Dirichlet series
| L(s) = 1 | + 33·2-s − 73·3-s − 1.67e3·4-s − 2.40e3·6-s + 1.06e4·7-s − 7.66e4·8-s − 1.40e5·9-s − 1.30e5·11-s + 1.22e5·12-s − 4.85e5·13-s + 3.52e5·14-s + 7.10e5·16-s + 1.93e5·17-s − 4.64e6·18-s − 1.24e6·19-s − 7.78e5·21-s − 4.32e6·22-s + 2.59e6·23-s + 5.59e6·24-s − 1.60e7·26-s + 8.64e6·27-s − 1.78e7·28-s − 2.24e6·29-s − 7.94e6·31-s + 7.12e7·32-s + 9.55e6·33-s + 6.40e6·34-s + ⋯ |
| L(s) = 1 | + 1.45·2-s − 0.520·3-s − 3.26·4-s − 0.758·6-s + 1.67·7-s − 6.61·8-s − 7.15·9-s − 2.69·11-s + 1.70·12-s − 4.71·13-s + 2.44·14-s + 2.70·16-s + 0.563·17-s − 10.4·18-s − 2.19·19-s − 0.873·21-s − 3.93·22-s + 1.93·23-s + 3.44·24-s − 6.87·26-s + 3.12·27-s − 5.49·28-s − 0.589·29-s − 1.54·31-s + 12.0·32-s + 1.40·33-s + 0.821·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{34} \cdot 13^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{34} \cdot 13^{17}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{17} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(5^{34} \cdot 13^{17}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(6.35702\times 10^{37}\) |
| Root analytic conductor: | \(12.9377\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(17\) |
| Selberg data: | \((34,\ 5^{34} \cdot 13^{17} ,\ ( \ : [9/2]^{17} ),\ -1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( 1 \) |
| 13 | \( ( 1 + p^{4} T )^{17} \) | |
| good | 2 | \( 1 - 33 T + 2763 T^{2} - 69741 T^{3} + 230389 p^{4} T^{4} - 37166061 p T^{5} + 828550961 p^{2} T^{6} - 6726353013 p^{3} T^{7} + 146499128401 p^{4} T^{8} - 240011021181 p^{7} T^{9} + 11206539735443 p^{7} T^{10} - 29682080078553 p^{9} T^{11} + 200564170274969 p^{12} T^{12} - 114405842142537 p^{16} T^{13} + 14147141140546761 p^{15} T^{14} - 30627390902336199 p^{17} T^{15} + 60603144319530489 p^{22} T^{16} - 7995624134544501 p^{28} T^{17} + 60603144319530489 p^{31} T^{18} - 30627390902336199 p^{35} T^{19} + 14147141140546761 p^{42} T^{20} - 114405842142537 p^{52} T^{21} + 200564170274969 p^{57} T^{22} - 29682080078553 p^{63} T^{23} + 11206539735443 p^{70} T^{24} - 240011021181 p^{79} T^{25} + 146499128401 p^{85} T^{26} - 6726353013 p^{93} T^{27} + 828550961 p^{101} T^{28} - 37166061 p^{109} T^{29} + 230389 p^{121} T^{30} - 69741 p^{126} T^{31} + 2763 p^{135} T^{32} - 33 p^{144} T^{33} + p^{153} T^{34} \) |
| 3 | \( 1 + 73 T + 48725 p T^{2} + 4103824 p T^{3} + 3765975286 p T^{4} + 12917565782 p^{4} T^{5} + 612063386607770 T^{6} + 59560574792697860 T^{7} + 2880104190257448769 p^{2} T^{8} + \)\(28\!\cdots\!61\)\( p^{2} T^{9} + \)\(37\!\cdots\!53\)\( p^{5} T^{10} + \)\(10\!\cdots\!84\)\( p^{4} T^{11} + \)\(36\!\cdots\!74\)\( p^{6} T^{12} + \)\(34\!\cdots\!38\)\( p^{6} T^{13} + \)\(10\!\cdots\!78\)\( p^{8} T^{14} + \)\(94\!\cdots\!52\)\( p^{8} T^{15} + \)\(29\!\cdots\!40\)\( p^{12} T^{16} + \)\(22\!\cdots\!92\)\( p^{10} T^{17} + \)\(29\!\cdots\!40\)\( p^{21} T^{18} + \)\(94\!\cdots\!52\)\( p^{26} T^{19} + \)\(10\!\cdots\!78\)\( p^{35} T^{20} + \)\(34\!\cdots\!38\)\( p^{42} T^{21} + \)\(36\!\cdots\!74\)\( p^{51} T^{22} + \)\(10\!\cdots\!84\)\( p^{58} T^{23} + \)\(37\!\cdots\!53\)\( p^{68} T^{24} + \)\(28\!\cdots\!61\)\( p^{74} T^{25} + 2880104190257448769 p^{83} T^{26} + 59560574792697860 p^{90} T^{27} + 612063386607770 p^{99} T^{28} + 12917565782 p^{112} T^{29} + 3765975286 p^{118} T^{30} + 4103824 p^{127} T^{31} + 48725 p^{136} T^{32} + 73 p^{144} T^{33} + p^{153} T^{34} \) | |
