Dirichlet series
| L(s) = 1 | − 288·2-s + 2.18e3·3-s + 3.68e4·4-s − 3.19e3·5-s − 6.29e5·6-s − 1.41e5·7-s − 2.45e6·8-s + 2.12e6·9-s + 9.18e5·10-s − 2.96e5·11-s + 8.06e7·12-s + 2.23e6·13-s + 4.07e7·14-s − 6.97e6·15-s + 4.71e7·16-s − 3.53e6·17-s − 6.12e8·18-s − 1.64e7·19-s − 1.17e8·20-s − 3.09e8·21-s + 8.53e7·22-s + 5.54e7·23-s − 5.37e9·24-s + 1.53e8·25-s − 6.43e8·26-s + 1.07e9·27-s − 5.22e9·28-s + ⋯ |
| L(s) = 1 | − 6.36·2-s + 5.19·3-s + 18·4-s − 0.456·5-s − 33.0·6-s − 3.18·7-s − 26.5·8-s + 12·9-s + 2.90·10-s − 0.555·11-s + 93.5·12-s + 1.66·13-s + 20.2·14-s − 2.37·15-s + 45/4·16-s − 0.603·17-s − 76.3·18-s − 1.52·19-s − 8.21·20-s − 16.5·21-s + 3.53·22-s + 1.79·23-s − 137.·24-s + 3.14·25-s − 10.6·26-s + 14.4·27-s − 57.3·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{18} \cdot 19^{18}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{18} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(36\) |
| Conductor: | \(2^{18} \cdot 3^{18} \cdot 19^{18}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.21046\times 10^{34}\) |
| Root analytic conductor: | \(9.35901\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((36,\ 2^{18} \cdot 3^{18} \cdot 19^{18} ,\ ( \ : [11/2]^{18} ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(0.1958313426\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1958313426\) |
| \(L(\frac{13}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{5} T + p^{10} T^{2} )^{9} \) |
| 3 | \( ( 1 - p^{5} T + p^{10} T^{2} )^{9} \) | |
| 19 | \( 1 + 16479124 T + 5667725649605 p T^{2} - 163410238412927936 p^{3} T^{3} - \)\(51\!\cdots\!28\)\( p^{5} T^{4} - \)\(15\!\cdots\!00\)\( p^{7} T^{5} + \)\(18\!\cdots\!36\)\( p^{10} T^{6} + \)\(38\!\cdots\!12\)\( p^{13} T^{7} + \)\(26\!\cdots\!32\)\( p^{17} T^{8} - \)\(11\!\cdots\!04\)\( p^{22} T^{9} + \)\(26\!\cdots\!32\)\( p^{28} T^{10} + \)\(38\!\cdots\!12\)\( p^{35} T^{11} + \)\(18\!\cdots\!36\)\( p^{43} T^{12} - \)\(15\!\cdots\!00\)\( p^{51} T^{13} - \)\(51\!\cdots\!28\)\( p^{60} T^{14} - 163410238412927936 p^{69} T^{15} + 5667725649605 p^{78} T^{16} + 16479124 p^{88} T^{17} + p^{99} T^{18} \) | |
| good | 5 | \( 1 + 638 p T - 143521319 T^{2} - 260449555462 T^{3} + 10461795463915251 T^{4} - 24460310249484089116 T^{5} - \)\(61\!\cdots\!34\)\( T^{6} + \)\(10\!\cdots\!44\)\( p T^{7} + \)\(47\!\cdots\!58\)\( p^{4} T^{8} - \)\(69\!\cdots\!44\)\( p^{4} T^{9} - \)\(75\!\cdots\!32\)\( p^{4} T^{10} + \)\(86\!\cdots\!16\)\( p^{5} T^{11} - \)\(35\!\cdots\!76\)\( p^{6} T^{12} - \)\(16\!\cdots\!16\)\( p^{7} T^{13} + \)\(16\!\cdots\!44\)\( p^{8} T^{14} + \)\(20\!\cdots\!48\)\( p^{9} T^{15} - \)\(50\!\cdots\!99\)\( p^{10} T^{16} - \)\(12\!\cdots\!58\)\( p^{11} T^{17} + \)\(11\!\cdots\!29\)\( p^{12} T^{18} - \)\(12\!\cdots\!58\)\( p^{22} T^{19} - \)\(50\!\cdots\!99\)\( p^{32} T^{20} + \)\(20\!\cdots\!48\)\( p^{42} T^{21} + \)\(16\!\cdots\!44\)\( p^{52} T^{22} - \)\(16\!\cdots\!16\)\( p^{62} T^{23} - \)\(35\!\cdots\!76\)\( p^{72} T^{24} + \)\(86\!\cdots\!16\)\( p^{82} T^{25} - \)\(75\!\cdots\!32\)\( p^{92} T^{26} - \)\(69\!\cdots\!44\)\( p^{103} T^{27} + \)\(47\!\cdots\!58\)\( p^{114} T^{28} + \)\(10\!\cdots\!44\)\( p^{122} T^{29} - \)\(61\!\cdots\!34\)\( p^{132} T^{30} - 24460310249484089116 p^{143} T^{31} + 10461795463915251 p^{154} T^{32} - 260449555462 p^{165} T^{33} - 143521319 p^{176} T^{34} + 638 p^{188} T^{35} + p^{198} T^{36} \) |
