)\(51\!\cdots\!78\)\( T^{13} + \)\(20\!\cdots\!48\)\( T^{14} + \)\(57\!\cdots\!66\)\( T^{15} - \)\(35\!\cdots\!02\)\( T^{16} - \)\(28\!\cdots\!92\)\( T^{17} + \)\(47\!\cdots\!00\)\( T^{18} - \)\(28\!\cdots\!92\)\( p^{9} T^{19} - \)\(35\!\cdots\!02\)\( p^{18} T^{20} + \)\(57\!\cdots\!66\)\( p^{27} T^{21} + \)\(20\!\cdots\!48\)\( p^{36} T^{22} - \)\(51\!\cdots\!78\)\( p^{45} T^{23} - \)\(13\!\cdots\!05\)\( p^{54} T^{24} + \)\(43\!\cdots\!61\)\( p^{63} T^{25} + \)\(95\!\cdots\!22\)\( p^{72} T^{26} - \)\(46\!\cdots\!49\)\( p^{81} T^{27} - \)\(36\!\cdots\!89\)\( p^{90} T^{28} + \)\(43\!\cdots\!58\)\( p^{99} T^{29} - \)\(25\!\cdots\!12\)\( p^{108} T^{30} - \)\(25\!\cdots\!54\)\( p^{117} T^{31} + \)\(47\!\cdots\!99\)\( p^{126} T^{32} + \)\(55\!\cdots\!77\)\( p^{135} T^{33} - 3339024543208094 p^{144} T^{34} + 6482527 p^{153} T^{35} + p^{162} T^{36} \) | |
| 53 | \( 1 - 15314401 T - 17333731588980144 T^{2} + \)\(43\!\cdots\!91\)\( T^{3} + \)\(14\!\cdots\!03\)\( T^{4} - \)\(46\!\cdots\!06\)\( T^{5} - \)\(79\!\cdots\!72\)\( T^{6} + \)\(54\!\cdots\!22\)\( p T^{7} + \)\(38\!\cdots\!99\)\( T^{8} - \)\(13\!\cdots\!81\)\( T^{9} - \)\(17\!\cdots\!08\)\( T^{10} + \)\(51\!\cdots\!19\)\( T^{11} + \)\(68\!\cdots\!95\)\( T^{12} - \)\(16\!\cdots\!34\)\( T^{13} - \)\(25\!\cdots\!80\)\( T^{14} + \)\(37\!\cdots\!70\)\( T^{15} + \)\(91\!\cdots\!86\)\( T^{16} - \)\(42\!\cdots\!88\)\( T^{17} - \)\(31\!\cdots\!56\)\( T^{18} - \)\(42\!\cdots\!88\)\( p^{9} T^{19} + \)\(91\!\cdots\!86\)\( p^{18} T^{20} + \)\(37\!\cdots\!70\)\( p^{27} T^{21} - \)\(25\!\cdots\!80\)\( p^{36} T^{22} - \)\(16\!\cdots\!34\)\( p^{45} T^{23} + \)\(68\!\cdots\!95\)\( p^{54} T^{24} + \)\(51\!\cdots\!19\)\( p^{63} T^{25} - \)\(17\!\cdots\!08\)\( p^{72} T^{26} - \)\(13\!\cdots\!81\)\( p^{81} T^{27} + \)\(38\!\cdots\!99\)\( p^{90} T^{28} + \)\(54\!\cdots\!22\)\( p^{100} T^{29} - \)\(79\!\cdots\!72\)\( p^{108} T^{30} - \)\(46\!\cdots\!06\)\( p^{117} T^{31} + \)\(14\!\cdots\!03\)\( p^{126} T^{32} + \)\(43\!\cdots\!91\)\( p^{135} T^{33} - 17333731588980144 p^{144} T^{34} - 15314401 p^{153} T^{35} + p^{162} T^{36} \) | |
| 59 | \( 1 - 84965083 T - 39587058125800066 T^{2} + \)\(25\!\cdots\!99\)\( T^{3} + \)\(84\!\cdots\!51\)\( T^{4} - \)\(34\!\cdots\!82\)\( T^{5} - \)\(12\!\cdots\!96\)\( T^{6} + \)\(17\!\cdots\!10\)\( T^{7} + \)\(14\!\cdots\!71\)\( T^{8} + \)\(17\!\cdots\!73\)\( T^{9} - \)\(13\!\cdots\!42\)\( T^{10} - \)\(49\!\cdots\!81\)\( T^{11} + \)\(10\!\cdots\!95\)\( T^{12} + \)\(59\!\cdots\!98\)\( T^{13} - \)\(70\!\cdots\!76\)\( T^{14} - \)\(44\!\cdots\!22\)\( T^{15} + \)\(51\!\cdots\!90\)\( T^{16} + \)\(15\!\cdots\!28\)\( T^{17} - \)\(41\!