| 7 | \( 1 - 10670 T + 355744721 T^{2} - 3563420316798 T^{3} + 63439034138683286 T^{4} - \)\(58\!\cdots\!86\)\( T^{5} + \)\(10\!\cdots\!16\)\( p T^{6} - \)\(13\!\cdots\!20\)\( p^{2} T^{7} + \)\(19\!\cdots\!05\)\( p^{3} T^{8} - \)\(43\!\cdots\!60\)\( p^{6} T^{9} + \)\(27\!\cdots\!35\)\( p^{5} T^{10} - \)\(28\!\cdots\!84\)\( p^{6} T^{11} + \)\(32\!\cdots\!76\)\( p^{7} T^{12} - \)\(31\!\cdots\!78\)\( p^{8} T^{13} + \)\(47\!\cdots\!18\)\( p^{10} T^{14} - \)\(43\!\cdots\!54\)\( p^{11} T^{15} + \)\(30\!\cdots\!10\)\( p^{11} T^{16} - \)\(26\!\cdots\!92\)\( p^{12} T^{17} + \)\(30\!\cdots\!10\)\( p^{20} T^{18} - \)\(43\!\cdots\!54\)\( p^{29} T^{19} + \)\(47\!\cdots\!18\)\( p^{37} T^{20} - \)\(31\!\cdots\!78\)\( p^{44} T^{21} + \)\(32\!\cdots\!76\)\( p^{52} T^{22} - \)\(28\!\cdots\!84\)\( p^{60} T^{23} + \)\(27\!\cdots\!35\)\( p^{68} T^{24} - \)\(43\!\cdots\!60\)\( p^{78} T^{25} + \)\(19\!\cdots\!05\)\( p^{84} T^{26} - \)\(13\!\cdots\!20\)\( p^{92} T^{27} + \)\(10\!\cdots\!16\)\( p^{100} T^{28} - \)\(58\!\cdots\!86\)\( p^{108} T^{29} + 63439034138683286 p^{117} T^{30} - 3563420316798 p^{126} T^{31} + 355744721 p^{135} T^{32} - 10670 p^{144} T^{33} + p^{153} T^{34} \) | |
| 11 | \( 1 + 130917 T + 22911419450 T^{2} + 2454003180613263 T^{3} + \)\(27\!\cdots\!65\)\( T^{4} + \)\(24\!\cdots\!56\)\( T^{5} + \)\(21\!\cdots\!55\)\( T^{6} + \)\(16\!\cdots\!51\)\( T^{7} + \)\(11\!\cdots\!54\)\( p T^{8} + \)\(83\!\cdots\!65\)\( T^{9} + \)\(55\!\cdots\!11\)\( T^{10} + \)\(34\!\cdots\!60\)\( T^{11} + \)\(20\!\cdots\!98\)\( T^{12} + \)\(11\!\cdots\!78\)\( T^{13} + \)\(63\!\cdots\!08\)\( T^{14} + \)\(33\!\cdots\!06\)\( T^{15} + \)\(17\!\cdots\!50\)\( T^{16} + \)\(83\!\cdots\!84\)\( T^{17} + \)\(17\!\cdots\!50\)\( p^{9} T^{18} + \)\(33\!\cdots\!06\)\( p^{18} T^{19} + \)\(63\!\cdots\!08\)\( p^{27} T^{20} + \)\(11\!\cdots\!78\)\( p^{36} T^{21} + \)\(20\!\cdots\!98\)\( p^{45} T^{22} + \)\(34\!\cdots\!60\)\( p^{54} T^{23} + \)\(55\!\cdots\!11\)\( p^{63} T^{24} + \)\(83\!\cdots\!65\)\( p^{72} T^{25} + \)\(11\!\cdots\!54\)\( p^{82} T^{26} + \)\(16\!\cdots\!51\)\( p^{90} T^{27} + \)\(21\!\cdots\!55\)\( p^{99} T^{28} + \)\(24\!\cdots\!56\)\( p^{108} T^{29} + \)\(27\!\cdots\!65\)\( p^{117} T^{30} + 2454003180613263 p^{126} T^{31} + 22911419450 p^{135} T^{32} + 130917 p^{144} T^{33} + p^{153} T^{34} \) | |
| 17 | \( 1 - 11409 p T + 1141879939200 T^{2} - 183567487881583947 T^{3} + \)\(64\!\cdots\!23\)\( T^{4} - \)\(86\!\cdots\!76\)\( T^{5} + \)\(24\!\cdots\!59\)\( T^{6} - \)\(26\!\cdots\!19\)\( T^{7} + \)\(68\!\cdots\!40\)\( T^{8} - \)\(63\!\cdots\!33\)\( T^{9} + \)\(91\!\cdots\!89\)\( p T^{10} - \)\(12\!\cdots\!04\)\( T^{11} + \)\(29\!\cdots\!66\)\( T^{12} - \)\(19\!\cdots\!38\)\( T^{13} + \)\(46\!\cdots\!40\)\( T^{14} - \)\(28\!\cdots\!82\)\( T^{15} + \)\(64\!\cdots\!06\)\( T^{16} - \)\(35\!\cdots\!04\)\( T^{17} + \)\(64\!\cdots\!06\)\( p^{9} T^{18} - \)\(28\!\cdots\!82\)\( p^{18} T^{19} + \)\(46\!\cdots\!40\)\( p^{27} T^{20} - \)\(19\!\cdots\!38\)\( p^{36} T^{21} + \)\(29\!\cdots\!66\)\( p^{45} T^{22} - \)\(12\!\cdots\!04\)\( p^{54} T^{23} + \)\(91\!\cdots\!89\)\( p^{64} T^{24} - \)\(63\!\cdots\!33\)\( p^{72} T^{25} + \)\(68\!\cdots\!40\)\( p^{81} T^{26} - \)\(26\!\cdots\!19\)\( p^{90} T^{27} + \)\(24\!\cdots\!59\)\( p^{99} T^{28} - \)\(86\!\cdots\!76\)\( p^{108} T^{29} + \)\(64\!\cdots\!23\)\( p^{117} T^{30} - 183567487881583947 p^{126} T^{31} + 1141879939200 p^{135} T^{32} - 11409 p^{145} T^{33} + p^{153} T^{34} \) | |
| 19 | \( 1 + 1244111 T + 4182249348634 T^{2} + 4366031139489210105 T^{3} + \)\(83\!\cdots\!39\)\( T^{4} + \)\(75\!\cdots\!70\)\( T^{5} + \)\(10\!\cdots\!83\)\( T^{6} + \)\(85\!\cdots\!23\)\( T^{7} + \)\(96\!\cdots\!31\)\( T^{8} + \)\(70\!\cdots\!52\)\( T^{9} + \)\(68\!\cdots\!03\)\( T^{10} + \)\(45\!\cdots\!37\)\( T^{11} + \)\(38\!\cdots\!51\)\( T^{12} + \)\(23\!\cdots\!70\)\( T^{13} + \)\(17\!\cdots\!71\)\( T^{14} + \)\(99\!\cdots\!35\)\( T^{15} + \)\(69\!\cdots\!09\)\( T^{16} + \)\(35\!\cdots\!54\)\( T^{17} + \)\(69\!\cdots\!09\)\( p^{9} T^{18} + \)\(99\!