| 7 | \( ( 1 + 70827 T + 10096055369 T^{2} + 603309391628746 T^{3} + 6644675370574144070 p T^{4} + \)\(34\!\cdots\!28\)\( p T^{5} + \)\(27\!\cdots\!72\)\( p^{2} T^{6} + \)\(18\!\cdots\!44\)\( p^{3} T^{7} + \)\(12\!\cdots\!25\)\( p^{4} T^{8} + \)\(78\!\cdots\!37\)\( p^{5} T^{9} + \)\(12\!\cdots\!25\)\( p^{15} T^{10} + \)\(18\!\cdots\!44\)\( p^{25} T^{11} + \)\(27\!\cdots\!72\)\( p^{35} T^{12} + \)\(34\!\cdots\!28\)\( p^{45} T^{13} + 6644675370574144070 p^{56} T^{14} + 603309391628746 p^{66} T^{15} + 10096055369 p^{77} T^{16} + 70827 p^{88} T^{17} + p^{99} T^{18} )^{2} \) | |
| 11 | \( ( 1 + 148240 T + 1131096233005 T^{2} + 396680835155737848 T^{3} + \)\(65\!\cdots\!78\)\( T^{4} + \)\(31\!\cdots\!16\)\( T^{5} + \)\(29\!\cdots\!90\)\( T^{6} + \)\(13\!\cdots\!40\)\( T^{7} + \)\(10\!\cdots\!80\)\( T^{8} + \)\(43\!\cdots\!44\)\( T^{9} + \)\(10\!\cdots\!80\)\( p^{11} T^{10} + \)\(13\!\cdots\!40\)\( p^{22} T^{11} + \)\(29\!\cdots\!90\)\( p^{33} T^{12} + \)\(31\!\cdots\!16\)\( p^{44} T^{13} + \)\(65\!\cdots\!78\)\( p^{55} T^{14} + 396680835155737848 p^{66} T^{15} + 1131096233005 p^{77} T^{16} + 148240 p^{88} T^{17} + p^{99} T^{18} )^{2} \) | |
| 13 | \( 1 - 2235167 T - 8703467447892 T^{2} + 18821333025836215621 T^{3} + \)\(46\!\cdots\!59\)\( T^{4} - \)\(84\!\cdots\!66\)\( T^{5} - \)\(15\!\cdots\!68\)\( p T^{6} + \)\(27\!\cdots\!18\)\( T^{7} + \)\(70\!\cdots\!83\)\( T^{8} - \)\(69\!\cdots\!59\)\( T^{9} - \)\(21\!\cdots\!52\)\( T^{10} + \)\(14\!\cdots\!45\)\( T^{11} + \)\(56\!\cdots\!51\)\( T^{12} - \)\(25\!\cdots\!66\)\( T^{13} - \)\(13\!\cdots\!32\)\( T^{14} + \)\(34\!\cdots\!74\)\( T^{15} + \)\(27\!\cdots\!30\)\( T^{16} - \)\(23\!\cdots\!08\)\( T^{17} - \)\(52\!\cdots\!28\)\( T^{18} - \)\(23\!\cdots\!08\)\( p^{11} T^{19} + \)\(27\!\cdots\!30\)\( p^{22} T^{20} + \)\(34\!\cdots\!74\)\( p^{33} T^{21} - \)\(13\!\cdots\!32\)\( p^{44} T^{22} - \)\(25\!\cdots\!66\)\( p^{55} T^{23} + \)\(56\!\cdots\!51\)\( p^{66} T^{24} + \)\(14\!\cdots\!45\)\( p^{77} T^{25} - \)\(21\!\cdots\!52\)\( p^{88} T^{26} - \)\(69\!\cdots\!59\)\( p^{99} T^{27} + \)\(70\!\cdots\!83\)\( p^{110} T^{28} + \)\(27\!\cdots\!18\)\( p^{121} T^{29} - \)\(15\!\cdots\!68\)\( p^{133} T^{30} - \)\(84\!\cdots\!66\)\( p^{143} T^{31} + \)\(46\!\cdots\!59\)\( p^{154} T^{32} + 18821333025836215621 p^{165} T^{33} - 8703467447892 p^{176} T^{34} - 2235167 p^{187} T^{35} + p^{198} T^{36} \) | |
| 17 | \( 1 + 3532132 T - 10203107627185 p T^{2} - \)\(91\!\cdots\!72\)\( T^{3} + \)\(16\!\cdots\!93\)\( T^{4} + \)\(11\!\cdots\!84\)\( T^{5} - \)\(90\!\cdots\!52\)\( T^{6} - \)\(10\!\cdots\!40\)\( T^{7} + \)\(28\!\cdots\!38\)\( T^{8} + \)\(63\!\cdots\!76\)\( T^{9} + \)\(17\!\cdots\!50\)\( T^{10} - \)\(30\!\cdots\!12\)\( T^{11} - \)\(87\!\cdots\!94\)\( T^{12} + \)\(66\!\cdots\!92\)\( p T^{13} + \)\(64\!\cdots\!76\)\( T^{14} - \)\(30\!\cdots\!00\)\( T^{15} - \)\(32\!\cdots\!25\)\( T^{16} + \)\(22\!\cdots\!64\)\( p T^{17} + \)\(42\!\cdots\!25\)\( p^{2} T^{18} + \)\(22\!\cdots\!64\)\( p^{12} T^{19} - \)\(32\!\cdots\!25\)\( p^{22} T^{20} - \)\(30\!\cdots\!00\)\( p^{33} T^{21} + \)\(64\!\cdots\!76\)\( p^{44} T^{22} + \)\(66\!\cdots\!92\)\( p^{56} T^{23} - \)\(87\!\cdots\!94\)\( p^{66} T^{24} - \)\(30\!\cdots\!12\)\( p^{77} T^{25} + \)\(17\!\cdots\!50\)\( p^{88} T^{26} + \)\(63\!\cdots\!76\)\( p^{99} T^{27} + \)\(28\!\cdots\!38\)\( p^{110} T^{28} - \)\(10\!\cdots\!40\)\( p^{121} T^{29} - \)\(90\!\cdots\!52\)\( p^{132} T^{30} + \)\(11\!\cdots\!84\)\( p^{143} T^{31} + \)\(16\!\cdots\!93\)\( p^{154} T^{32} - \)\(91\!\cdots\!72\)\( p^{165} T^{33} - 10203107627185 p^{177} T^{34} + 3532132 p^{187} T^{35} + p^{198} T^{36} \) | |
