Dirichlet series
| L(s) = 1 | + 125·5-s − 1.24e3·7-s + 6.10e3·11-s + 4.93e3·13-s − 4.87e4·17-s + 2.68e4·19-s + 1.93e4·23-s + 2.88e5·25-s + 4.67e4·29-s + 1.85e5·31-s − 1.55e5·35-s + 1.08e5·37-s + 6.38e5·41-s − 8.92e5·43-s − 2.30e5·47-s + 4.37e6·49-s + 2.87e6·53-s + 7.63e5·55-s + 2.17e6·59-s − 1.87e6·61-s + 6.17e5·65-s − 5.31e5·67-s + 3.72e6·71-s − 1.80e6·73-s − 7.60e6·77-s − 3.60e6·79-s + 1.07e7·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.37·7-s + 1.38·11-s + 0.623·13-s − 2.40·17-s + 0.899·19-s + 0.332·23-s + 3.69·25-s + 0.356·29-s + 1.11·31-s − 0.613·35-s + 0.351·37-s + 1.44·41-s − 1.71·43-s − 0.324·47-s + 5.31·49-s + 2.65·53-s + 0.618·55-s + 1.37·59-s − 1.05·61-s + 0.278·65-s − 0.215·67-s + 1.23·71-s − 0.542·73-s − 1.89·77-s − 0.823·79-s + 2.07·83-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{60}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 3^{60}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{80} \cdot 3^{60}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.01309\times 10^{42}\) |
| Root analytic conductor: | \(11.6168\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{80} \cdot 3^{60} ,\ ( \ : [7/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(32.96922696\) |
| \(L(\frac12)\) | \(\approx\) | \(32.96922696\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) | |
| good | 5 | \( 1 - p^{3} T - 273334 T^{2} + 33319601 T^{3} + 7756201523 p T^{4} - 2689343583204 T^{5} - 136933519513656 p^{2} T^{6} - 150781590831300792 T^{7} + 7835541570584421156 p^{2} T^{8} + \)\(73\!\cdots\!08\)\( T^{9} - \)\(82\!\cdots\!56\)\( T^{10} - \)\(41\!\cdots\!44\)\( p^{2} T^{11} + \)\(15\!\cdots\!29\)\( p^{2} T^{12} + \)\(74\!\cdots\!03\)\( p^{3} T^{13} + \)\(88\!\cdots\!26\)\( p^{4} T^{14} - \)\(18\!\cdots\!87\)\( p^{5} T^{15} - \)\(52\!\cdots\!93\)\( p^{6} T^{16} + \)\(60\!\cdots\!16\)\( p^{8} T^{17} + \)\(15\!\cdots\!04\)\( p^{10} T^{18} - \)\(26\!\cdots\!12\)\( p^{9} T^{19} - \)\(15\!\cdots\!56\)\( p^{10} T^{20} - \)\(26\!\cdots\!12\)\( p^{16} T^{21} + \)\(15\!\cdots\!04\)\( p^{24} T^{22} + \)\(60\!\cdots\!16\)\( p^{29} T^{23} - \)\(52\!\cdots\!93\)\( p^{34} T^{24} - \)\(18\!\cdots\!87\)\( p^{40} T^{25} + \)\(88\!\cdots\!26\)\( p^{46} T^{26} + \)\(74\!\cdots\!03\)\( p^{52} T^{27} + \)\(15\!\cdots\!29\)\( p^{58} T^{28} - \)\(41\!\cdots\!44\)\( p^{65} T^{29} - \)\(82\!\cdots\!56\)\( p^{70} T^{30} + \)\(73\!\cdots\!08\)\( p^{77} T^{31} + 7835541570584421156 p^{86} T^{32} - 150781590831300792 p^{91} T^{33} - 136933519513656 p^{100} T^{34} - 2689343583204 p^{105} T^{35} + 7756201523 p^{113} T^{36} + 33319601 p^{119} T^{37} - 273334 p^{126} T^{38} - p^{136} T^{39} + p^{140} T^{40} \) |
| 7 | \( 1 + 1245 T - 2825332 T^{2} - 5552199951 T^{3} + 206331285553 p T^{4} + 9077986298667528 T^{5} + 560813825056292880 p T^{6} - \)\(53\!\cdots\!28\)\( T^{7} - \)\(51\!\cdots\!56\)\( T^{8} - \)\(13\!\cdots\!24\)\( T^{9} - \)\(13\!\cdots\!96\)\( T^{10} + \)\(55\!\cdots\!52\)\( T^{11} + \)\(53\!\cdots\!93\)\( T^{12} + \)\(47\!\cdots\!69\)\( T^{13} - \)\(14\!\cdots\!52\)\( p T^{14} - \)\(41\!\cdots\!79\)\( T^{15} - \)\(39\!\cdots\!79\)\( p T^{16} - \)\(73\!\cdots\!44\)\( T^{17} + \)\(45\!\cdots\!36\)\( T^{18} + \)\(12\!\cdots\!76\)\( T^{19} + \)\(15\!\cdots\!28\)\( T^{20} + \)\(12\!\cdots\!76\)\( p^{7} T^{21} + \)\(45\!\cdots\!36\)\( p^{14} T^{22} - \)\(73\!\cdots\!44\)\( p^{21} T^{23} - \)\(39\!\cdots\!79\)\( p^{29} T^{24} - \)\(41\!\cdots\!79\)\( p^{35} T^{25} - \)\(14\!\cdots\!52\)\( p^{43} T^{26} + \)\(47\!\cdots\!69\)\( p^{49} T^{27} + \)\(53\!\cdots\!93\)\( p^{56} T^{28} + \)\(55\!\cdots\!52\)\( p^{63} T^{29} - \)\(13\!\cdots\!96\)\( p^{70} T^{30} - \)\(13\!\cdots\!24\)\( p^{77} T^{31} - \)\(51\!\cdots\!56\)\( p^{84} T^{32} - \)\(53\!\cdots\!28\)\( p^{91} T^{33} + 560813825056292880 p^{99} T^{34} + 9077986298667528 p^{105} T^{35} + 206331285553 p^{113} T^{36} - 5552199951 p^{119} T^{37} - 2825332 p^{126} T^{38} + 1245 p^{133} T^{39} + p^{140} T^{40} \) | |