\cdots\!35\)\( p^{18} T^{19} + \)\(17\!\cdots\!71\)\( p^{27} T^{20} + \)\(23\!\cdots\!70\)\( p^{36} T^{21} + \)\(38\!\cdots\!51\)\( p^{45} T^{22} + \)\(45\!\cdots\!37\)\( p^{54} T^{23} + \)\(68\!\cdots\!03\)\( p^{63} T^{24} + \)\(70\!\cdots\!52\)\( p^{72} T^{25} + \)\(96\!\cdots\!31\)\( p^{81} T^{26} + \)\(85\!\cdots\!23\)\( p^{90} T^{27} + \)\(10\!\cdots\!83\)\( p^{99} T^{28} + \)\(75\!\cdots\!70\)\( p^{108} T^{29} + \)\(83\!\cdots\!39\)\( p^{117} T^{30} + 4366031139489210105 p^{126} T^{31} + 4182249348634 p^{135} T^{32} + 1244111 p^{144} T^{33} + p^{153} T^{34} \) | |
| 23 | \( 1 - 2599170 T + 15613811715806 T^{2} - 31894001287851876156 T^{3} + \)\(11\!\cdots\!58\)\( T^{4} - \)\(19\!\cdots\!02\)\( T^{5} + \)\(52\!\cdots\!17\)\( T^{6} - \)\(79\!\cdots\!88\)\( T^{7} + \)\(80\!\cdots\!50\)\( p T^{8} - \)\(25\!\cdots\!12\)\( T^{9} + \)\(54\!\cdots\!20\)\( T^{10} - \)\(70\!\cdots\!36\)\( T^{11} + \)\(13\!\cdots\!55\)\( T^{12} - \)\(16\!\cdots\!74\)\( T^{13} + \)\(30\!\cdots\!84\)\( T^{14} - \)\(35\!\cdots\!08\)\( T^{15} + \)\(60\!\cdots\!47\)\( T^{16} - \)\(66\!\cdots\!64\)\( T^{17} + \)\(60\!\cdots\!47\)\( p^{9} T^{18} - \)\(35\!\cdots\!08\)\( p^{18} T^{19} + \)\(30\!\cdots\!84\)\( p^{27} T^{20} - \)\(16\!\cdots\!74\)\( p^{36} T^{21} + \)\(13\!\cdots\!55\)\( p^{45} T^{22} - \)\(70\!\cdots\!36\)\( p^{54} T^{23} + \)\(54\!\cdots\!20\)\( p^{63} T^{24} - \)\(25\!\cdots\!12\)\( p^{72} T^{25} + \)\(80\!\cdots\!50\)\( p^{82} T^{26} - \)\(79\!\cdots\!88\)\( p^{90} T^{27} + \)\(52\!\cdots\!17\)\( p^{99} T^{28} - \)\(19\!\cdots\!02\)\( p^{108} T^{29} + \)\(11\!\cdots\!58\)\( p^{117} T^{30} - 31894001287851876156 p^{126} T^{31} + 15613811715806 p^{135} T^{32} - 2599170 p^{144} T^{33} + p^{153} T^{34} \) | |
| 29 | \( 1 + 2247174 T + 118480224760124 T^{2} + \)\(19\!\cdots\!32\)\( T^{3} + \)\(69\!\cdots\!38\)\( T^{4} + \)\(94\!\cdots\!22\)\( T^{5} + \)\(27\!\cdots\!15\)\( T^{6} + \)\(35\!\cdots\!68\)\( T^{7} + \)\(79\!\cdots\!15\)\( T^{8} + \)\(11\!\cdots\!30\)\( T^{9} + \)\(18\!\cdots\!42\)\( T^{10} + \)\(32\!\cdots\!80\)\( T^{11} + \)\(36\!\cdots\!41\)\( T^{12} + \)\(76\!\cdots\!68\)\( T^{13} + \)\(61\!\cdots\!57\)\( T^{14} + \)\(14\!\cdots\!28\)\( T^{15} + \)\(95\!\cdots\!67\)\( T^{16} + \)\(23\!\cdots\!36\)\( T^{17} + \)\(95\!\cdots\!67\)\( p^{9} T^{18} + \)\(14\!\cdots\!28\)\( p^{18} T^{19} + \)\(61\!\cdots\!57\)\( p^{27} T^{20} + \)\(76\!\cdots\!68\)\( p^{36} T^{21} + \)\(36\!\cdots\!41\)\( p^{45} T^{22} + \)\(32\!\cdots\!80\)\( p^{54} T^{23} + \)\(18\!\cdots\!42\)\( p^{63} T^{24} + \)\(11\!\cdots\!30\)\( p^{72} T^{25} + \)\(79\!\cdots\!15\)\( p^{81} T^{26} + \)\(35\!\cdots\!68\)\( p^{90} T^{27} + \)\(27\!\cdots\!15\)\( p^{99} T^{28} + \)\(94\!\cdots\!22\)\( p^{108} T^{29} + \)\(69\!\cdots\!38\)\( p^{117} T^{30} + \)\(19\!\cdots\!32\)\( p^{126} T^{31} + 118480224760124 p^{135} T^{32} + 2247174 p^{144} T^{33} + p^{153} T^{34} \) | |
| 31 | \( 1 + 256416 p T + 283284071642225 T^{2} + \)\(18\!\cdots\!38\)\( T^{3} + \)\(38\!\cdots\!54\)\( T^{4} + \)\(22\!\cdots\!22\)\( T^{5} + \)\(34\!\cdots\!40\)\( T^{6} + \)\(17\!\cdots\!76\)\( T^{7} + \)\(22\!\cdots\!15\)\( T^{8} + \)\(10\!\cdots\!02\)\( T^{9} + \)\(37\!\cdots\!75\)\( p T^{10} + \)\(51\!\cdots\!96\)\( T^{11} + \)\(50\!\cdots\!12\)\( T^{12} + \)\(20\!\cdots\!70\)\( T^{13} + \)\(18\!\cdots\!90\)\( T^{14} + \)\(22\!\cdots\!34\)\( p T^{15} + \)\(56\!\cdots\!30\)\( T^{16} + \)\(19\!\cdots\!28\)\( T^{17} + \)\(56\!\cdots\!30\)\( p^{9} T^{18} + \)\(22\!\cdots\!34\)\( p^{19} T^{19} + \)\(18\!\cdots\!90\)\( p^{27} T^{20} + \)\(20\!\cdots\!70\)\( p^{36} T^{21} + \)\(50\!\cdots\!12\)\( p^{45} T^{22} + \)\(51\!\cdots\!96\)\( p^{54} T^{23} + \)\(37\!\cdots\!75\)\( p^{64} T^{24} + \)\(10\!\cdots\!02\)\( p^{72} T^{25} + \)\(22\!\cdots\!15\)\( p^{81} T^{26} + \)\(17\!\cdots\!76\)\( p^{90} T^{27} + \)\(34\!\cdots\!40\)\( p^{99} T^{28} + \)\(22\!\cdots\!22\)\( p^{108} T^{29} + \)\(38\!\cdots\!54\)\( p^{117} T^{30} + \)\(18\!\cdots\!38\)\( p^{126} T^{31} + 283284071642225 p^{135} T^{32} + 256416 p^{145} T^{33} + p^{153} T^{34} \) | |