\cdots\!12\)\( T^{18} + \)\(15\!\cdots\!28\)\( p^{9} T^{19} + \)\(51\!\cdots\!90\)\( p^{18} T^{20} - \)\(44\!\cdots\!22\)\( p^{27} T^{21} - \)\(70\!\cdots\!76\)\( p^{36} T^{22} + \)\(59\!\cdots\!98\)\( p^{45} T^{23} + \)\(10\!\cdots\!95\)\( p^{54} T^{24} - \)\(49\!\cdots\!81\)\( p^{63} T^{25} - \)\(13\!\cdots\!42\)\( p^{72} T^{26} + \)\(17\!\cdots\!73\)\( p^{81} T^{27} + \)\(14\!\cdots\!71\)\( p^{90} T^{28} + \)\(17\!\cdots\!10\)\( p^{99} T^{29} - \)\(12\!\cdots\!96\)\( p^{108} T^{30} - \)\(34\!\cdots\!82\)\( p^{117} T^{31} + \)\(84\!\cdots\!51\)\( p^{126} T^{32} + \)\(25\!\cdots\!99\)\( p^{135} T^{33} - 39587058125800066 p^{144} T^{34} - 84965083 p^{153} T^{35} + p^{162} T^{36} \) | |
| 61 | \( 1 - 122850341 T - 47416990182447040 T^{2} + \)\(68\!\cdots\!91\)\( T^{3} + \)\(97\!\cdots\!31\)\( T^{4} - \)\(17\!\cdots\!90\)\( T^{5} - \)\(11\!\cdots\!04\)\( T^{6} + \)\(26\!\cdots\!54\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} - \)\(28\!\cdots\!81\)\( T^{9} - \)\(19\!\cdots\!32\)\( T^{10} + \)\(24\!\cdots\!39\)\( T^{11} + \)\(40\!\cdots\!27\)\( T^{12} - \)\(23\!\cdots\!54\)\( T^{13} - \)\(64\!\cdots\!08\)\( T^{14} + \)\(23\!\cdots\!58\)\( T^{15} + \)\(76\!\cdots\!58\)\( T^{16} - \)\(11\!\cdots\!32\)\( T^{17} - \)\(85\!\cdots\!64\)\( T^{18} - \)\(11\!\cdots\!32\)\( p^{9} T^{19} + \)\(76\!\cdots\!58\)\( p^{18} T^{20} + \)\(23\!\cdots\!58\)\( p^{27} T^{21} - \)\(64\!\cdots\!08\)\( p^{36} T^{22} - \)\(23\!\cdots\!54\)\( p^{45} T^{23} + \)\(40\!\cdots\!27\)\( p^{54} T^{24} + \)\(24\!\cdots\!39\)\( p^{63} T^{25} - \)\(19\!\cdots\!32\)\( p^{72} T^{26} - \)\(28\!\cdots\!81\)\( p^{81} T^{27} + \)\(11\!\cdots\!15\)\( p^{90} T^{28} + \)\(26\!\cdots\!54\)\( p^{99} T^{29} - \)\(11\!\cdots\!04\)\( p^{108} T^{30} - \)\(17\!\cdots\!90\)\( p^{117} T^{31} + \)\(97\!\cdots\!31\)\( p^{126} T^{32} + \)\(68\!\cdots\!91\)\( p^{135} T^{33} - 47416990182447040 p^{144} T^{34} - 122850341 p^{153} T^{35} + p^{162} T^{36} \) | |
| 67 | \( 1 - 27388339 T - 157700788141064442 T^{2} + \)\(13\!\cdots\!19\)\( T^{3} + \)\(12\!\cdots\!67\)\( T^{4} - \)\(16\!\cdots\!66\)\( T^{5} - \)\(63\!\cdots\!24\)\( T^{6} + \)\(15\!\cdots\!90\)\( p T^{7} + \)\(21\!\cdots\!35\)\( T^{8} - \)\(41\!\cdots\!07\)\( T^{9} - \)\(59\!\cdots\!90\)\( T^{10} + \)\(10\!\cdots\!71\)\( T^{11} + \)\(18\!\cdots\!71\)\( T^{12} - \)\(10\!\cdots\!54\)\( T^{13} - \)\(77\!\cdots\!08\)\( T^{14} - \)\(10\!\cdots\!22\)\( T^{15} + \)\(31\!\cdots\!70\)\( T^{16} + \)\(31\!\cdots\!32\)\( T^{17} - \)\(10\!\cdots\!48\)\( T^{18} + \)\(31\!\cdots\!32\)\( p^{9} T^{19} + \)\(31\!\cdots\!70\)\( p^{18} T^{20} - \)\(10\!\cdots\!22\)\( p^{27} T^{21} - \)\(77\!