| 23 | \( 1 - 55446634 T - 496916613087597 T^{2} + \)\(11\!\cdots\!78\)\( T^{3} - \)\(28\!\cdots\!13\)\( T^{4} - \)\(67\!\cdots\!44\)\( T^{5} + \)\(43\!\cdots\!42\)\( T^{6} - \)\(21\!\cdots\!20\)\( T^{7} - \)\(15\!\cdots\!54\)\( p T^{8} + \)\(80\!\cdots\!88\)\( T^{9} + \)\(21\!\cdots\!28\)\( T^{10} - \)\(11\!\cdots\!60\)\( T^{11} - \)\(25\!\cdots\!64\)\( p T^{12} + \)\(14\!\cdots\!48\)\( T^{13} - \)\(19\!\cdots\!20\)\( T^{14} - \)\(14\!\cdots\!16\)\( T^{15} + \)\(48\!\cdots\!61\)\( T^{16} + \)\(54\!\cdots\!38\)\( T^{17} - \)\(57\!\cdots\!73\)\( T^{18} + \)\(54\!\cdots\!38\)\( p^{11} T^{19} + \)\(48\!\cdots\!61\)\( p^{22} T^{20} - \)\(14\!\cdots\!16\)\( p^{33} T^{21} - \)\(19\!\cdots\!20\)\( p^{44} T^{22} + \)\(14\!\cdots\!48\)\( p^{55} T^{23} - \)\(25\!\cdots\!64\)\( p^{67} T^{24} - \)\(11\!\cdots\!60\)\( p^{77} T^{25} + \)\(21\!\cdots\!28\)\( p^{88} T^{26} + \)\(80\!\cdots\!88\)\( p^{99} T^{27} - \)\(15\!\cdots\!54\)\( p^{111} T^{28} - \)\(21\!\cdots\!20\)\( p^{121} T^{29} + \)\(43\!\cdots\!42\)\( p^{132} T^{30} - \)\(67\!\cdots\!44\)\( p^{143} T^{31} - \)\(28\!\cdots\!13\)\( p^{154} T^{32} + \)\(11\!\cdots\!78\)\( p^{165} T^{33} - 496916613087597 p^{176} T^{34} - 55446634 p^{187} T^{35} + p^{198} T^{36} \) | |
| 29 | \( 1 - 85033996 T - 46947071442645701 T^{2} + \)\(59\!\cdots\!52\)\( T^{3} + \)\(87\!\cdots\!65\)\( T^{4} - \)\(17\!\cdots\!92\)\( T^{5} - \)\(65\!\cdots\!28\)\( T^{6} + \)\(30\!\cdots\!44\)\( T^{7} - \)\(38\!\cdots\!34\)\( T^{8} - \)\(39\!\cdots\!60\)\( T^{9} + \)\(19\!\cdots\!50\)\( T^{10} + \)\(39\!\cdots\!28\)\( T^{11} - \)\(36\!\cdots\!98\)\( T^{12} - \)\(31\!\cdots\!16\)\( T^{13} + \)\(50\!\cdots\!84\)\( T^{14} + \)\(22\!\cdots\!68\)\( T^{15} - \)\(64\!\cdots\!25\)\( T^{16} - \)\(94\!\cdots\!44\)\( T^{17} + \)\(81\!\cdots\!09\)\( T^{18} - \)\(94\!\cdots\!44\)\( p^{11} T^{19} - \)\(64\!\cdots\!25\)\( p^{22} T^{20} + \)\(22\!\cdots\!68\)\( p^{33} T^{21} + \)\(50\!\cdots\!84\)\( p^{44} T^{22} - \)\(31\!\cdots\!16\)\( p^{55} T^{23} - \)\(36\!\cdots\!98\)\( p^{66} T^{24} + \)\(39\!\cdots\!28\)\( p^{77} T^{25} + \)\(19\!\cdots\!50\)\( p^{88} T^{26} - \)\(39\!\cdots\!60\)\( p^{99} T^{27} - \)\(38\!\cdots\!34\)\( p^{110} T^{28} + \)\(30\!\cdots\!44\)\( p^{121} T^{29} - \)\(65\!\cdots\!28\)\( p^{132} T^{30} - \)\(17\!\cdots\!92\)\( p^{143} T^{31} + \)\(87\!\cdots\!65\)\( p^{154} T^{32} + \)\(59\!\cdots\!52\)\( p^{165} T^{33} - 46947071442645701 p^{176} T^{34} - 85033996 p^{187} T^{35} + p^{198} T^{36} \) | |
| 31 | \( ( 1 + 107688003 T + 144145814986484621 T^{2} + \)\(98\!\cdots\!46\)\( T^{3} + \)\(10\!\cdots\!06\)\( T^{4} + \)\(51\!\cdots\!08\)\( T^{5} + \)\(48\!\cdots\!84\)\( T^{6} + \)\(19\!\cdots\!24\)\( T^{7} + \)\(16\!\cdots\!13\)\( T^{8} + \)\(55\!\cdots\!63\)\( T^{9} + \)\(16\!\cdots\!13\)\( p^{11} T^{10} + \)\(19\!\cdots\!24\)\( p^{22} T^{11} + \)\(48\!\cdots\!84\)\( p^{33} T^{12} + \)\(51\!\cdots\!08\)\( p^{44} T^{13} + \)\(10\!\cdots\!06\)\( p^{55} T^{14} + \)\(98\!\cdots\!46\)\( p^{66} T^{15} + 144145814986484621 p^{77} T^{16} + 107688003 p^{88} T^{17} + p^{99} T^{18} )^{2} \) | |
| 37 | \( ( 1 + 304105817 T + 989420971583971769 T^{2} + \)\(35\!\cdots\!92\)\( T^{3} + \)\(49\!\cdots\!98\)\( T^{4} + \)\(18\!\cdots\!78\)\( T^{5} + \)\(16\!\cdots\!66\)\( T^{6} + \)\(60\!\cdots\!88\)\( T^{7} + \)\(39\!\cdots\!71\)\( T^{8} + \)\(35\!\cdots\!35\)\( p T^{9} + \)\(39\!\cdots\!71\)\( p^{11} T^{10} + \)\(60\!\cdots\!88\)\( p^{22} T^{11} + \)\(16\!\cdots\!66\)\( p^{33} T^{12} + \)\(18\!\cdots\!78\)\( p^{44} T^{13} + \)\(49\!\cdots\!98\)\( p^{55} T^{14} + \)\(35\!\cdots\!92\)\( p^{66} T^{15} + 989420971583971769 p^{77} T^{16} + 304105817 p^{88} T^{17} + p^{99} T^{18} )^{2} \) | |