| 37 | \( 1 + 17455504 T + 1404206716342705 T^{2} + \)\(22\!\cdots\!48\)\( T^{3} + \)\(96\!\cdots\!08\)\( T^{4} + \)\(13\!\cdots\!80\)\( T^{5} + \)\(42\!\cdots\!20\)\( T^{6} + \)\(55\!\cdots\!72\)\( T^{7} + \)\(13\!\cdots\!24\)\( T^{8} + \)\(44\!\cdots\!88\)\( p T^{9} + \)\(34\!\cdots\!88\)\( T^{10} + \)\(38\!\cdots\!52\)\( T^{11} + \)\(71\!\cdots\!76\)\( T^{12} + \)\(72\!\cdots\!60\)\( T^{13} + \)\(12\!\cdots\!20\)\( T^{14} + \)\(11\!\cdots\!12\)\( T^{15} + \)\(18\!\cdots\!62\)\( T^{16} + \)\(16\!\cdots\!80\)\( T^{17} + \)\(18\!\cdots\!62\)\( p^{9} T^{18} + \)\(11\!\cdots\!12\)\( p^{18} T^{19} + \)\(12\!\cdots\!20\)\( p^{27} T^{20} + \)\(72\!\cdots\!60\)\( p^{36} T^{21} + \)\(71\!\cdots\!76\)\( p^{45} T^{22} + \)\(38\!\cdots\!52\)\( p^{54} T^{23} + \)\(34\!\cdots\!88\)\( p^{63} T^{24} + \)\(44\!\cdots\!88\)\( p^{73} T^{25} + \)\(13\!\cdots\!24\)\( p^{81} T^{26} + \)\(55\!\cdots\!72\)\( p^{90} T^{27} + \)\(42\!\cdots\!20\)\( p^{99} T^{28} + \)\(13\!\cdots\!80\)\( p^{108} T^{29} + \)\(96\!\cdots\!08\)\( p^{117} T^{30} + \)\(22\!\cdots\!48\)\( p^{126} T^{31} + 1404206716342705 p^{135} T^{32} + 17455504 p^{144} T^{33} + p^{153} T^{34} \) | |
| 41 | \( 1 + 28253763 T + 2164126317589796 T^{2} + \)\(44\!\cdots\!81\)\( T^{3} + \)\(21\!\cdots\!97\)\( T^{4} + \)\(30\!\cdots\!30\)\( T^{5} + \)\(12\!\cdots\!67\)\( T^{6} + \)\(86\!\cdots\!75\)\( T^{7} + \)\(46\!\cdots\!57\)\( T^{8} - \)\(18\!\cdots\!40\)\( T^{9} + \)\(12\!\cdots\!27\)\( T^{10} - \)\(28\!\cdots\!19\)\( T^{11} + \)\(34\!\cdots\!81\)\( T^{12} - \)\(14\!\cdots\!42\)\( T^{13} + \)\(15\!\cdots\!31\)\( T^{14} - \)\(46\!\cdots\!85\)\( T^{15} + \)\(74\!\cdots\!55\)\( T^{16} - \)\(14\!\cdots\!30\)\( T^{17} + \)\(74\!\cdots\!55\)\( p^{9} T^{18} - \)\(46\!\cdots\!85\)\( p^{18} T^{19} + \)\(15\!\cdots\!31\)\( p^{27} T^{20} - \)\(14\!\cdots\!42\)\( p^{36} T^{21} + \)\(34\!\cdots\!81\)\( p^{45} T^{22} - \)\(28\!\cdots\!19\)\( p^{54} T^{23} + \)\(12\!\cdots\!27\)\( p^{63} T^{24} - \)\(18\!\cdots\!40\)\( p^{72} T^{25} + \)\(46\!\cdots\!57\)\( p^{81} T^{26} + \)\(86\!\cdots\!75\)\( p^{90} T^{27} + \)\(12\!\cdots\!67\)\( p^{99} T^{28} + \)\(30\!\cdots\!30\)\( p^{108} T^{29} + \)\(21\!\cdots\!97\)\( p^{117} T^{30} + \)\(44\!\cdots\!81\)\( p^{126} T^{31} + 2164126317589796 p^{135} T^{32} + 28253763 p^{144} T^{33} + p^{153} T^{34} \) | |
| 43 | \( 1 - 28631126 T + 3825810944292034 T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + \)\(74\!\cdots\!58\)\( T^{4} - \)\(21\!\cdots\!10\)\( T^{5} + \)\(10\!\cdots\!49\)\( T^{6} - \)\(29\!\cdots\!72\)\( T^{7} + \)\(11\!\cdots\!22\)\( T^{8} - \)\(31\!\cdots\!12\)\( T^{9} + \)\(99\!\cdots\!32\)\( T^{10} - \)\(27\!\cdots\!68\)\( T^{11} + \)\(76\!\cdots\!59\)\( T^{12} - \)\(19\!\cdots\!02\)\( T^{13} + \)\(51\!\cdots\!72\)\( T^{14} - \)\(12\!\cdots\!16\)\( T^{15} + \)\(29\!\cdots\!79\)\( T^{16} - \)\(64\!\cdots\!68\)\( T^{17} + \)\(29\!\cdots\!79\)\( p^{9} T^{18} - \)\(12\!\cdots\!16\)\( p^{18} T^{19} + \)\(51\!\cdots\!72\)\( p^{27} T^{20} - \)\(19\!\cdots\!02\)\( p^{36} T^{21} + \)\(76\!\cdots\!59\)\( p^{45} T^{22} - \)\(27\!\cdots\!68\)\( p^{54} T^{23} + \)\(99\!\cdots\!32\)\( p^{63} T^{24} - \)\(31\!\cdots\!12\)\( p^{72} T^{25} + \)\(11\!\cdots\!22\)\( p^{81} T^{26} - \)\(29\!\cdots\!72\)\( p^{90} T^{27} + \)\(10\!\cdots\!49\)\( p^{99} T^{28} - \)\(21\!\cdots\!10\)\( p^{108} T^{29} + \)\(74\!\cdots\!58\)\( p^{117} T^{30} - \)\(10\!\cdots\!40\)\( p^{126} T^{31} + 3825810944292034 p^{135} T^{32} - 28631126 p^{144} T^{33} + p^{153} T^{34} \) | |