| 11 | \( 1 - 6106 T - 77933983 T^{2} + 414887140222 T^{3} + 3620187890902222 T^{4} - 1232961757386830310 p T^{5} - \)\(12\!\cdots\!77\)\( T^{6} + \)\(27\!\cdots\!18\)\( T^{7} + \)\(31\!\cdots\!74\)\( p T^{8} - \)\(20\!\cdots\!42\)\( T^{9} - \)\(70\!\cdots\!21\)\( T^{10} - \)\(99\!\cdots\!98\)\( T^{11} + \)\(10\!\cdots\!52\)\( T^{12} + \)\(49\!\cdots\!74\)\( T^{13} - \)\(57\!\cdots\!77\)\( T^{14} - \)\(11\!\cdots\!18\)\( p T^{15} - \)\(28\!\cdots\!19\)\( T^{16} + \)\(21\!\cdots\!32\)\( T^{17} + \)\(11\!\cdots\!70\)\( T^{18} - \)\(16\!\cdots\!64\)\( T^{19} - \)\(28\!\cdots\!64\)\( T^{20} - \)\(16\!\cdots\!64\)\( p^{7} T^{21} + \)\(11\!\cdots\!70\)\( p^{14} T^{22} + \)\(21\!\cdots\!32\)\( p^{21} T^{23} - \)\(28\!\cdots\!19\)\( p^{28} T^{24} - \)\(11\!\cdots\!18\)\( p^{36} T^{25} - \)\(57\!\cdots\!77\)\( p^{42} T^{26} + \)\(49\!\cdots\!74\)\( p^{49} T^{27} + \)\(10\!\cdots\!52\)\( p^{56} T^{28} - \)\(99\!\cdots\!98\)\( p^{63} T^{29} - \)\(70\!\cdots\!21\)\( p^{70} T^{30} - \)\(20\!\cdots\!42\)\( p^{77} T^{31} + \)\(31\!\cdots\!74\)\( p^{85} T^{32} + \)\(27\!\cdots\!18\)\( p^{91} T^{33} - \)\(12\!\cdots\!77\)\( p^{98} T^{34} - 1232961757386830310 p^{106} T^{35} + 3620187890902222 p^{112} T^{36} + 414887140222 p^{119} T^{37} - 77933983 p^{126} T^{38} - 6106 p^{133} T^{39} + p^{140} T^{40} \) | |
| 13 | \( 1 - 4937 T - 308181696 T^{2} + 2676593089315 T^{3} + 46998517878344521 T^{4} - \)\(60\!\cdots\!24\)\( T^{5} - \)\(33\!\cdots\!34\)\( T^{6} + \)\(85\!\cdots\!46\)\( T^{7} - \)\(75\!\cdots\!36\)\( T^{8} - \)\(57\!\cdots\!70\)\( p T^{9} + \)\(47\!\cdots\!90\)\( T^{10} + \)\(34\!\cdots\!16\)\( T^{11} - \)\(56\!\cdots\!21\)\( T^{12} + \)\(68\!\cdots\!49\)\( T^{13} + \)\(27\!\cdots\!60\)\( p T^{14} - \)\(25\!\cdots\!11\)\( T^{15} - \)\(79\!\cdots\!41\)\( T^{16} + \)\(19\!\cdots\!96\)\( T^{17} - \)\(79\!\cdots\!64\)\( T^{18} - \)\(55\!\cdots\!80\)\( T^{19} + \)\(89\!\cdots\!44\)\( T^{20} - \)\(55\!\cdots\!80\)\( p^{7} T^{21} - \)\(79\!\cdots\!64\)\( p^{14} T^{22} + \)\(19\!\cdots\!96\)\( p^{21} T^{23} - \)\(79\!\cdots\!41\)\( p^{28} T^{24} - \)\(25\!\cdots\!11\)\( p^{35} T^{25} + \)\(27\!\cdots\!60\)\( p^{43} T^{26} + \)\(68\!\cdots\!49\)\( p^{49} T^{27} - \)\(56\!\cdots\!21\)\( p^{56} T^{28} + \)\(34\!\cdots\!16\)\( p^{63} T^{29} + \)\(47\!\cdots\!90\)\( p^{70} T^{30} - \)\(57\!\cdots\!70\)\( p^{78} T^{31} - \)\(75\!\cdots\!36\)\( p^{84} T^{32} + \)\(85\!\cdots\!46\)\( p^{91} T^{33} - \)\(33\!\cdots\!34\)\( p^{98} T^{34} - \)\(60\!\cdots\!24\)\( p^{105} T^{35} + 46998517878344521 p^{112} T^{36} + 2676593089315 p^{119} T^{37} - 308181696 p^{126} T^{38} - 4937 p^{133} T^{39} + p^{140} T^{40} \) | |
| 17 | \( ( 1 + 1433 p T + 3235528727 T^{2} + 71578269121440 T^{3} + 5012683639630581459 T^{4} + \)\(98\!\cdots\!37\)\( T^{5} + \)\(48\!\cdots\!24\)\( T^{6} + \)\(84\!\cdots\!11\)\( T^{7} + \)\(19\!\cdots\!76\)\( p T^{8} + \)\(49\!\cdots\!11\)\( T^{9} + \)\(15\!\cdots\!94\)\( T^{10} + \)\(49\!\cdots\!11\)\( p^{7} T^{11} + \)\(19\!\cdots\!76\)\( p^{15} T^{12} + \)\(84\!\cdots\!11\)\( p^{21} T^{13} + \)\(48\!\cdots\!24\)\( p^{28} T^{14} + \)\(98\!\cdots\!37\)\( p^{35} T^{15} + 5012683639630581459 p^{42} T^{16} + 71578269121440 p^{49} T^{17} + 3235528727 p^{56} T^{18} + 1433 p^{64} T^{19} + p^{70} T^{20} )^{2} \) | |