\cdots\!08\)\( p^{36} T^{22} - \)\(10\!\cdots\!54\)\( p^{45} T^{23} + \)\(18\!\cdots\!71\)\( p^{54} T^{24} + \)\(10\!\cdots\!71\)\( p^{63} T^{25} - \)\(59\!\cdots\!90\)\( p^{72} T^{26} - \)\(41\!\cdots\!07\)\( p^{81} T^{27} + \)\(21\!\cdots\!35\)\( p^{90} T^{28} + \)\(15\!\cdots\!90\)\( p^{100} T^{29} - \)\(63\!\cdots\!24\)\( p^{108} T^{30} - \)\(16\!\cdots\!66\)\( p^{117} T^{31} + \)\(12\!\cdots\!67\)\( p^{126} T^{32} + \)\(13\!\cdots\!19\)\( p^{135} T^{33} - 157700788141064442 p^{144} T^{34} - 27388339 p^{153} T^{35} + p^{162} T^{36} \) | |
| 71 | \( ( 1 - 236642208 T + 219822322534748991 T^{2} - \)\(51\!\cdots\!40\)\( T^{3} + \)\(27\!\cdots\!36\)\( T^{4} - \)\(57\!\cdots\!00\)\( T^{5} + \)\(22\!\cdots\!76\)\( T^{6} - \)\(43\!\cdots\!76\)\( T^{7} + \)\(13\!\cdots\!62\)\( T^{8} - \)\(23\!\cdots\!84\)\( T^{9} + \)\(13\!\cdots\!62\)\( p^{9} T^{10} - \)\(43\!\cdots\!76\)\( p^{18} T^{11} + \)\(22\!\cdots\!76\)\( p^{27} T^{12} - \)\(57\!\cdots\!00\)\( p^{36} T^{13} + \)\(27\!\cdots\!36\)\( p^{45} T^{14} - \)\(51\!\cdots\!40\)\( p^{54} T^{15} + 219822322534748991 p^{63} T^{16} - 236642208 p^{72} T^{17} + p^{81} T^{18} )^{2} \) | |
| 73 | \( 1 - 217201553 T - 145379268013884244 T^{2} + \)\(12\!\cdots\!43\)\( T^{3} + \)\(81\!\cdots\!71\)\( T^{4} + \)\(16\!\cdots\!30\)\( T^{5} - \)\(43\!\cdots\!24\)\( T^{6} - \)\(13\!\cdots\!78\)\( T^{7} - \)\(15\!\cdots\!05\)\( T^{8} - \)\(64\!\cdots\!89\)\( T^{9} - \)\(16\!\cdots\!44\)\( T^{10} + \)\(83\!\cdots\!43\)\( T^{11} + \)\(16\!\cdots\!75\)\( T^{12} + \)\(15\!\cdots\!26\)\( T^{13} - \)\(10\!\cdots\!32\)\( T^{14} - \)\(31\!\cdots\!62\)\( T^{15} - \)\(11\!\cdots\!18\)\( T^{16} + \)\(10\!\cdots\!52\)\( T^{17} + \)\(39\!\cdots\!76\)\( T^{18} + \)\(10\!\cdots\!52\)\( p^{9} T^{19} - \)\(11\!\cdots\!18\)\( p^{18} T^{20} - \)\(31\!\cdots\!62\)\( p^{27} T^{21} - \)\(10\!\cdots\!32\)\( p^{36} T^{22} + \)\(15\!\cdots\!26\)\( p^{45} T^{23} + \)\(16\!\cdots\!75\)\( p^{54} T^{24} + \)\(83\!\cdots\!43\)\( p^{63} T^{25} - \)\(16\!\cdots\!44\)\( p^{72} T^{26} - \)\(64\!\cdots\!89\)\( p^{81} T^{27} - \)\(15\!\cdots\!05\)\( p^{90} T^{28} - \)\(13\!\cdots\!78\)\( p^{99} T^{29} - \)\(43\!\cdots\!24\)\( p^{108} T^{30} + \)\(16\!\cdots\!30\)\( p^{117} T^{31} + \)\(81\!\cdots\!71\)\( p^{126} T^{32} + \)\(12\!\cdots\!43\)\( p^{135} T^{33} - 145379268013884244 p^{144} T^{34} - 217201553 p^{153} T^{35} + p^{162} T^{36} \) | |