| 47 | \( 1 - 41505678 T + 11292038827158801 T^{2} - \)\(45\!\cdots\!10\)\( T^{3} + \)\(66\!\cdots\!18\)\( T^{4} - \)\(25\!\cdots\!90\)\( T^{5} + \)\(26\!\cdots\!04\)\( T^{6} - \)\(93\!\cdots\!56\)\( T^{7} + \)\(77\!\cdots\!35\)\( T^{8} - \)\(25\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!65\)\( T^{10} - \)\(56\!\cdots\!24\)\( T^{11} + \)\(34\!\cdots\!28\)\( T^{12} - \)\(99\!\cdots\!90\)\( T^{13} + \)\(54\!\cdots\!54\)\( T^{14} - \)\(30\!\cdots\!46\)\( p T^{15} + \)\(72\!\cdots\!54\)\( T^{16} - \)\(17\!\cdots\!40\)\( T^{17} + \)\(72\!\cdots\!54\)\( p^{9} T^{18} - \)\(30\!\cdots\!46\)\( p^{19} T^{19} + \)\(54\!\cdots\!54\)\( p^{27} T^{20} - \)\(99\!\cdots\!90\)\( p^{36} T^{21} + \)\(34\!\cdots\!28\)\( p^{45} T^{22} - \)\(56\!\cdots\!24\)\( p^{54} T^{23} + \)\(18\!\cdots\!65\)\( p^{63} T^{24} - \)\(25\!\cdots\!00\)\( p^{72} T^{25} + \)\(77\!\cdots\!35\)\( p^{81} T^{26} - \)\(93\!\cdots\!56\)\( p^{90} T^{27} + \)\(26\!\cdots\!04\)\( p^{99} T^{28} - \)\(25\!\cdots\!90\)\( p^{108} T^{29} + \)\(66\!\cdots\!18\)\( p^{117} T^{30} - \)\(45\!\cdots\!10\)\( p^{126} T^{31} + 11292038827158801 p^{135} T^{32} - 41505678 p^{144} T^{33} + p^{153} T^{34} \) | |
| 53 | \( 1 - 20918940 T + 16546509846723076 T^{2} - \)\(38\!\cdots\!10\)\( T^{3} + \)\(13\!\cdots\!10\)\( T^{4} - \)\(47\!\cdots\!68\)\( T^{5} + \)\(83\!\cdots\!31\)\( T^{6} - \)\(41\!\cdots\!36\)\( T^{7} + \)\(40\!\cdots\!43\)\( T^{8} - \)\(50\!\cdots\!08\)\( p T^{9} + \)\(18\!\cdots\!98\)\( T^{10} - \)\(12\!\cdots\!98\)\( T^{11} + \)\(80\!\cdots\!93\)\( T^{12} - \)\(50\!\cdots\!84\)\( T^{13} + \)\(33\!\cdots\!05\)\( T^{14} - \)\(17\!\cdots\!02\)\( T^{15} + \)\(12\!\cdots\!23\)\( T^{16} - \)\(57\!\cdots\!68\)\( T^{17} + \)\(12\!\cdots\!23\)\( p^{9} T^{18} - \)\(17\!\cdots\!02\)\( p^{18} T^{19} + \)\(33\!\cdots\!05\)\( p^{27} T^{20} - \)\(50\!\cdots\!84\)\( p^{36} T^{21} + \)\(80\!\cdots\!93\)\( p^{45} T^{22} - \)\(12\!\cdots\!98\)\( p^{54} T^{23} + \)\(18\!\cdots\!98\)\( p^{63} T^{24} - \)\(50\!\cdots\!08\)\( p^{73} T^{25} + \)\(40\!\cdots\!43\)\( p^{81} T^{26} - \)\(41\!\cdots\!36\)\( p^{90} T^{27} + \)\(83\!\cdots\!31\)\( p^{99} T^{28} - \)\(47\!\cdots\!68\)\( p^{108} T^{29} + \)\(13\!\cdots\!10\)\( p^{117} T^{30} - \)\(38\!\cdots\!10\)\( p^{126} T^{31} + 16546509846723076 p^{135} T^{32} - 20918940 p^{144} T^{33} + p^{153} T^{34} \) | |
| 59 | \( 1 + 242672658 T + 104858254740307485 T^{2} + \)\(21\!\cdots\!58\)\( T^{3} + \)\(54\!\cdots\!82\)\( T^{4} + \)\(97\!\cdots\!66\)\( T^{5} + \)\(18\!\cdots\!32\)\( T^{6} + \)\(29\!\cdots\!40\)\( T^{7} + \)\(46\!\cdots\!31\)\( T^{8} + \)\(64\!\cdots\!16\)\( T^{9} + \)\(88\!\cdots\!93\)\( T^{10} + \)\(11\!\cdots\!52\)\( T^{11} + \)\(13\!\cdots\!04\)\( T^{12} + \)\(15\!\cdots\!58\)\( T^{13} + \)\(17\!\cdots\!06\)\( T^{14} + \)\(17\!\cdots\!50\)\( T^{15} + \)\(17\!\cdots\!78\)\( T^{16} + \)\(16\!\cdots\!44\)\( T^{17} + \)\(17\!\cdots\!78\)\( p^{9} T^{18} + \)\(17\!\cdots\!50\)\( p^{18} T^{19} + \)\(17\!\cdots\!06\)\( p^{27} T^{20} + \)\(15\!\cdots\!58\)\( p^{36} T^{21} + \)\(13\!\cdots\!04\)\( p^{45} T^{22} + \)\(11\!\cdots\!52\)\( p^{54} T^{23} + \)\(88\!\cdots\!93\)\( p^{63} T^{24} + \)\(64\!\cdots\!16\)\( p^{72} T^{25} + \)\(46\!\cdots\!31\)\( p^{81} T^{26} + \)\(29\!\cdots\!40\)\( p^{90} T^{27} + \)\(18\!\cdots\!32\)\( p^{99} T^{28} + \)\(97\!\cdots\!66\)\( p^{108} T^{29} + \)\(54\!\cdots\!82\)\( p^{117} T^{30} + \)\(21\!\cdots\!58\)\( p^{126} T^{31} + 104858254740307485 p^{135} T^{32} + 242672658 p^{144} T^{33} + p^{153} T^{34} \) | |