| 19 | \( ( 1 - 13441 T + 5825177209 T^{2} - 50755469372634 T^{3} + 15632970900473601273 T^{4} - \)\(51\!\cdots\!79\)\( T^{5} + \)\(25\!\cdots\!76\)\( T^{6} + \)\(26\!\cdots\!19\)\( p T^{7} + \)\(31\!\cdots\!74\)\( T^{8} + \)\(16\!\cdots\!15\)\( T^{9} + \)\(30\!\cdots\!38\)\( T^{10} + \)\(16\!\cdots\!15\)\( p^{7} T^{11} + \)\(31\!\cdots\!74\)\( p^{14} T^{12} + \)\(26\!\cdots\!19\)\( p^{22} T^{13} + \)\(25\!\cdots\!76\)\( p^{28} T^{14} - \)\(51\!\cdots\!79\)\( p^{35} T^{15} + 15632970900473601273 p^{42} T^{16} - 50755469372634 p^{49} T^{17} + 5825177209 p^{56} T^{18} - 13441 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 23 | \( 1 - 19387 T - 12506206216 T^{2} + 249399314526205 T^{3} + 72073454454396886987 T^{4} - \)\(23\!\cdots\!76\)\( T^{5} - \)\(23\!\cdots\!88\)\( T^{6} + \)\(17\!\cdots\!88\)\( T^{7} + \)\(43\!\cdots\!20\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{9} + \)\(39\!\cdots\!64\)\( T^{10} + \)\(51\!\cdots\!44\)\( T^{11} - \)\(80\!\cdots\!63\)\( T^{12} - \)\(18\!\cdots\!87\)\( T^{13} + \)\(56\!\cdots\!48\)\( T^{14} + \)\(52\!\cdots\!49\)\( T^{15} - \)\(33\!\cdots\!53\)\( T^{16} - \)\(11\!\cdots\!48\)\( T^{17} + \)\(16\!\cdots\!72\)\( T^{18} + \)\(12\!\cdots\!08\)\( T^{19} - \)\(65\!\cdots\!88\)\( T^{20} + \)\(12\!\cdots\!08\)\( p^{7} T^{21} + \)\(16\!\cdots\!72\)\( p^{14} T^{22} - \)\(11\!\cdots\!48\)\( p^{21} T^{23} - \)\(33\!\cdots\!53\)\( p^{28} T^{24} + \)\(52\!\cdots\!49\)\( p^{35} T^{25} + \)\(56\!\cdots\!48\)\( p^{42} T^{26} - \)\(18\!\cdots\!87\)\( p^{49} T^{27} - \)\(80\!\cdots\!63\)\( p^{56} T^{28} + \)\(51\!\cdots\!44\)\( p^{63} T^{29} + \)\(39\!\cdots\!64\)\( p^{70} T^{30} - \)\(10\!\cdots\!48\)\( p^{77} T^{31} + \)\(43\!\cdots\!20\)\( p^{84} T^{32} + \)\(17\!\cdots\!88\)\( p^{91} T^{33} - \)\(23\!\cdots\!88\)\( p^{98} T^{34} - \)\(23\!\cdots\!76\)\( p^{105} T^{35} + 72073454454396886987 p^{112} T^{36} + 249399314526205 p^{119} T^{37} - 12506206216 p^{126} T^{38} - 19387 p^{133} T^{39} + p^{140} T^{40} \) | |
| 29 | \( 1 - 46791 T - 74401739804 T^{2} + 6428230474090905 T^{3} + \)\(22\!\cdots\!21\)\( T^{4} - \)\(29\!\cdots\!16\)\( T^{5} - \)\(34\!\cdots\!74\)\( T^{6} + \)\(60\!\cdots\!38\)\( T^{7} + \)\(42\!\cdots\!36\)\( T^{8} - \)\(59\!\cdots\!70\)\( T^{9} - \)\(11\!\cdots\!74\)\( T^{10} + \)\(84\!\cdots\!76\)\( T^{11} + \)\(25\!\cdots\!39\)\( T^{12} - \)\(28\!\cdots\!45\)\( T^{13} - \)\(17\!\cdots\!20\)\( T^{14} + \)\(43\!\cdots\!19\)\( T^{15} - \)\(20\!\cdots\!05\)\( T^{16} - \)\(57\!\cdots\!84\)\( T^{17} + \)\(22\!\cdots\!88\)\( T^{18} - \)\(28\!\cdots\!96\)\( T^{19} + \)\(19\!\cdots\!56\)\( T^{20} - \)\(28\!\cdots\!96\)\( p^{7} T^{21} + \)\(22\!\cdots\!88\)\( p^{14} T^{22} - \)\(57\!\cdots\!84\)\( p^{21} T^{23} - \)\(20\!\cdots\!05\)\( p^{28} T^{24} + \)\(43\!\cdots\!19\)\( p^{35} T^{25} - \)\(17\!\cdots\!20\)\( p^{42} T^{26} - \)\(28\!\cdots\!45\)\( p^{49} T^{27} + \)\(25\!\cdots\!39\)\( p^{56} T^{28} + \)\(84\!\cdots\!76\)\( p^{63} T^{29} - \)\(11\!\cdots\!74\)\( p^{70} T^{30} - \)\(59\!\cdots\!70\)\( p^{77} T^{31} + \)\(42\!\cdots\!36\)\( p^{84} T^{32} + \)\(60\!\cdots\!38\)\( p^{91} T^{33} - \)\(34\!\cdots\!74\)\( p^{98} T^{34} - \)\(29\!\cdots\!16\)\( p^{105} T^{35} + \)\(22\!\cdots\!21\)\( p^{112} T^{36} + 6428230474090905 p^{119} T^{37} - 74401739804 p^{126} T^{38} - 46791 p^{133} T^{39} + p^{140} T^{40} \) | |