| 79 | \( 1 - 326283565 T - 460044676308206086 T^{2} + \)\(19\!\cdots\!05\)\( T^{3} + \)\(93\!\cdots\!19\)\( T^{4} - \)\(51\!\cdots\!78\)\( T^{5} - \)\(86\!\cdots\!72\)\( T^{6} + \)\(73\!\cdots\!14\)\( T^{7} - \)\(73\!\cdots\!69\)\( T^{8} - \)\(45\!\cdots\!37\)\( T^{9} + \)\(11\!\cdots\!78\)\( T^{10} - \)\(40\!\cdots\!87\)\( T^{11} - \)\(12\!\cdots\!09\)\( T^{12} + \)\(13\!\cdots\!58\)\( T^{13} - \)\(15\!\cdots\!80\)\( T^{14} - \)\(17\!\cdots\!14\)\( T^{15} + \)\(24\!\cdots\!02\)\( T^{16} + \)\(88\!\cdots\!96\)\( T^{17} - \)\(41\!\cdots\!52\)\( T^{18} + \)\(88\!\cdots\!96\)\( p^{9} T^{19} + \)\(24\!\cdots\!02\)\( p^{18} T^{20} - \)\(17\!\cdots\!14\)\( p^{27} T^{21} - \)\(15\!\cdots\!80\)\( p^{36} T^{22} + \)\(13\!\cdots\!58\)\( p^{45} T^{23} - \)\(12\!\cdots\!09\)\( p^{54} T^{24} - \)\(40\!\cdots\!87\)\( p^{63} T^{25} + \)\(11\!\cdots\!78\)\( p^{72} T^{26} - \)\(45\!\cdots\!37\)\( p^{81} T^{27} - \)\(73\!\cdots\!69\)\( p^{90} T^{28} + \)\(73\!\cdots\!14\)\( p^{99} T^{29} - \)\(86\!\cdots\!72\)\( p^{108} T^{30} - \)\(51\!\cdots\!78\)\( p^{117} T^{31} + \)\(93\!\cdots\!19\)\( p^{126} T^{32} + \)\(19\!\cdots\!05\)\( p^{135} T^{33} - 460044676308206086 p^{144} T^{34} - 326283565 p^{153} T^{35} + p^{162} T^{36} \) | |
| 83 | \( ( 1 + 705674796 T + 1174128756128137227 T^{2} + \)\(60\!\cdots\!88\)\( T^{3} + \)\(57\!\cdots\!60\)\( T^{4} + \)\(21\!\cdots\!28\)\( T^{5} + \)\(16\!\cdots\!56\)\( T^{6} + \)\(47\!\cdots\!28\)\( T^{7} + \)\(32\!\cdots\!34\)\( T^{8} + \)\(85\!\cdots\!28\)\( T^{9} + \)\(32\!\cdots\!34\)\( p^{9} T^{10} + \)\(47\!\cdots\!28\)\( p^{18} T^{11} + \)\(16\!\cdots\!56\)\( p^{27} T^{12} + \)\(21\!\cdots\!28\)\( p^{36} T^{13} + \)\(57\!\cdots\!60\)\( p^{45} T^{14} + \)\(60\!\cdots\!88\)\( p^{54} T^{15} + 1174128756128137227 p^{63} T^{16} + 705674796 p^{72} T^{17} + p^{81} T^{18} )^{2} \) | |
| 89 | \( 1 - 153096753 T - 1111161786671068452 T^{2} + \)\(41\!\cdots\!75\)\( T^{3} + \)\(62\!\cdots\!27\)\( T^{4} - \)\(41\!\cdots\!02\)\( T^{5} - \)\(21\!\cdots\!52\)\( T^{6} + \)\(23\!\cdots\!22\)\( T^{7} + \)\(30\!\cdots\!19\)\( T^{8} - \)\(93\!\cdots\!93\)\( T^{9} + \)\(12\!\cdots\!72\)\( T^{10} + \)\(26\!\cdots\!39\)\( T^{11} - \)\(91\!\cdots\!45\)\( T^{12} - \)\(46\!\cdots\!62\)\( T^{13} + \)\(27\!\cdots\!92\)\( T^{14} + \)\(24\!\cdots\!54\)\( T^{15} - \)\(33\!\cdots\!50\)\( T^{16} + \)\(44\!\cdots\!00\)\( T^{17} - \)\(10\!\cdots\!60\)\( T^{18} + \)\(44\!\cdots\!00\)\( p^{9} T^{19} - \)\(33\!\cdots\!50\)\( p^{18} T^{20} + \)\(24\!\cdots\!54\)\( p^{27} T^{21} + \)\(27\!\cdots\!92\)\( p^{36} T^{22} - \)\(46\!\cdots\!62\)\( p^{45} T^{23} - \)\(91\!\cdots\!45\)\( p^{54} T^{24} + \)\(26\!\cdots\!39\)\( p^{63} T^{25} + \)\(12\!\cdots\!72\)\( p^{72} T^{26} - \)\(93\!\cdots\!93\)\( p^{81} T^{27} + \)\(30\!\cdots\!19\)\( p^{90} T^{28} + \)\(23\!\cdots\!22\)\( p^{99} T^{29} - \)\(21\!\cdots\!52\)\( p^{108} T^{30} - \)\(41\!\cdots\!02\)\( p^{117} T^{31} + \)\(62\!\cdots\!27\)\( p^{126} T^{32} + \)\(41\!\cdots\!75\)\( p^{135} T^{33} - 1111161786671068452 p^{144} T^{34} - 153096753 p^{153} T^{35} + p^{162} T^{36} \) | |