| 61 | \( 1 + 236730546 T + 126567538341630836 T^{2} + \)\(23\!\cdots\!60\)\( T^{3} + \)\(71\!\cdots\!34\)\( T^{4} + \)\(11\!\cdots\!18\)\( T^{5} + \)\(24\!\cdots\!19\)\( T^{6} + \)\(31\!\cdots\!08\)\( T^{7} + \)\(56\!\cdots\!95\)\( T^{8} + \)\(59\!\cdots\!90\)\( T^{9} + \)\(91\!\cdots\!74\)\( T^{10} + \)\(75\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!77\)\( T^{12} + \)\(57\!\cdots\!44\)\( T^{13} + \)\(86\!\cdots\!97\)\( T^{14} + \)\(12\!\cdots\!76\)\( T^{15} + \)\(57\!\cdots\!51\)\( T^{16} - \)\(16\!\cdots\!32\)\( T^{17} + \)\(57\!\cdots\!51\)\( p^{9} T^{18} + \)\(12\!\cdots\!76\)\( p^{18} T^{19} + \)\(86\!\cdots\!97\)\( p^{27} T^{20} + \)\(57\!\cdots\!44\)\( p^{36} T^{21} + \)\(10\!\cdots\!77\)\( p^{45} T^{22} + \)\(75\!\cdots\!80\)\( p^{54} T^{23} + \)\(91\!\cdots\!74\)\( p^{63} T^{24} + \)\(59\!\cdots\!90\)\( p^{72} T^{25} + \)\(56\!\cdots\!95\)\( p^{81} T^{26} + \)\(31\!\cdots\!08\)\( p^{90} T^{27} + \)\(24\!\cdots\!19\)\( p^{99} T^{28} + \)\(11\!\cdots\!18\)\( p^{108} T^{29} + \)\(71\!\cdots\!34\)\( p^{117} T^{30} + \)\(23\!\cdots\!60\)\( p^{126} T^{31} + 126567538341630836 p^{135} T^{32} + 236730546 p^{144} T^{33} + p^{153} T^{34} \) | |
| 67 | \( 1 + 33870225 T + 221114606814373462 T^{2} + \)\(67\!\cdots\!91\)\( T^{3} + \)\(24\!\cdots\!77\)\( T^{4} + \)\(79\!\cdots\!28\)\( T^{5} + \)\(18\!\cdots\!07\)\( T^{6} + \)\(67\!\cdots\!75\)\( T^{7} + \)\(10\!\cdots\!38\)\( T^{8} + \)\(43\!\cdots\!29\)\( T^{9} + \)\(51\!\cdots\!31\)\( T^{10} + \)\(21\!\cdots\!36\)\( T^{11} + \)\(20\!\cdots\!74\)\( T^{12} + \)\(88\!\cdots\!06\)\( T^{13} + \)\(73\!\cdots\!32\)\( T^{14} + \)\(30\!\cdots\!66\)\( T^{15} + \)\(22\!\cdots\!14\)\( T^{16} + \)\(89\!\cdots\!24\)\( T^{17} + \)\(22\!\cdots\!14\)\( p^{9} T^{18} + \)\(30\!\cdots\!66\)\( p^{18} T^{19} + \)\(73\!\cdots\!32\)\( p^{27} T^{20} + \)\(88\!\cdots\!06\)\( p^{36} T^{21} + \)\(20\!\cdots\!74\)\( p^{45} T^{22} + \)\(21\!\cdots\!36\)\( p^{54} T^{23} + \)\(51\!\cdots\!31\)\( p^{63} T^{24} + \)\(43\!\cdots\!29\)\( p^{72} T^{25} + \)\(10\!\cdots\!38\)\( p^{81} T^{26} + \)\(67\!\cdots\!75\)\( p^{90} T^{27} + \)\(18\!\cdots\!07\)\( p^{99} T^{28} + \)\(79\!\cdots\!28\)\( p^{108} T^{29} + \)\(24\!\cdots\!77\)\( p^{117} T^{30} + \)\(67\!\cdots\!91\)\( p^{126} T^{31} + 221114606814373462 p^{135} T^{32} + 33870225 p^{144} T^{33} + p^{153} T^{34} \) | |
| 71 | \( 1 + 11396412 p T + 654053028994363203 T^{2} + \)\(36\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!40\)\( T^{4} + \)\(79\!\cdots\!08\)\( T^{5} + \)\(31\!\cdots\!72\)\( T^{6} + \)\(11\!\cdots\!92\)\( T^{7} + \)\(38\!\cdots\!92\)\( T^{8} + \)\(12\!\cdots\!08\)\( T^{9} + \)\(36\!\cdots\!56\)\( T^{10} + \)\(10\!\cdots\!96\)\( T^{11} + \)\(27\!\cdots\!76\)\( T^{12} + \)\(69\!\cdots\!32\)\( T^{13} + \)\(17\!\cdots\!44\)\( T^{14} + \)\(40\!\cdots\!60\)\( T^{15} + \)\(91\!\cdots\!38\)\( T^{16} + \)\(19\!\cdots\!44\)\( T^{17} + \)\(91\!\cdots\!38\)\( p^{9} T^{18} + \)\(40\!\cdots\!60\)\( p^{18} T^{19} + \)\(17\!\cdots\!44\)\( p^{27} T^{20} + \)\(69\!\cdots\!32\)\( p^{36} T^{21} + \)\(27\!\cdots\!76\)\( p^{45} T^{22} + \)\(10\!\cdots\!96\)\( p^{54} T^{23} + \)\(36\!\cdots\!56\)\( p^{63} T^{24} + \)\(12\!\cdots\!08\)\( p^{72} T^{25} + \)\(38\!\cdots\!92\)\( p^{81} T^{26} + \)\(11\!\cdots\!92\)\( p^{90} T^{27} + \)\(31\!\cdots\!72\)\( p^{99} T^{28} + \)\(79\!\cdots\!08\)\( p^{108} T^{29} + \)\(18\!\cdots\!40\)\( p^{117} T^{30} + \)\(36\!\cdots\!88\)\( p^{126} T^{31} + 654053028994363203 p^{135} T^{32} + 11396412 p^{145} T^{33} + p^{153} T^{34} \) | |