| 41 | \( 1 + 321480024 T - 2469367994146771025 T^{2} - \)\(94\!\cdots\!08\)\( T^{3} + \)\(28\!\cdots\!65\)\( T^{4} + \)\(12\!\cdots\!16\)\( T^{5} - \)\(21\!\cdots\!16\)\( T^{6} - \)\(11\!\cdots\!16\)\( T^{7} + \)\(11\!\cdots\!46\)\( T^{8} + \)\(66\!\cdots\!88\)\( T^{9} - \)\(44\!\cdots\!62\)\( T^{10} - \)\(21\!\cdots\!68\)\( T^{11} + \)\(99\!\cdots\!02\)\( T^{12} - \)\(16\!\cdots\!96\)\( T^{13} - \)\(13\!\cdots\!04\)\( T^{14} + \)\(45\!\cdots\!80\)\( T^{15} + \)\(93\!\cdots\!95\)\( T^{16} - \)\(12\!\cdots\!40\)\( T^{17} - \)\(77\!\cdots\!19\)\( T^{18} - \)\(12\!\cdots\!40\)\( p^{11} T^{19} + \)\(93\!\cdots\!95\)\( p^{22} T^{20} + \)\(45\!\cdots\!80\)\( p^{33} T^{21} - \)\(13\!\cdots\!04\)\( p^{44} T^{22} - \)\(16\!\cdots\!96\)\( p^{55} T^{23} + \)\(99\!\cdots\!02\)\( p^{66} T^{24} - \)\(21\!\cdots\!68\)\( p^{77} T^{25} - \)\(44\!\cdots\!62\)\( p^{88} T^{26} + \)\(66\!\cdots\!88\)\( p^{99} T^{27} + \)\(11\!\cdots\!46\)\( p^{110} T^{28} - \)\(11\!\cdots\!16\)\( p^{121} T^{29} - \)\(21\!\cdots\!16\)\( p^{132} T^{30} + \)\(12\!\cdots\!16\)\( p^{143} T^{31} + \)\(28\!\cdots\!65\)\( p^{154} T^{32} - \)\(94\!\cdots\!08\)\( p^{165} T^{33} - 2469367994146771025 p^{176} T^{34} + 321480024 p^{187} T^{35} + p^{198} T^{36} \) | |
| 43 | \( 1 + 835642311 T - 6168899719180220552 T^{2} - \)\(56\!\cdots\!97\)\( T^{3} + \)\(20\!\cdots\!33\)\( T^{4} + \)\(19\!\cdots\!74\)\( T^{5} - \)\(46\!\cdots\!02\)\( T^{6} - \)\(44\!\cdots\!00\)\( T^{7} + \)\(82\!\cdots\!35\)\( T^{8} + \)\(73\!\cdots\!89\)\( T^{9} - \)\(12\!\cdots\!18\)\( T^{10} - \)\(91\!\cdots\!21\)\( T^{11} + \)\(16\!\cdots\!51\)\( T^{12} + \)\(87\!\cdots\!08\)\( T^{13} - \)\(20\!\cdots\!70\)\( T^{14} - \)\(60\!\cdots\!74\)\( T^{15} + \)\(22\!\cdots\!48\)\( T^{16} + \)\(20\!\cdots\!74\)\( T^{17} - \)\(22\!\cdots\!04\)\( T^{18} + \)\(20\!\cdots\!74\)\( p^{11} T^{19} + \)\(22\!\cdots\!48\)\( p^{22} T^{20} - \)\(60\!\cdots\!74\)\( p^{33} T^{21} - \)\(20\!\cdots\!70\)\( p^{44} T^{22} + \)\(87\!\cdots\!08\)\( p^{55} T^{23} + \)\(16\!\cdots\!51\)\( p^{66} T^{24} - \)\(91\!\cdots\!21\)\( p^{77} T^{25} - \)\(12\!\cdots\!18\)\( p^{88} T^{26} + \)\(73\!\cdots\!89\)\( p^{99} T^{27} + \)\(82\!\cdots\!35\)\( p^{110} T^{28} - \)\(44\!\cdots\!00\)\( p^{121} T^{29} - \)\(46\!\cdots\!02\)\( p^{132} T^{30} + \)\(19\!\cdots\!74\)\( p^{143} T^{31} + \)\(20\!\cdots\!33\)\( p^{154} T^{32} - \)\(56\!\cdots\!97\)\( p^{165} T^{33} - 6168899719180220552 p^{176} T^{34} + 835642311 p^{187} T^{35} + p^{198} T^{36} \) | |
| 47 | \( 1 - 269536978 T - 11228592954992620731 T^{2} - \)\(14\!\cdots\!70\)\( T^{3} + \)\(57\!\cdots\!97\)\( T^{4} + \)\(31\!\cdots\!12\)\( T^{5} - \)\(20\!\cdots\!84\)\( T^{6} - \)\(15\!\cdots\!24\)\( T^{7} + \)\(62\!\cdots\!78\)\( T^{8} + \)\(53\!\cdots\!08\)\( T^{9} - \)\(17\!\cdots\!74\)\( T^{10} - \)\(19\!\cdots\!64\)\( T^{11} + \)\(85\!\cdots\!98\)\( p T^{12} + \)\(56\!\cdots\!52\)\( T^{13} - \)\(75\!\cdots\!84\)\( T^{14} - \)\(91\!\cdots\!96\)\( T^{15} + \)\(16\!\cdots\!87\)\( T^{16} + \)\(61\!\cdots\!66\)\( T^{17} - \)\(41\!\cdots\!33\)\( T^{18} + \)\(61\!\cdots\!66\)\( p^{11} T^{19} + \)\(16\!\cdots\!87\)\( p^{22} T^{20} - \)\(91\!\cdots\!96\)\( p^{33} T^{21} - \)\(75\!\cdots\!84\)\( p^{44} T^{22} + \)\(56\!\cdots\!52\)\( p^{55} T^{23} + \)\(85\!\cdots\!98\)\( p^{67} T^{24} - \)\(19\!\cdots\!64\)\( p^{77} T^{25} - \)\(17\!\cdots\!74\)\( p^{88} T^{26} + \)\(53\!\cdots\!08\)\( p^{99} T^{27} + \)\(62\!\cdots\!78\)\( p^{110} T^{28} - \)\(15\!\cdots\!24\)\( p^{121} T^{29} - \)\(20\!\cdots\!84\)\( p^{132} T^{30} + \)\(31\!\cdots\!12\)\( p^{143} T^{31} + \)\(57\!\cdots\!97\)\( p^{154} T^{32} - \)\(14\!\cdots\!70\)\( p^{165} T^{33} - 11228592954992620731 p^{176} T^{34} - 269536978 p^{187} T^{35} + p^{198} T^{36} \) | |