| 31 | \( 1 - 5969 p T - 129866900394 T^{2} + 41869090586140027 T^{3} + \)\(64\!\cdots\!61\)\( T^{4} - \)\(40\!\cdots\!32\)\( T^{5} + \)\(61\!\cdots\!14\)\( p T^{6} + \)\(22\!\cdots\!74\)\( T^{7} - \)\(22\!\cdots\!60\)\( T^{8} - \)\(86\!\cdots\!94\)\( T^{9} + \)\(16\!\cdots\!38\)\( T^{10} + \)\(21\!\cdots\!80\)\( T^{11} - \)\(71\!\cdots\!89\)\( T^{12} - \)\(26\!\cdots\!13\)\( T^{13} + \)\(22\!\cdots\!70\)\( T^{14} - \)\(30\!\cdots\!67\)\( T^{15} - \)\(57\!\cdots\!53\)\( T^{16} + \)\(20\!\cdots\!72\)\( T^{17} + \)\(12\!\cdots\!32\)\( T^{18} - \)\(94\!\cdots\!64\)\( p T^{19} - \)\(30\!\cdots\!92\)\( p^{2} T^{20} - \)\(94\!\cdots\!64\)\( p^{8} T^{21} + \)\(12\!\cdots\!32\)\( p^{14} T^{22} + \)\(20\!\cdots\!72\)\( p^{21} T^{23} - \)\(57\!\cdots\!53\)\( p^{28} T^{24} - \)\(30\!\cdots\!67\)\( p^{35} T^{25} + \)\(22\!\cdots\!70\)\( p^{42} T^{26} - \)\(26\!\cdots\!13\)\( p^{49} T^{27} - \)\(71\!\cdots\!89\)\( p^{56} T^{28} + \)\(21\!\cdots\!80\)\( p^{63} T^{29} + \)\(16\!\cdots\!38\)\( p^{70} T^{30} - \)\(86\!\cdots\!94\)\( p^{77} T^{31} - \)\(22\!\cdots\!60\)\( p^{84} T^{32} + \)\(22\!\cdots\!74\)\( p^{91} T^{33} + \)\(61\!\cdots\!14\)\( p^{99} T^{34} - \)\(40\!\cdots\!32\)\( p^{105} T^{35} + \)\(64\!\cdots\!61\)\( p^{112} T^{36} + 41869090586140027 p^{119} T^{37} - 129866900394 p^{126} T^{38} - 5969 p^{134} T^{39} + p^{140} T^{40} \) | |
| 37 | \( ( 1 - 54210 T + 456322860766 T^{2} - 52052423379323634 T^{3} + \)\(11\!\cdots\!85\)\( T^{4} - \)\(16\!\cdots\!76\)\( T^{5} + \)\(20\!\cdots\!36\)\( T^{6} - \)\(33\!\cdots\!92\)\( T^{7} + \)\(27\!\cdots\!14\)\( T^{8} - \)\(44\!\cdots\!40\)\( T^{9} + \)\(28\!\cdots\!24\)\( T^{10} - \)\(44\!\cdots\!40\)\( p^{7} T^{11} + \)\(27\!\cdots\!14\)\( p^{14} T^{12} - \)\(33\!\cdots\!92\)\( p^{21} T^{13} + \)\(20\!\cdots\!36\)\( p^{28} T^{14} - \)\(16\!\cdots\!76\)\( p^{35} T^{15} + \)\(11\!\cdots\!85\)\( p^{42} T^{16} - 52052423379323634 p^{49} T^{17} + 456322860766 p^{56} T^{18} - 54210 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 41 | \( 1 - 638112 T - 506921768897 T^{2} + 293497203350429040 T^{3} + \)\(11\!\cdots\!22\)\( T^{4} - \)\(34\!\cdots\!16\)\( T^{5} - \)\(25\!\cdots\!03\)\( T^{6} + \)\(59\!\cdots\!84\)\( T^{7} + \)\(31\!\cdots\!50\)\( T^{8} - \)\(31\!\cdots\!92\)\( T^{9} + \)\(23\!\cdots\!97\)\( T^{10} + \)\(40\!\cdots\!32\)\( T^{11} + \)\(34\!\cdots\!40\)\( T^{12} + \)\(11\!\cdots\!28\)\( T^{13} - \)\(17\!\cdots\!51\)\( T^{14} + \)\(87\!\cdots\!28\)\( T^{15} - \)\(49\!\cdots\!19\)\( T^{16} + \)\(28\!\cdots\!04\)\( T^{17} + \)\(24\!\cdots\!66\)\( T^{18} - \)\(12\!\cdots\!48\)\( T^{19} + \)\(27\!\cdots\!88\)\( T^{20} - \)\(12\!\cdots\!48\)\( p^{7} T^{21} + \)\(24\!\cdots\!66\)\( p^{14} T^{22} + \)\(28\!\cdots\!04\)\( p^{21} T^{23} - \)\(49\!\cdots\!19\)\( p^{28} T^{24} + \)\(87\!\cdots\!28\)\( p^{35} T^{25} - \)\(17\!\cdots\!51\)\( p^{42} T^{26} + \)\(11\!\cdots\!28\)\( p^{49} T^{27} + \)\(34\!\cdots\!40\)\( p^{56} T^{28} + \)\(40\!\cdots\!32\)\( p^{63} T^{29} + \)\(23\!\cdots\!97\)\( p^{70} T^{30} - \)\(31\!\cdots\!92\)\( p^{77} T^{31} + \)\(31\!\cdots\!50\)\( p^{84} T^{32} + \)\(59\!\cdots\!84\)\( p^{91} T^{33} - \)\(25\!\cdots\!03\)\( p^{98} T^{34} - \)\(34\!\cdots\!16\)\( p^{105} T^{35} + \)\(11\!\cdots\!22\)\( p^{112} T^{36} + 293497203350429040 p^{119} T^{37} - 506921768897 p^{126} T^{38} - 638112 p^{133} T^{39} + p^{140} T^{40} \) | |
| 43 | \( 1 + 892628 T - 663627037209 T^{2} - 804966205769887636 T^{3} + \)\(17\!\cdots\!70\)\( T^{4} + \)\(36\!\cdots\!92\)\( T^{5} + \)\(25\!\cdots\!93\)\( T^{6} - \)\(10\!\cdots\!20\)\( T^{7} - \)\(46\!\cdots\!14\)\( T^{8} + \)\(10\!\cdots\!40\)\( T^{9} + \)\(22\!\cdots\!69\)\( T^{10} + \)\(57\!\cdots\!32\)\( T^{11} - \)\(61\!\cdots\!16\)\( T^{12} - \)\(37\!\cdots\!80\)\( T^{13} + \)\(73\!\cdots\!21\)\( T^{14} + \)\(12\!\cdots\!72\)\( T^{15} + \)\(19\!\cdots\!49\)\( T^{16} - \)\(28\!\cdots\!88\)\( T^{17} - \)\(14\!\cdots\!98\)\( T^{18} + \)\(30\!\cdots\!72\)\( T^{19} + \)\(50\!\cdots\!76\)\( T^{20} + \)\(30\!\cdots\!72\)\( p^{7} T^{21} - \)\(14\!\cdots\!98\)\( p^{14} T^{22} - \)\(28\!\cdots\!88\)\( p^{21} T^{23} + \)\(19\!