| 73 | \( 1 + 460218191 T + 534486333444687612 T^{2} + \)\(19\!\cdots\!49\)\( T^{3} + \)\(13\!\cdots\!89\)\( T^{4} + \)\(40\!\cdots\!82\)\( T^{5} + \)\(28\!\cdots\!91\)\( p T^{6} + \)\(52\!\cdots\!67\)\( T^{7} + \)\(22\!\cdots\!29\)\( T^{8} + \)\(48\!\cdots\!28\)\( T^{9} + \)\(18\!\cdots\!23\)\( T^{10} + \)\(33\!\cdots\!69\)\( T^{11} + \)\(12\!\cdots\!01\)\( T^{12} + \)\(17\!\cdots\!14\)\( T^{13} + \)\(71\!\cdots\!87\)\( T^{14} + \)\(81\!\cdots\!35\)\( T^{15} + \)\(39\!\cdots\!75\)\( T^{16} + \)\(41\!\cdots\!70\)\( T^{17} + \)\(39\!\cdots\!75\)\( p^{9} T^{18} + \)\(81\!\cdots\!35\)\( p^{18} T^{19} + \)\(71\!\cdots\!87\)\( p^{27} T^{20} + \)\(17\!\cdots\!14\)\( p^{36} T^{21} + \)\(12\!\cdots\!01\)\( p^{45} T^{22} + \)\(33\!\cdots\!69\)\( p^{54} T^{23} + \)\(18\!\cdots\!23\)\( p^{63} T^{24} + \)\(48\!\cdots\!28\)\( p^{72} T^{25} + \)\(22\!\cdots\!29\)\( p^{81} T^{26} + \)\(52\!\cdots\!67\)\( p^{90} T^{27} + \)\(28\!\cdots\!91\)\( p^{100} T^{28} + \)\(40\!\cdots\!82\)\( p^{108} T^{29} + \)\(13\!\cdots\!89\)\( p^{117} T^{30} + \)\(19\!\cdots\!49\)\( p^{126} T^{31} + 534486333444687612 p^{135} T^{32} + 460218191 p^{144} T^{33} + p^{153} T^{34} \) | |
| 79 | \( 1 + 295742582 T + 975608746699837370 T^{2} + \)\(32\!\cdots\!40\)\( T^{3} + \)\(50\!\cdots\!14\)\( T^{4} + \)\(17\!\cdots\!98\)\( T^{5} + \)\(18\!\cdots\!61\)\( T^{6} + \)\(62\!\cdots\!00\)\( T^{7} + \)\(51\!\cdots\!70\)\( T^{8} + \)\(17\!\cdots\!60\)\( T^{9} + \)\(11\!\cdots\!12\)\( T^{10} + \)\(37\!\cdots\!72\)\( T^{11} + \)\(22\!\cdots\!75\)\( T^{12} + \)\(68\!\cdots\!30\)\( T^{13} + \)\(36\!\cdots\!48\)\( T^{14} + \)\(10\!\cdots\!48\)\( T^{15} + \)\(50\!\cdots\!91\)\( T^{16} + \)\(13\!\cdots\!80\)\( T^{17} + \)\(50\!\cdots\!91\)\( p^{9} T^{18} + \)\(10\!\cdots\!48\)\( p^{18} T^{19} + \)\(36\!\cdots\!48\)\( p^{27} T^{20} + \)\(68\!\cdots\!30\)\( p^{36} T^{21} + \)\(22\!\cdots\!75\)\( p^{45} T^{22} + \)\(37\!\cdots\!72\)\( p^{54} T^{23} + \)\(11\!\cdots\!12\)\( p^{63} T^{24} + \)\(17\!\cdots\!60\)\( p^{72} T^{25} + \)\(51\!\cdots\!70\)\( p^{81} T^{26} + \)\(62\!\cdots\!00\)\( p^{90} T^{27} + \)\(18\!\cdots\!61\)\( p^{99} T^{28} + \)\(17\!\cdots\!98\)\( p^{108} T^{29} + \)\(50\!\cdots\!14\)\( p^{117} T^{30} + \)\(32\!\cdots\!40\)\( p^{126} T^{31} + 975608746699837370 p^{135} T^{32} + 295742582 p^{144} T^{33} + p^{153} T^{34} \) | |
| 83 | \( 1 + 298090917 T + 1324263163309004686 T^{2} + \)\(46\!\cdots\!43\)\( T^{3} + \)\(87\!\cdots\!29\)\( T^{4} + \)\(34\!\cdots\!68\)\( T^{5} + \)\(39\!\cdots\!39\)\( T^{6} + \)\(16\!\cdots\!23\)\( T^{7} + \)\(14\!\cdots\!18\)\( T^{8} + \)\(60\!\cdots\!09\)\( T^{9} + \)\(43\!\cdots\!59\)\( T^{10} + \)\(18\!\cdots\!20\)\( T^{11} + \)\(11\!\cdots\!02\)\( T^{12} + \)\(47\!\cdots\!38\)\( T^{13} + \)\(27\!\cdots\!04\)\( T^{14} + \)\(10\!\cdots\!58\)\( T^{15} + \)\(58\!\cdots\!34\)\( T^{16} + \)\(21\!\cdots\!16\)\( T^{17} + \)\(58\!\cdots\!34\)\( p^{9} T^{18} + \)\(10\!\cdots\!58\)\( p^{18} T^{19} + \)\(27\!\cdots\!04\)\( p^{27} T^{20} + \)\(47\!\cdots\!38\)\( p^{36} T^{21} + \)\(11\!\cdots\!02\)\( p^{45} T^{22} + \)\(18\!\cdots\!20\)\( p^{54} T^{23} + \)\(43\!\cdots\!59\)\( p^{63} T^{24} + \)\(60\!\cdots\!09\)\( p^{72} T^{25} + \)\(14\!\cdots\!18\)\( p^{81} T^{26} + \)\(16\!\cdots\!23\)\( p^{90} T^{27} + \)\(39\!\cdots\!39\)\( p^{99} T^{28} + \)\(34\!\cdots\!68\)\( p^{108} T^{29} + \)\(87\!\cdots\!29\)\( p^{117} T^{30} + \)\(46\!\cdots\!43\)\( p^{126} T^{31} + 1324263163309004686 p^{135} T^{32} + 298090917 p^{144} T^{33} + p^{153} T^{34} \) | |