| 53 | \( 1 - 1559946318 T - 31584344501459400071 T^{2} + \)\(31\!\cdots\!54\)\( T^{3} + \)\(53\!\cdots\!31\)\( T^{4} - \)\(41\!\cdots\!00\)\( T^{5} - \)\(44\!\cdots\!50\)\( T^{6} + \)\(28\!\cdots\!40\)\( T^{7} + \)\(18\!\cdots\!06\)\( T^{8} - \)\(52\!\cdots\!96\)\( T^{9} + \)\(55\!\cdots\!60\)\( T^{10} + \)\(11\!\cdots\!48\)\( T^{11} - \)\(74\!\cdots\!16\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{13} + \)\(47\!\cdots\!24\)\( T^{14} + \)\(16\!\cdots\!80\)\( T^{15} + \)\(33\!\cdots\!45\)\( T^{16} - \)\(69\!\cdots\!34\)\( T^{17} - \)\(22\!\cdots\!67\)\( T^{18} - \)\(69\!\cdots\!34\)\( p^{11} T^{19} + \)\(33\!\cdots\!45\)\( p^{22} T^{20} + \)\(16\!\cdots\!80\)\( p^{33} T^{21} + \)\(47\!\cdots\!24\)\( p^{44} T^{22} - \)\(18\!\cdots\!40\)\( p^{55} T^{23} - \)\(74\!\cdots\!16\)\( p^{66} T^{24} + \)\(11\!\cdots\!48\)\( p^{77} T^{25} + \)\(55\!\cdots\!60\)\( p^{88} T^{26} - \)\(52\!\cdots\!96\)\( p^{99} T^{27} + \)\(18\!\cdots\!06\)\( p^{110} T^{28} + \)\(28\!\cdots\!40\)\( p^{121} T^{29} - \)\(44\!\cdots\!50\)\( p^{132} T^{30} - \)\(41\!\cdots\!00\)\( p^{143} T^{31} + \)\(53\!\cdots\!31\)\( p^{154} T^{32} + \)\(31\!\cdots\!54\)\( p^{165} T^{33} - 31584344501459400071 p^{176} T^{34} - 1559946318 p^{187} T^{35} + p^{198} T^{36} \) | |
| 59 | \( 1 - 3956983832 T - \)\(10\!\cdots\!77\)\( T^{2} + \)\(85\!\cdots\!12\)\( T^{3} + \)\(32\!\cdots\!11\)\( T^{4} - \)\(56\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!42\)\( T^{6} + \)\(12\!\cdots\!08\)\( T^{7} - \)\(73\!\cdots\!34\)\( T^{8} + \)\(28\!\cdots\!92\)\( T^{9} - \)\(12\!\cdots\!00\)\( T^{10} - \)\(14\!\cdots\!32\)\( T^{11} + \)\(34\!\cdots\!16\)\( p T^{12} - \)\(35\!\cdots\!80\)\( T^{13} - \)\(62\!\cdots\!44\)\( T^{14} + \)\(35\!\cdots\!40\)\( T^{15} + \)\(13\!\cdots\!57\)\( T^{16} - \)\(61\!\cdots\!60\)\( T^{17} + \)\(30\!\cdots\!51\)\( T^{18} - \)\(61\!\cdots\!60\)\( p^{11} T^{19} + \)\(13\!\cdots\!57\)\( p^{22} T^{20} + \)\(35\!\cdots\!40\)\( p^{33} T^{21} - \)\(62\!\cdots\!44\)\( p^{44} T^{22} - \)\(35\!\cdots\!80\)\( p^{55} T^{23} + \)\(34\!\cdots\!16\)\( p^{67} T^{24} - \)\(14\!\cdots\!32\)\( p^{77} T^{25} - \)\(12\!\cdots\!00\)\( p^{88} T^{26} + \)\(28\!\cdots\!92\)\( p^{99} T^{27} - \)\(73\!\cdots\!34\)\( p^{110} T^{28} + \)\(12\!\cdots\!08\)\( p^{121} T^{29} + \)\(11\!\cdots\!42\)\( p^{132} T^{30} - \)\(56\!\cdots\!72\)\( p^{143} T^{31} + \)\(32\!\cdots\!11\)\( p^{154} T^{32} + \)\(85\!\cdots\!12\)\( p^{165} T^{33} - \)\(10\!\cdots\!77\)\( p^{176} T^{34} - 3956983832 p^{187} T^{35} + p^{198} T^{36} \) | |
| 61 | \( 1 + 1945929435 T - \)\(17\!\cdots\!64\)\( T^{2} - \)\(64\!\cdots\!09\)\( T^{3} + \)\(12\!\cdots\!07\)\( T^{4} + \)\(73\!\cdots\!94\)\( T^{5} - \)\(45\!\cdots\!80\)\( T^{6} - \)\(44\!\cdots\!90\)\( T^{7} + \)\(10\!\cdots\!79\)\( T^{8} + \)\(16\!\cdots\!71\)\( T^{9} - \)\(45\!\cdots\!76\)\( T^{10} - \)\(33\!\cdots\!69\)\( T^{11} - \)\(10\!\cdots\!17\)\( T^{12} - \)\(64\!\cdots\!46\)\( T^{13} + \)\(34\!\cdots\!52\)\( T^{14} + \)\(73\!\cdots\!98\)\( T^{15} + \)\(29\!\cdots\!62\)\( T^{16} - \)\(16\!\cdots\!24\)\( T^{17} - \)\(25\!\cdots\!68\)\( T^{18} - \)\(16\!\cdots\!24\)\( p^{11} T^{19} + \)\(29\!\cdots\!62\)\( p^{22} T^{20} + \)\(73\!\cdots\!98\)\( p^{33} T^{21} + \)\(34\!