| 97 | \( ( 1 - 459990686 T + 4630996277768473817 T^{2} - \)\(28\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!64\)\( T^{4} - \)\(69\!\cdots\!56\)\( T^{5} + \)\(15\!\cdots\!84\)\( T^{6} - \)\(97\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!06\)\( T^{8} - \)\(90\!\cdots\!44\)\( T^{9} + \)\(16\!\cdots\!06\)\( p^{9} T^{10} - \)\(97\!\cdots\!20\)\( p^{18} T^{11} + \)\(15\!\cdots\!84\)\( p^{27} T^{12} - \)\(69\!\cdots\!56\)\( p^{36} T^{13} + \)\(10\!\cdots\!64\)\( p^{45} T^{14} - \)\(28\!\cdots\!80\)\( p^{54} T^{15} + 4630996277768473817 p^{63} T^{16} - 459990686 p^{72} T^{17} + p^{81} T^{18} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.09012841552664537500374336821, −1.97767891201822972921273410547, −1.90298692303580942847857487375, −1.85699746909611298463448671256, −1.56934617582833418847036669966, −1.50694275482546486707133007932, −1.44710418866584764018702625718, −1.39866064904950480261906036560, −1.30258103619004674758326145802, −1.29411422703702464975538865094, −1.26420333943874621347418208121, −1.20398443225547339351920223939, −1.13026121331375858171038020051, −1.08703921599467211063749840104, −1.04403126751083237011478248775, −1.00573492997192930005832963244, −0.70276762832331480562583672375, −0.68203283984062551445519279520, −0.58610998537211277961640202215, −0.57790994918194647813046310842, −0.57345095819528961045608844167, −0.53843780160642971634749204439, −0.31484535772413564086119823631, −0.27004540806172883492675531748, −0.01212352253732360648978550832, 0.01212352253732360648978550832, 0.27004540806172883492675531748, 0.31484535772413564086119823631, 0.53843780160642971634749204439, 0.57345095819528961045608844167, 0.57790994918194647813046310842, 0.58610998537211277961640202215, 0.68203283984062551445519279520, 0.70276762832331480562583672375, 1.00573492997192930005832963244, 1.04403126751083237011478248775, 1.08703921599467211063749840104, 1.13026121331375858171038020051, 1.20398443225547339351920223939, 1.26420333943874621347418208121, 1.29411422703702464975538865094, 1.30258103619004674758326145802, 1.39866064904950480261906036560, 1.44710418866584764018702625718, 1.50694275482546486707133007932, 1.56934617582833418847036669966, 1.85699746909611298463448671256, 1.90298692303580942847857487375, 1.97767891201822972921273410547, 2.09012841552664537500374336821
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.