\cdots\!49\)\( p^{28} T^{24} + \)\(12\!\cdots\!72\)\( p^{35} T^{25} + \)\(73\!\cdots\!21\)\( p^{42} T^{26} - \)\(37\!\cdots\!80\)\( p^{49} T^{27} - \)\(61\!\cdots\!16\)\( p^{56} T^{28} + \)\(57\!\cdots\!32\)\( p^{63} T^{29} + \)\(22\!\cdots\!69\)\( p^{70} T^{30} + \)\(10\!\cdots\!40\)\( p^{77} T^{31} - \)\(46\!\cdots\!14\)\( p^{84} T^{32} - \)\(10\!\cdots\!20\)\( p^{91} T^{33} + \)\(25\!\cdots\!93\)\( p^{98} T^{34} + \)\(36\!\cdots\!92\)\( p^{105} T^{35} + \)\(17\!\cdots\!70\)\( p^{112} T^{36} - 804966205769887636 p^{119} T^{37} - 663627037209 p^{126} T^{38} + 892628 p^{133} T^{39} + p^{140} T^{40} \) | |
| 47 | \( 1 + 230883 T - 3111768396236 T^{2} - 1616547402912129945 T^{3} + \)\(49\!\cdots\!47\)\( T^{4} + \)\(39\!\cdots\!48\)\( T^{5} - \)\(50\!\cdots\!28\)\( T^{6} - \)\(57\!\cdots\!12\)\( T^{7} + \)\(34\!\cdots\!00\)\( T^{8} + \)\(13\!\cdots\!16\)\( p T^{9} - \)\(13\!\cdots\!76\)\( T^{10} - \)\(52\!\cdots\!44\)\( T^{11} - \)\(40\!\cdots\!87\)\( T^{12} + \)\(35\!\cdots\!39\)\( T^{13} + \)\(12\!\cdots\!28\)\( T^{14} - \)\(18\!\cdots\!37\)\( T^{15} - \)\(12\!\cdots\!33\)\( T^{16} + \)\(73\!\cdots\!68\)\( T^{17} + \)\(89\!\cdots\!68\)\( T^{18} - \)\(29\!\cdots\!08\)\( p T^{19} - \)\(22\!\cdots\!72\)\( p^{2} T^{20} - \)\(29\!\cdots\!08\)\( p^{8} T^{21} + \)\(89\!\cdots\!68\)\( p^{14} T^{22} + \)\(73\!\cdots\!68\)\( p^{21} T^{23} - \)\(12\!\cdots\!33\)\( p^{28} T^{24} - \)\(18\!\cdots\!37\)\( p^{35} T^{25} + \)\(12\!\cdots\!28\)\( p^{42} T^{26} + \)\(35\!\cdots\!39\)\( p^{49} T^{27} - \)\(40\!\cdots\!87\)\( p^{56} T^{28} - \)\(52\!\cdots\!44\)\( p^{63} T^{29} - \)\(13\!\cdots\!76\)\( p^{70} T^{30} + \)\(13\!\cdots\!16\)\( p^{78} T^{31} + \)\(34\!\cdots\!00\)\( p^{84} T^{32} - \)\(57\!\cdots\!12\)\( p^{91} T^{33} - \)\(50\!\cdots\!28\)\( p^{98} T^{34} + \)\(39\!\cdots\!48\)\( p^{105} T^{35} + \)\(49\!\cdots\!47\)\( p^{112} T^{36} - 1616547402912129945 p^{119} T^{37} - 3111768396236 p^{126} T^{38} + 230883 p^{133} T^{39} + p^{140} T^{40} \) | |
| 53 | \( ( 1 - 1436470 T + 6680853196766 T^{2} - 4838141531949573414 T^{3} + \)\(16\!\cdots\!61\)\( T^{4} - \)\(36\!\cdots\!20\)\( T^{5} + \)\(26\!\cdots\!68\)\( T^{6} - \)\(67\!\cdots\!60\)\( T^{7} + \)\(44\!\cdots\!90\)\( T^{8} - \)\(61\!\cdots\!92\)\( T^{9} + \)\(63\!\cdots\!12\)\( T^{10} - \)\(61\!\cdots\!92\)\( p^{7} T^{11} + \)\(44\!\cdots\!90\)\( p^{14} T^{12} - \)\(67\!\cdots\!60\)\( p^{21} T^{13} + \)\(26\!\cdots\!68\)\( p^{28} T^{14} - \)\(36\!\cdots\!20\)\( p^{35} T^{15} + \)\(16\!\cdots\!61\)\( p^{42} T^{16} - 4838141531949573414 p^{49} T^{17} + 6680853196766 p^{56} T^{18} - 1436470 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 59 | \( 1 - 2172454 T - 10886534918191 T^{2} + 12982595429641623586 T^{3} + \)\(84\!\cdots\!34\)\( T^{4} - \)\(17\!\cdots\!62\)\( T^{5} - \)\(42\!\cdots\!37\)\( T^{6} - \)\(24\!\cdots\!06\)\( T^{7} + \)\(14\!\cdots\!74\)\( T^{8} + \)\(17\!\cdots\!62\)\( T^{9} - \)\(30\!\cdots\!61\)\( T^{10} - \)\(59\!\cdots\!10\)\( T^{11} + \)\(35\!\cdots\!16\)\( T^{12} + \)\(10\!\cdots\!66\)\( T^{13} - \)\(33\!\cdots\!25\)\( T^{14} + \)\(11\!\cdots\!90\)\( T^{15} + \)\(34\!\cdots\!05\)\( T^{16} - \)\(49\!\cdots\!32\)\( T^{17} - \)\(19\!\cdots\!62\)\( T^{18} + \)\(69\!\cdots\!12\)\( T^{19} + \)\(64\!\cdots\!16\)\( T^{20} + \)\(69\!\cdots\!12\)\( p^{7} T^{21} - \)\(19\!\cdots\!62\)\( p^{14} T^{22} - \)\(49\!\cdots\!32\)\( p^{21} T^{23} + \)\(34\!\cdots\!05\)\( p^{28} T^{24} + \)\(11\!\cdots\!90\)\( p^{35} T^{25} - \)\(33\!\cdots\!25\)\( p^{42} T^{26} + \)\(10\!\cdots\!66\)\( p^{49} T^{27} + \)\(35\!\cdots\!16\)\( p^{56} T^{28} - \)\(59\!\cdots\!10\)\( p^{63} T^{29} - \)\(30\!\cdots\!61\)\( p^{70} T^{30} + \)\(17\!\cdots\!62\)\( p^{77} T^{31} + \)\(14\!\cdots\!74\)\( p^{84} T^{32} - \)\(24\!\cdots\!06\)\( p^{91} T^{33} - \)\(42\!\cdots\!37\)\( p^{98} T^{34} - \)\(17\!\cdots\!62\)\( p^{105} T^{35} + \)\(84\!\cdots\!34\)\( p^{112} T^{36} + 12982595429641623586 p^{119} T^{37} - 10886534918191 p^{126} T^{38} - 2172454 p^{133} T^{39} + p^{140} T^{40} \) | |