\cdots\!52\)\( p^{44} T^{22} - \)\(64\!\cdots\!46\)\( p^{55} T^{23} - \)\(10\!\cdots\!17\)\( p^{66} T^{24} - \)\(33\!\cdots\!69\)\( p^{77} T^{25} - \)\(45\!\cdots\!76\)\( p^{88} T^{26} + \)\(16\!\cdots\!71\)\( p^{99} T^{27} + \)\(10\!\cdots\!79\)\( p^{110} T^{28} - \)\(44\!\cdots\!90\)\( p^{121} T^{29} - \)\(45\!\cdots\!80\)\( p^{132} T^{30} + \)\(73\!\cdots\!94\)\( p^{143} T^{31} + \)\(12\!\cdots\!07\)\( p^{154} T^{32} - \)\(64\!\cdots\!09\)\( p^{165} T^{33} - \)\(17\!\cdots\!64\)\( p^{176} T^{34} + 1945929435 p^{187} T^{35} + p^{198} T^{36} \) | |
| 67 | \( 1 + 4377825179 T - \)\(51\!\cdots\!08\)\( T^{2} + \)\(35\!\cdots\!09\)\( p T^{3} + \)\(14\!\cdots\!29\)\( T^{4} - \)\(16\!\cdots\!94\)\( T^{5} - \)\(21\!\cdots\!30\)\( T^{6} + \)\(47\!\cdots\!12\)\( T^{7} + \)\(78\!\cdots\!55\)\( T^{8} - \)\(81\!\cdots\!51\)\( T^{9} + \)\(31\!\cdots\!02\)\( T^{10} + \)\(93\!\cdots\!83\)\( T^{11} - \)\(82\!\cdots\!65\)\( T^{12} - \)\(83\!\cdots\!60\)\( T^{13} + \)\(12\!\cdots\!90\)\( T^{14} + \)\(62\!\cdots\!10\)\( T^{15} - \)\(17\!\cdots\!64\)\( T^{16} - \)\(25\!\cdots\!90\)\( T^{17} + \)\(21\!\cdots\!24\)\( T^{18} - \)\(25\!\cdots\!90\)\( p^{11} T^{19} - \)\(17\!\cdots\!64\)\( p^{22} T^{20} + \)\(62\!\cdots\!10\)\( p^{33} T^{21} + \)\(12\!\cdots\!90\)\( p^{44} T^{22} - \)\(83\!\cdots\!60\)\( p^{55} T^{23} - \)\(82\!\cdots\!65\)\( p^{66} T^{24} + \)\(93\!\cdots\!83\)\( p^{77} T^{25} + \)\(31\!\cdots\!02\)\( p^{88} T^{26} - \)\(81\!\cdots\!51\)\( p^{99} T^{27} + \)\(78\!\cdots\!55\)\( p^{110} T^{28} + \)\(47\!\cdots\!12\)\( p^{121} T^{29} - \)\(21\!\cdots\!30\)\( p^{132} T^{30} - \)\(16\!\cdots\!94\)\( p^{143} T^{31} + \)\(14\!\cdots\!29\)\( p^{154} T^{32} + \)\(35\!\cdots\!09\)\( p^{166} T^{33} - \)\(51\!\cdots\!08\)\( p^{176} T^{34} + 4377825179 p^{187} T^{35} + p^{198} T^{36} \) | |
| 71 | \( 1 - 879667146 T - \)\(11\!\cdots\!51\)\( T^{2} + \)\(50\!\cdots\!34\)\( T^{3} + \)\(66\!\cdots\!77\)\( T^{4} - \)\(51\!\cdots\!08\)\( T^{5} - \)\(24\!\cdots\!72\)\( T^{6} + \)\(29\!\cdots\!60\)\( T^{7} + \)\(94\!\cdots\!86\)\( p T^{8} - \)\(11\!\cdots\!24\)\( T^{9} - \)\(13\!\cdots\!62\)\( T^{10} + \)\(33\!\cdots\!96\)\( T^{11} + \)\(16\!\cdots\!82\)\( T^{12} - \)\(77\!\cdots\!68\)\( T^{13} + \)\(47\!\cdots\!16\)\( T^{14} + \)\(13\!\cdots\!92\)\( T^{15} - \)\(66\!\cdots\!17\)\( T^{16} - \)\(11\!\cdots\!90\)\( T^{17} + \)\(21\!\cdots\!87\)\( T^{18} - \)\(11\!\cdots\!90\)\( p^{11} T^{19} - \)\(66\!\cdots\!17\)\( p^{22} T^{20} + \)\(13\!\cdots\!92\)\( p^{33} T^{21} + \)\(47\!\cdots\!16\)\( p^{44} T^{22} - \)\(77\!\cdots\!68\)\( p^{55} T^{23} + \)\(16\!\cdots\!82\)\( p^{66} T^{24} + \)\(33\!\cdots\!96\)\( p^{77} T^{25} - \)\(13\!\cdots\!62\)\( p^{88} T^{26} - \)\(11\!\cdots\!24\)\( p^{99} T^{27} + \)\(94\!\cdots\!86\)\( p^{111} T^{28} + \)\(29\!\cdots\!60\)\( p^{121} T^{29} - \)\(24\!\cdots\!72\)\( p^{132} T^{30} - \)\(51\!\cdots\!08\)\( p^{143} T^{31} + \)\(66\!\cdots\!77\)\( p^{154} T^{32} + \)\(50\!\cdots\!34\)\( p^{165} T^{33} - \)\(11\!\cdots\!51\)\( p^{176} T^{34} - 879667146 p^{187} T^{35} + p^{198} T^{36} \) | |