| 89 | \( 1 + 448474185 T + 2357879909255413496 T^{2} + \)\(11\!\cdots\!75\)\( T^{3} + \)\(30\!\cdots\!77\)\( T^{4} + \)\(14\!\cdots\!54\)\( T^{5} + \)\(26\!\cdots\!35\)\( T^{6} + \)\(13\!\cdots\!45\)\( T^{7} + \)\(18\!\cdots\!33\)\( T^{8} + \)\(95\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!27\)\( T^{10} + \)\(56\!\cdots\!39\)\( T^{11} + \)\(54\!\cdots\!81\)\( T^{12} + \)\(27\!\cdots\!30\)\( T^{13} + \)\(23\!\cdots\!91\)\( T^{14} + \)\(12\!\cdots\!65\)\( T^{15} + \)\(91\!\cdots\!43\)\( T^{16} + \)\(44\!\cdots\!34\)\( T^{17} + \)\(91\!\cdots\!43\)\( p^{9} T^{18} + \)\(12\!\cdots\!65\)\( p^{18} T^{19} + \)\(23\!\cdots\!91\)\( p^{27} T^{20} + \)\(27\!\cdots\!30\)\( p^{36} T^{21} + \)\(54\!\cdots\!81\)\( p^{45} T^{22} + \)\(56\!\cdots\!39\)\( p^{54} T^{23} + \)\(10\!\cdots\!27\)\( p^{63} T^{24} + \)\(95\!\cdots\!00\)\( p^{72} T^{25} + \)\(18\!\cdots\!33\)\( p^{81} T^{26} + \)\(13\!\cdots\!45\)\( p^{90} T^{27} + \)\(26\!\cdots\!35\)\( p^{99} T^{28} + \)\(14\!\cdots\!54\)\( p^{108} T^{29} + \)\(30\!\cdots\!77\)\( p^{117} T^{30} + \)\(11\!\cdots\!75\)\( p^{126} T^{31} + 2357879909255413496 p^{135} T^{32} + 448474185 p^{144} T^{33} + p^{153} T^{34} \) | |
| 97 | \( 1 - 2630881746 T + 9577785926924285013 T^{2} - \)\(18\!\cdots\!88\)\( T^{3} + \)\(40\!\cdots\!32\)\( T^{4} - \)\(63\!\cdots\!44\)\( T^{5} + \)\(10\!\cdots\!08\)\( T^{6} - \)\(14\!\cdots\!44\)\( T^{7} + \)\(20\!\cdots\!00\)\( T^{8} - \)\(24\!\cdots\!64\)\( T^{9} + \)\(31\!\cdots\!16\)\( T^{10} - \)\(34\!\cdots\!36\)\( T^{11} + \)\(39\!\cdots\!32\)\( T^{12} - \)\(38\!\cdots\!92\)\( T^{13} + \)\(40\!\cdots\!72\)\( T^{14} - \)\(37\!\cdots\!44\)\( T^{15} + \)\(35\!\cdots\!38\)\( T^{16} - \)\(30\!\cdots\!88\)\( T^{17} + \)\(35\!\cdots\!38\)\( p^{9} T^{18} - \)\(37\!\cdots\!44\)\( p^{18} T^{19} + \)\(40\!\cdots\!72\)\( p^{27} T^{20} - \)\(38\!\cdots\!92\)\( p^{36} T^{21} + \)\(39\!\cdots\!32\)\( p^{45} T^{22} - \)\(34\!\cdots\!36\)\( p^{54} T^{23} + \)\(31\!\cdots\!16\)\( p^{63} T^{24} - \)\(24\!\cdots\!64\)\( p^{72} T^{25} + \)\(20\!\cdots\!00\)\( p^{81} T^{26} - \)\(14\!\cdots\!44\)\( p^{90} T^{27} + \)\(10\!\cdots\!08\)\( p^{99} T^{28} - \)\(63\!\cdots\!44\)\( p^{108} T^{29} + \)\(40\!\cdots\!32\)\( p^{117} T^{30} - \)\(18\!\cdots\!88\)\( p^{126} T^{31} + 9577785926924285013 p^{135} T^{32} - 2630881746 p^{144} T^{33} + p^{153} T^{34} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{34} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.48583037001715134364384998160, −2.40106823334002024654214876209, −2.38709384883827401805248051567, −2.10135647133638596562085359800, −2.06623315008679414041260347517, −1.96159729239311232972800250402, −1.93725639739614208382145044859, −1.91714512759464438358219800081, −1.90222029768232593768026203010, −1.88331214902924093465550238909, −1.86243076705255806185832321932, −1.56693959111184649649306178737, −1.46985360092759688908544691381, −1.36041744966319402581897119592, −1.25861006563011863202358955798, −1.25702019660894274285008602332, −1.25286130581863631572231530663, −1.19065175567519913931257568695, −0.889745518035467306514835561868, −0.888233182981643279406534018969, −0.848962091186449725000995033890, −0.834260650159524168577394670349, −0.823187617939514049793777048767, −0.72381224152965585105586341287, −0.65663600134473050066570634049, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.65663600134473050066570634049, 0.72381224152965585105586341287, 0.823187617939514049793777048767, 0.834260650159524168577394670349, 0.848962091186449725000995033890, 0.888233182981643279406534018969, 0.889745518035467306514835561868, 1.19065175567519913931257568695, 1.25286130581863631572231530663, 1.25702019660894274285008602332, 1.25861006563011863202358955798, 1.36041744966319402581897119592, 1.46985360092759688908544691381, 1.56693959111184649649306178737, 1.86243076705255806185832321932, 1.88331214902924093465550238909, 1.90222029768232593768026203010, 1.91714512759464438358219800081, 1.93725639739614208382145044859, 1.96159729239311232972800250402, 2.06623315008679414041260347517, 2.10135647133638596562085359800, 2.38709384883827401805248051567, 2.40106823334002024654214876209, 2.48583037001715134364384998160
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.