| 61 | \( 1 + 1878325 T - 20634110030190 T^{2} - 33939383176088976593 T^{3} + \)\(23\!\cdots\!75\)\( T^{4} + \)\(31\!\cdots\!96\)\( T^{5} - \)\(18\!\cdots\!72\)\( T^{6} - \)\(20\!\cdots\!44\)\( T^{7} + \)\(12\!\cdots\!88\)\( T^{8} + \)\(10\!\cdots\!84\)\( T^{9} - \)\(67\!\cdots\!88\)\( T^{10} - \)\(47\!\cdots\!28\)\( T^{11} + \)\(32\!\cdots\!33\)\( T^{12} + \)\(17\!\cdots\!41\)\( T^{13} - \)\(14\!\cdots\!18\)\( T^{14} - \)\(51\!\cdots\!41\)\( T^{15} + \)\(55\!\cdots\!99\)\( T^{16} + \)\(11\!\cdots\!92\)\( T^{17} - \)\(20\!\cdots\!40\)\( T^{18} - \)\(13\!\cdots\!32\)\( T^{19} + \)\(65\!\cdots\!68\)\( T^{20} - \)\(13\!\cdots\!32\)\( p^{7} T^{21} - \)\(20\!\cdots\!40\)\( p^{14} T^{22} + \)\(11\!\cdots\!92\)\( p^{21} T^{23} + \)\(55\!\cdots\!99\)\( p^{28} T^{24} - \)\(51\!\cdots\!41\)\( p^{35} T^{25} - \)\(14\!\cdots\!18\)\( p^{42} T^{26} + \)\(17\!\cdots\!41\)\( p^{49} T^{27} + \)\(32\!\cdots\!33\)\( p^{56} T^{28} - \)\(47\!\cdots\!28\)\( p^{63} T^{29} - \)\(67\!\cdots\!88\)\( p^{70} T^{30} + \)\(10\!\cdots\!84\)\( p^{77} T^{31} + \)\(12\!\cdots\!88\)\( p^{84} T^{32} - \)\(20\!\cdots\!44\)\( p^{91} T^{33} - \)\(18\!\cdots\!72\)\( p^{98} T^{34} + \)\(31\!\cdots\!96\)\( p^{105} T^{35} + \)\(23\!\cdots\!75\)\( p^{112} T^{36} - 33939383176088976593 p^{119} T^{37} - 20634110030190 p^{126} T^{38} + 1878325 p^{133} T^{39} + p^{140} T^{40} \) | |
| 67 | \( 1 + 531496 T - 35441234441373 T^{2} + 25233421371767901160 T^{3} + \)\(71\!\cdots\!58\)\( T^{4} - \)\(12\!\cdots\!68\)\( T^{5} - \)\(93\!\cdots\!55\)\( T^{6} + \)\(27\!\cdots\!08\)\( T^{7} + \)\(84\!\cdots\!94\)\( T^{8} - \)\(41\!\cdots\!64\)\( T^{9} - \)\(43\!\cdots\!83\)\( T^{10} + \)\(45\!\cdots\!92\)\( T^{11} - \)\(93\!\cdots\!28\)\( T^{12} - \)\(39\!\cdots\!72\)\( T^{13} + \)\(49\!\cdots\!41\)\( T^{14} + \)\(26\!\cdots\!60\)\( T^{15} - \)\(64\!\cdots\!51\)\( T^{16} - \)\(12\!\cdots\!72\)\( T^{17} + \)\(57\!\cdots\!98\)\( T^{18} + \)\(29\!\cdots\!44\)\( T^{19} - \)\(38\!\cdots\!56\)\( T^{20} + \)\(29\!\cdots\!44\)\( p^{7} T^{21} + \)\(57\!\cdots\!98\)\( p^{14} T^{22} - \)\(12\!\cdots\!72\)\( p^{21} T^{23} - \)\(64\!\cdots\!51\)\( p^{28} T^{24} + \)\(26\!\cdots\!60\)\( p^{35} T^{25} + \)\(49\!\cdots\!41\)\( p^{42} T^{26} - \)\(39\!\cdots\!72\)\( p^{49} T^{27} - \)\(93\!\cdots\!28\)\( p^{56} T^{28} + \)\(45\!\cdots\!92\)\( p^{63} T^{29} - \)\(43\!\cdots\!83\)\( p^{70} T^{30} - \)\(41\!\cdots\!64\)\( p^{77} T^{31} + \)\(84\!\cdots\!94\)\( p^{84} T^{32} + \)\(27\!\cdots\!08\)\( p^{91} T^{33} - \)\(93\!\cdots\!55\)\( p^{98} T^{34} - \)\(12\!\cdots\!68\)\( p^{105} T^{35} + \)\(71\!\cdots\!58\)\( p^{112} T^{36} + 25233421371767901160 p^{119} T^{37} - 35441234441373 p^{126} T^{38} + 531496 p^{133} T^{39} + p^{140} T^{40} \) | |
| 71 | \( ( 1 - 1861528 T + 31121113201046 T^{2} + 8230951185002352024 T^{3} + \)\(55\!\cdots\!85\)\( T^{4} + \)\(42\!\cdots\!60\)\( T^{5} + \)\(90\!\cdots\!20\)\( T^{6} + \)\(91\!\cdots\!80\)\( T^{7} + \)\(10\!\cdots\!50\)\( T^{8} + \)\(11\!\cdots\!68\)\( T^{9} + \)\(11\!\cdots\!20\)\( T^{10} + \)\(11\!\cdots\!68\)\( p^{7} T^{11} + \)\(10\!\cdots\!50\)\( p^{14} T^{12} + \)\(91\!\cdots\!80\)\( p^{21} T^{13} + \)\(90\!\cdots\!20\)\( p^{28} T^{14} + \)\(42\!\cdots\!60\)\( p^{35} T^{15} + \)\(55\!