| 73 | \( 1 - 26437885119 T - \)\(13\!\cdots\!32\)\( T^{2} + \)\(27\!\cdots\!89\)\( T^{3} + \)\(12\!\cdots\!03\)\( T^{4} - \)\(14\!\cdots\!74\)\( T^{5} - \)\(84\!\cdots\!52\)\( T^{6} + \)\(35\!\cdots\!26\)\( T^{7} + \)\(44\!\cdots\!47\)\( T^{8} + \)\(22\!\cdots\!37\)\( T^{9} - \)\(18\!\cdots\!84\)\( T^{10} - \)\(70\!\cdots\!59\)\( T^{11} + \)\(61\!\cdots\!91\)\( T^{12} + \)\(36\!\cdots\!74\)\( T^{13} - \)\(18\!\cdots\!72\)\( T^{14} - \)\(10\!\cdots\!02\)\( T^{15} + \)\(50\!\cdots\!94\)\( T^{16} + \)\(13\!\cdots\!64\)\( T^{17} - \)\(14\!\cdots\!64\)\( T^{18} + \)\(13\!\cdots\!64\)\( p^{11} T^{19} + \)\(50\!\cdots\!94\)\( p^{22} T^{20} - \)\(10\!\cdots\!02\)\( p^{33} T^{21} - \)\(18\!\cdots\!72\)\( p^{44} T^{22} + \)\(36\!\cdots\!74\)\( p^{55} T^{23} + \)\(61\!\cdots\!91\)\( p^{66} T^{24} - \)\(70\!\cdots\!59\)\( p^{77} T^{25} - \)\(18\!\cdots\!84\)\( p^{88} T^{26} + \)\(22\!\cdots\!37\)\( p^{99} T^{27} + \)\(44\!\cdots\!47\)\( p^{110} T^{28} + \)\(35\!\cdots\!26\)\( p^{121} T^{29} - \)\(84\!\cdots\!52\)\( p^{132} T^{30} - \)\(14\!\cdots\!74\)\( p^{143} T^{31} + \)\(12\!\cdots\!03\)\( p^{154} T^{32} + \)\(27\!\cdots\!89\)\( p^{165} T^{33} - \)\(13\!\cdots\!32\)\( p^{176} T^{34} - 26437885119 p^{187} T^{35} + p^{198} T^{36} \) | |
| 79 | \( 1 - 13332294079 T - \)\(38\!\cdots\!92\)\( T^{2} + \)\(93\!\cdots\!09\)\( T^{3} + \)\(81\!\cdots\!77\)\( T^{4} + \)\(51\!\cdots\!06\)\( T^{5} - \)\(11\!\cdots\!22\)\( T^{6} - \)\(14\!\cdots\!20\)\( T^{7} + \)\(11\!\cdots\!51\)\( T^{8} + \)\(20\!\cdots\!19\)\( T^{9} - \)\(96\!\cdots\!86\)\( T^{10} - \)\(20\!\cdots\!47\)\( T^{11} + \)\(70\!\cdots\!31\)\( T^{12} + \)\(16\!\cdots\!88\)\( T^{13} - \)\(44\!\cdots\!94\)\( T^{14} - \)\(10\!\cdots\!26\)\( T^{15} + \)\(25\!\cdots\!76\)\( T^{16} + \)\(30\!\cdots\!30\)\( T^{17} - \)\(15\!\cdots\!68\)\( T^{18} + \)\(30\!\cdots\!30\)\( p^{11} T^{19} + \)\(25\!\cdots\!76\)\( p^{22} T^{20} - \)\(10\!\cdots\!26\)\( p^{33} T^{21} - \)\(44\!\cdots\!94\)\( p^{44} T^{22} + \)\(16\!\cdots\!88\)\( p^{55} T^{23} + \)\(70\!\cdots\!31\)\( p^{66} T^{24} - \)\(20\!\cdots\!47\)\( p^{77} T^{25} - \)\(96\!\cdots\!86\)\( p^{88} T^{26} + \)\(20\!\cdots\!19\)\( p^{99} T^{27} + \)\(11\!\cdots\!51\)\( p^{110} T^{28} - \)\(14\!\cdots\!20\)\( p^{121} T^{29} - \)\(11\!\cdots\!22\)\( p^{132} T^{30} + \)\(51\!\cdots\!06\)\( p^{143} T^{31} + \)\(81\!\cdots\!77\)\( p^{154} T^{32} + \)\(93\!\cdots\!09\)\( p^{165} T^{33} - \)\(38\!\cdots\!92\)\( p^{176} T^{34} - 13332294079 p^{187} T^{35} + p^{198} T^{36} \) | |
| 83 | \( ( 1 - 102075863192 T + \)\(73\!\cdots\!19\)\( T^{2} - \)\(38\!\cdots\!52\)\( T^{3} + \)\(17\!\cdots\!72\)\( T^{4} - \)\(60\!\cdots\!36\)\( T^{5} + \)\(17\!\cdots\!16\)\( T^{6} - \)\(31\!\cdots\!20\)\( T^{7} + \)\(34\!\cdots\!30\)\( T^{8} + \)\(50\!\cdots\!52\)\( T^{9} + \)\(34\!\cdots\!30\)\( p^{11} T^{10} - \)\(31\!\cdots\!20\)\( p^{22} T^{11} + \)\(17\!\cdots\!16\)\( p^{33} T^{12} - \)\(60\!\cdots\!36\)\( p^{44} T^{13} + \)\(17\!\cdots\!72\)\( p^{55} T^{14} - \)\(38\!\cdots\!52\)\( p^{66} T^{15} + \)\(73\!\cdots\!19\)\( p^{77} T^{16} - 102075863192 p^{88} T^{17} + p^{99} T^{18} )^{2} \) | |
| 89 | \( 1 - 97199089952 T - \)\(10\!\cdots\!31\)\( T^{2} + \)\(14\!\cdots\!08\)\( T^{3} + \)\(41\!\cdots\!19\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{5} + \)\(19\!\cdots\!30\)\( T^{6} + \)\(66\!\cdots\!96\)\( T^{7} - \)\(13\!\cdots\!14\)\( T^{8} - \)\(28\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!28\)\( T^{10} + \)\(99\!\cdots\!00\)\( T^{11} - \)\(56\!\cdots\!00\)\( T^{12} - \)\(27\!\cdots\!48\)\( T^{13} + \)\(23\!\cdots\!80\)\( T^{14} + \)\(56\!\cdots\!72\)\( T^{15} - \)\(81\!\cdots\!43\)\( T^{16} - \)\(57\!\cdots\!16\)\( T^{17} + \)\(24\!\cdots\!37\)\( T^{18} - \)\(57\!\cdots\!16\)\( p^{11} T^{19} - \)\(81\!\cdots\!43\)\( p^{22} T^{20} + \)\(56\!\cdots\!72\)\( p^{33} T^{21} + \)\(23\!\cdots\!80\)\( p^{44} T^{22} - \)\(27\!\cdots\!48\)\( p^{55} T^{23} - \)\(56\!\cdots\!00\)\( p^{66} T^{24} + \)\(99\!\cdots\!00\)\( p^{77} T^{25} + \)\(10\!\cdots\!28\)\( p^{88} T^{26} - \)\(28\!\cdots\!00\)\( p^{99} T^{27} - \)\(13\!\cdots\!14\)\( p^{110} T^{28} + \)\(66\!\cdots\!96\)\( p^{121} T^{29} + \)\(19\!\cdots\!30\)\( p^{132} T^{30} - \)\(11\!\cdots\!28\)\( p^{143} T^{31} + \)\(41\!\cdots\!19\)\( p^{154} T^{32} + \)\(14\!\cdots\!08\)\( p^{165} T^{33} - \)\(10\!\cdots\!31\)\( p^{176} T^{34} - 97199089952 p^{187} T^{35} + p^{198} T^{36} \) | |