\cdots\!85\)\( p^{42} T^{16} + 8230951185002352024 p^{49} T^{17} + 31121113201046 p^{56} T^{18} - 1861528 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 902261 T + 69161812265467 T^{2} + 41740609211965087080 T^{3} + \)\(22\!\cdots\!19\)\( T^{4} + \)\(39\!\cdots\!93\)\( T^{5} + \)\(44\!\cdots\!24\)\( T^{6} - \)\(12\!\cdots\!29\)\( T^{7} + \)\(65\!\cdots\!80\)\( T^{8} - \)\(38\!\cdots\!89\)\( T^{9} + \)\(77\!\cdots\!14\)\( T^{10} - \)\(38\!\cdots\!89\)\( p^{7} T^{11} + \)\(65\!\cdots\!80\)\( p^{14} T^{12} - \)\(12\!\cdots\!29\)\( p^{21} T^{13} + \)\(44\!\cdots\!24\)\( p^{28} T^{14} + \)\(39\!\cdots\!93\)\( p^{35} T^{15} + \)\(22\!\cdots\!19\)\( p^{42} T^{16} + 41740609211965087080 p^{49} T^{17} + 69161812265467 p^{56} T^{18} + 902261 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 79 | \( 1 + 3607847 T - 31379316907218 T^{2} - \)\(49\!\cdots\!27\)\( T^{3} - \)\(14\!\cdots\!31\)\( T^{4} + \)\(77\!\cdots\!08\)\( T^{5} + \)\(11\!\cdots\!98\)\( T^{6} + \)\(45\!\cdots\!86\)\( T^{7} - \)\(38\!\cdots\!40\)\( T^{8} - \)\(17\!\cdots\!86\)\( T^{9} - \)\(94\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!08\)\( T^{11} + \)\(16\!\cdots\!83\)\( T^{12} + \)\(12\!\cdots\!73\)\( T^{13} + \)\(38\!\cdots\!22\)\( T^{14} - \)\(69\!\cdots\!97\)\( T^{15} - \)\(12\!\cdots\!25\)\( T^{16} - \)\(57\!\cdots\!44\)\( T^{17} - \)\(57\!\cdots\!36\)\( T^{18} + \)\(80\!\cdots\!28\)\( T^{19} + \)\(56\!\cdots\!48\)\( T^{20} + \)\(80\!\cdots\!28\)\( p^{7} T^{21} - \)\(57\!\cdots\!36\)\( p^{14} T^{22} - \)\(57\!\cdots\!44\)\( p^{21} T^{23} - \)\(12\!\cdots\!25\)\( p^{28} T^{24} - \)\(69\!\cdots\!97\)\( p^{35} T^{25} + \)\(38\!\cdots\!22\)\( p^{42} T^{26} + \)\(12\!\cdots\!73\)\( p^{49} T^{27} + \)\(16\!\cdots\!83\)\( p^{56} T^{28} - \)\(12\!\cdots\!08\)\( p^{63} T^{29} - \)\(94\!\cdots\!74\)\( p^{70} T^{30} - \)\(17\!\cdots\!86\)\( p^{77} T^{31} - \)\(38\!\cdots\!40\)\( p^{84} T^{32} + \)\(45\!\cdots\!86\)\( p^{91} T^{33} + \)\(11\!\cdots\!98\)\( p^{98} T^{34} + \)\(77\!\cdots\!08\)\( p^{105} T^{35} - \)\(14\!\cdots\!31\)\( p^{112} T^{36} - \)\(49\!\cdots\!27\)\( p^{119} T^{37} - 31379316907218 p^{126} T^{38} + 3607847 p^{133} T^{39} + p^{140} T^{40} \) | |
| 83 | \( 1 - 10794491 T - 47253781940122 T^{2} + \)\(87\!\cdots\!27\)\( T^{3} + \)\(11\!\cdots\!13\)\( T^{4} - \)\(28\!\cdots\!16\)\( T^{5} + \)\(35\!\cdots\!50\)\( T^{6} + \)\(25\!\cdots\!98\)\( T^{7} + \)\(40\!\cdots\!72\)\( T^{8} + \)\(12\!\cdots\!50\)\( T^{9} - \)\(98\!\cdots\!50\)\( T^{10} - \)\(45\!\cdots\!28\)\( T^{11} + \)\(32\!\cdots\!35\)\( T^{12} + \)\(75\!\cdots\!35\)\( T^{13} - \)\(32\!\cdots\!02\)\( T^{14} - \)\(10\!\cdots\!79\)\( T^{15} - \)\(13\!\cdots\!25\)\( T^{16} + \)\(18\!\cdots\!88\)\( T^{17} + \)\(80\!\cdots\!00\)\( T^{18} - \)\(23\!\cdots\!08\)\( T^{19} - \)\(25\!\cdots\!64\)\( T^{20} - \)\(23\!\cdots\!08\)\( p^{7} T^{21} + \)\(80\!\cdots\!00\)\( p^{14} T^{22} + \)\(18\!\cdots\!88\)\( p^{21} T^{23} - \)\(13\!\cdots\!25\)\( p^{28} T^{24} - \)\(10\!\cdots\!79\)\( p^{35} T^{25} - \)\(32\!\cdots\!02\)\( p^{42} T^{26} + \)\(75\!\cdots\!35\)\( p^{49} T^{27} + \)\(32\!\cdots\!35\)\( p^{56} T^{28} - \)\(45\!\cdots\!28\)\( p^{63} T^{29} - \)\(98\!\cdots\!50\)\( p^{70} T^{30} + \)\(12\!\cdots\!50\)\( p^{77} T^{31} + \)\(40\!\cdots\!72\)\( p^{84} T^{32} + \)\(25\!\cdots\!98\)\( p^{91} T^{33} + \)\(35\!\cdots\!50\)\( p^{98} T^{34} - \)\(28\!\cdots\!16\)\( p^{105} T^{35} + \)\(11\!\cdots\!13\)\( p^{112} T^{36} + \)\(87\!\cdots\!27\)\( p^{119} T^{37} - 47253781940122 p^{126} T^{38} - 10794491 p^{133} T^{39} + p^{140} T^{40} \) | |
| 89 | \( ( 1 + 16107444 T + 321703324522910 T^{2} + \)\(35\!\cdots\!04\)\( T^{3} + \)\(43\!\cdots\!05\)\( T^{4} + \)\(38\!\cdots\!80\)\( T^{5} + \)\(37\!\cdots\!04\)\( T^{6} + \)\(28\!\cdots\!20\)\( T^{7} + \)\(23\!\cdots\!26\)\( T^{8} + \)\(16\!\cdots\!44\)\( T^{9} + \)\(11\!\cdots\!48\)\( T^{10} + \)\(16\!\cdots\!44\)\( p^{7} T^{11} + \)\(23\!\cdots\!26\)\( p^{14} T^{12} + \)\(28\!\cdots\!20\)\( p^{21} T^{13} + \)\(37\!\cdots\!04\)\( p^{28} T^{14} + \)\(38\!\cdots\!80\)\( p^{35} T^{15} + \)\(43\!\cdots\!05\)\( p^{42} T^{16} + \)\(35\!\cdots\!04\)\( p^{49} T^{17} + 321703324522910 p^{56} T^{18} + 16107444 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 97 | \( 1 - 17951260 T - 275094546688125 T^{2} + \)\(35\!\cdots\!24\)\( T^{3} + \)\(84\!\cdots\!54\)\( T^{4} - \)\(39\!\cdots\!72\)\( T^{5} - \)\(15\!\cdots\!27\)\( T^{6} - \)\(12\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!94\)\( T^{8} + \)\(88\!\cdots\!52\)\( T^{9} - \)\(15\!\cdots\!79\)\( T^{10} - \)\(12\!\cdots\!16\)\( T^{11} + \)\(81\!\cdots\!64\)\( T^{12} + \)\(85\!\cdots\!12\)\( T^{13} - \)\(32\!\cdots\!35\)\( T^{14} - \)\(14\!\cdots\!20\)\( T^{15} + \)\(49\!\cdots\!85\)\( T^{16} - \)\(26\!\cdots\!44\)\( T^{17} - \)\(82\!\cdots\!62\)\( T^{18} + \)\(13\!\cdots\!20\)\( T^{19} + \)\(86\!\cdots\!20\)\( T^{20} + \)\(13\!\cdots\!20\)\( p^{7} T^{21} - \)\(82\!\cdots\!62\)\( p^{14} T^{22} - \)\(26\!\cdots\!44\)\( p^{21} T^{23} + \)\(49\!\cdots\!85\)\( p^{28} T^{24} - \)\(14\!\cdots\!20\)\( p^{35} T^{25} - \)\(32\!\cdots\!35\)\( p^{42} T^{26} + \)\(85\!\cdots\!12\)\( p^{49} T^{27} + \)\(81\!\cdots\!64\)\( p^{56} T^{28} - \)\(12\!\cdots\!16\)\( p^{63} T^{29} - \)\(15\!\cdots\!79\)\( p^{70} T^{30} + \)\(88\!\cdots\!52\)\( p^{77} T^{31} + \)\(19\!\cdots\!94\)\( p^{84} T^{32} - \)\(12\!\cdots\!36\)\( p^{91} T^{33} - \)\(15\!\cdots\!27\)\( p^{98} T^{34} - \)\(39\!\cdots\!72\)\( p^{105} T^{35} + \)\(84\!\cdots\!54\)\( p^{112} T^{36} + \)\(35\!\cdots\!24\)\( p^{119} T^{37} - 275094546688125 p^{126} T^{38} - 17951260 p^{133} T^{39} + p^{140} T^{40} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−1.53551492534693855070576097785, −1.52251529564755217616725871843, −1.49363684621805860957443763167, −1.46335787558228347228346153943, −1.35218058683073451135929675661, −1.29665940299247189511846101859, −1.08977454542341155428500085318, −1.07853013352582536772672680162, −0.920688847626330632059021612388, −0.908493148424802354568705349344, −0.884978700201472039063198010730, −0.830249761765379436820007624526, −0.820952032522116582183484447062, −0.76603624685522235199117623861, −0.69915754970533849055064467924, −0.56345314156138636981989232959, −0.48904465520178049741438141906, −0.48022805263716946741693105146, −0.45412019599803271857584172763, −0.44793809827729524640923244536, −0.44001248827879927494393959951, −0.39166597916248853059844324257, −0.23364125140525914025819624841, −0.14380996549416970995783101723, −0.02300946457484224460840553369, 0.02300946457484224460840553369, 0.14380996549416970995783101723, 0.23364125140525914025819624841, 0.39166597916248853059844324257, 0.44001248827879927494393959951, 0.44793809827729524640923244536, 0.45412019599803271857584172763, 0.48022805263716946741693105146, 0.48904465520178049741438141906, 0.56345314156138636981989232959, 0.69915754970533849055064467924, 0.76603624685522235199117623861, 0.820952032522116582183484447062, 0.830249761765379436820007624526, 0.884978700201472039063198010730, 0.908493148424802354568705349344, 0.920688847626330632059021612388, 1.07853013352582536772672680162, 1.08977454542341155428500085318, 1.29665940299247189511846101859, 1.35218058683073451135929675661, 1.46335787558228347228346153943, 1.49363684621805860957443763167, 1.52251529564755217616725871843, 1.53551492534693855070576097785
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.