| 97 | \( 1 + 73924998086 T - \)\(32\!\cdots\!45\)\( T^{2} - \)\(11\!\cdots\!50\)\( T^{3} + \)\(66\!\cdots\!45\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} - \)\(88\!\cdots\!40\)\( T^{6} + \)\(31\!\cdots\!32\)\( T^{7} + \)\(81\!\cdots\!26\)\( T^{8} - \)\(69\!\cdots\!44\)\( T^{9} - \)\(45\!\cdots\!18\)\( T^{10} + \)\(89\!\cdots\!04\)\( T^{11} - \)\(14\!\cdots\!30\)\( T^{12} - \)\(80\!\cdots\!00\)\( T^{13} + \)\(37\!\cdots\!80\)\( T^{14} + \)\(49\!\cdots\!04\)\( T^{15} - \)\(50\!\cdots\!09\)\( T^{16} - \)\(13\!\cdots\!06\)\( T^{17} + \)\(42\!\cdots\!77\)\( T^{18} - \)\(13\!\cdots\!06\)\( p^{11} T^{19} - \)\(50\!\cdots\!09\)\( p^{22} T^{20} + \)\(49\!\cdots\!04\)\( p^{33} T^{21} + \)\(37\!\cdots\!80\)\( p^{44} T^{22} - \)\(80\!\cdots\!00\)\( p^{55} T^{23} - \)\(14\!\cdots\!30\)\( p^{66} T^{24} + \)\(89\!\cdots\!04\)\( p^{77} T^{25} - \)\(45\!\cdots\!18\)\( p^{88} T^{26} - \)\(69\!\cdots\!44\)\( p^{99} T^{27} + \)\(81\!\cdots\!26\)\( p^{110} T^{28} + \)\(31\!\cdots\!32\)\( p^{121} T^{29} - \)\(88\!\cdots\!40\)\( p^{132} T^{30} + \)\(11\!\cdots\!96\)\( p^{143} T^{31} + \)\(66\!\cdots\!45\)\( p^{154} T^{32} - \)\(11\!\cdots\!50\)\( p^{165} T^{33} - \)\(32\!\cdots\!45\)\( p^{176} T^{34} + 73924998086 p^{187} T^{35} + p^{198} T^{36} \) | |
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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
−1.84345934032981932778678308075, −1.84008113958636587102671952903, −1.81482309071172447493760566582, −1.62668594834647875010634383675, −1.56960433278611496609246078483, −1.44194624942299812785540059517, −1.37643980864699364529138562063, −1.37000592767750565414050310093, −1.28142795736281654904824718893, −1.23608057248905076485723058504, −0.963602342284708185068216762249, −0.875358791637046137623599145370, −0.872511630551264398118293962802, −0.849530275757719971001890537197, −0.78511519413636112705977015634, −0.74829896882365450494568388698, −0.59531905616314451126217102301, −0.48257093674846027582173702240, −0.46900253492045320653670922508, −0.41894582299093789000849783929, −0.37844094055538214577870604837, −0.28221650336043291607288415956, −0.14823191426268768979229656511, −0.13940948245783221183655312682, −0.05802431643216822339024054742, 0.05802431643216822339024054742, 0.13940948245783221183655312682, 0.14823191426268768979229656511, 0.28221650336043291607288415956, 0.37844094055538214577870604837, 0.41894582299093789000849783929, 0.46900253492045320653670922508, 0.48257093674846027582173702240, 0.59531905616314451126217102301, 0.74829896882365450494568388698, 0.78511519413636112705977015634, 0.849530275757719971001890537197, 0.872511630551264398118293962802, 0.875358791637046137623599145370, 0.963602342284708185068216762249, 1.23608057248905076485723058504, 1.28142795736281654904824718893, 1.37000592767750565414050310093, 1.37643980864699364529138562063, 1.44194624942299812785540059517, 1.56960433278611496609246078483, 1.62668594834647875010634383675, 1.81482309071172447493760566582, 1.84008113958636587102671952903, 1.84345934032981932778678308075
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.