Dirichlet series
| L(s) = 1 | − 12·2-s − 21·3-s + 28·4-s + 100·5-s + 252·6-s − 131·7-s + 236·8-s + 30·9-s − 1.20e3·10-s + 220·11-s − 588·12-s − 223·13-s + 1.57e3·14-s − 2.10e3·15-s − 1.33e3·16-s − 471·17-s − 360·18-s + 380·19-s + 2.80e3·20-s + 2.75e3·21-s − 2.64e3·22-s − 653·23-s − 4.95e3·24-s + 5.25e3·25-s + 2.67e3·26-s + 2.58e3·27-s − 3.66e3·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 4.04·3-s + 7/2·4-s + 8.94·5-s + 17.1·6-s − 7.07·7-s + 10.4·8-s + 10/9·9-s − 37.9·10-s + 6.03·11-s − 14.1·12-s − 4.75·13-s + 30.0·14-s − 36.1·15-s − 20.8·16-s − 6.71·17-s − 4.71·18-s + 4.58·19-s + 31.3·20-s + 28.5·21-s − 25.5·22-s − 5.91·23-s − 42.1·24-s + 42·25-s + 20.1·26-s + 18.4·27-s − 24.7·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 11^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{20} \cdot 11^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{20} \cdot 11^{20} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.30445\times 10^{35}\) |
| Root analytic conductor: | \(7.85219\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(20\) |
| Selberg data: | \((40,\ 5^{20} \cdot 11^{20} \cdot 19^{20} ,\ ( \ : [3/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 - p T )^{20} \) |
| 11 | \( ( 1 - p T )^{20} \) | |
| 19 | \( ( 1 - p T )^{20} \) | |
| good | 2 | \( 1 + 3 p^{2} T + 29 p^{2} T^{2} + 205 p^{2} T^{3} + 5097 T^{4} + 27317 T^{5} + 133939 T^{6} + 74605 p^{3} T^{7} + 2484433 T^{8} + 2407585 p^{2} T^{9} + 8835015 p^{2} T^{10} + 61280293 p T^{11} + 406746809 T^{12} + 1290810915 T^{13} + 3962270541 T^{14} + 5885068621 p T^{15} + 8561394975 p^{2} T^{16} + 12218287303 p^{3} T^{17} + 1082558005 p^{8} T^{18} + 24405553279 p^{5} T^{19} + 8629887767 p^{8} T^{20} + 24405553279 p^{8} T^{21} + 1082558005 p^{14} T^{22} + 12218287303 p^{12} T^{23} + 8561394975 p^{14} T^{24} + 5885068621 p^{16} T^{25} + 3962270541 p^{18} T^{26} + 1290810915 p^{21} T^{27} + 406746809 p^{24} T^{28} + 61280293 p^{28} T^{29} + 8835015 p^{32} T^{30} + 2407585 p^{35} T^{31} + 2484433 p^{36} T^{32} + 74605 p^{42} T^{33} + 133939 p^{42} T^{34} + 27317 p^{45} T^{35} + 5097 p^{48} T^{36} + 205 p^{53} T^{37} + 29 p^{56} T^{38} + 3 p^{59} T^{39} + p^{60} T^{40} \) |
| 3 | \( 1 + 7 p T + 137 p T^{2} + 5414 T^{3} + 66520 T^{4} + 676756 T^{5} + 6524822 T^{6} + 56020246 T^{7} + 461755423 T^{8} + 1171490299 p T^{9} + 25911087274 T^{10} + 60021360043 p T^{11} + 406150745611 p T^{12} + 7865044591138 T^{13} + 67983809911 p^{6} T^{14} + 299998498283791 T^{15} + 1774845490357372 T^{16} + 10129618494016913 T^{17} + 232692887468698 p^{5} T^{18} + 305231772167374669 T^{19} + 1612176832836496790 T^{20} + 305231772167374669 p^{3} T^{21} + 232692887468698 p^{11} T^{22} + 10129618494016913 p^{9} T^{23} + 1774845490357372 p^{12} T^{24} + 299998498283791 p^{15} T^{25} + 67983809911 p^{24} T^{26} + 7865044591138 p^{21} T^{27} + 406150745611 p^{25} T^{28} + 60021360043 p^{28} T^{29} + 25911087274 p^{30} T^{30} + 1171490299 p^{34} T^{31} + 461755423 p^{36} T^{32} + 56020246 p^{39} T^{33} + 6524822 p^{42} T^{34} + 676756 p^{45} T^{35} + 66520 p^{48} T^{36} + 5414 p^{51} T^{37} + 137 p^{55} T^{38} + 7 p^{58} T^{39} + p^{60} T^{40} \) | |
| 7 | \( 1 + 131 T + 11628 T^{2} + 764621 T^{3} + 41812238 T^{4} + 280319617 p T^{5} + 81875882051 T^{6} + 3087727796632 T^{7} + 15284702970332 p T^{8} + 3439435011107684 T^{9} + 103507419356267497 T^{10} + 2933327221468310613 T^{11} + 78699872420487495351 T^{12} + \)\(20\!\cdots\!66\)\( T^{13} + \)\(48\!\cdots\!15\)\( T^{14} + \)\(11\!\cdots\!05\)\( T^{15} + \)\(25\!\cdots\!19\)\( T^{16} + \)\(53\!\cdots\!62\)\( T^{17} + \)\(10\!\cdots\!01\)\( T^{18} + \)\(30\!\cdots\!61\)\( p T^{19} + \)\(40\!\cdots\!42\)\( T^{20} + \)\(30\!\cdots\!61\)\( p^{4} T^{21} + \)\(10\!\cdots\!01\)\( p^{6} T^{22} + \)\(53\!\cdots\!62\)\( p^{9} T^{23} + \)\(25\!\cdots\!19\)\( p^{12} T^{24} + \)\(11\!\cdots\!05\)\( p^{15} T^{25} + \)\(48\!\cdots\!15\)\( p^{18} T^{26} + \)\(20\!\cdots\!66\)\( p^{21} T^{27} + 78699872420487495351 p^{24} T^{28} + 2933327221468310613 p^{27} T^{29} + 103507419356267497 p^{30} T^{30} + 3439435011107684 p^{33} T^{31} + 15284702970332 p^{37} T^{32} + 3087727796632 p^{39} T^{33} + 81875882051 p^{42} T^{34} + 280319617 p^{46} T^{35} + 41812238 p^{48} T^{36} + 764621 p^{51} T^{37} + 11628 p^{54} T^{38} + 131 p^{57} T^{39} + p^{60} T^{40} \) | |
| 13 | \( 1 + 223 T + 49443 T^{2} + 6974376 T^{3} + 939027110 T^{4} + 100853923880 T^{5} + 10372256055435 T^{6} + 922368198843693 T^{7} + 79109487200748062 T^{8} + 6102742171517867051 T^{9} + 35133624850540503902 p T^{10} + \)\(31\!\cdots\!69\)\( T^{11} + \)\(21\!\cdots\!22\)\( T^{12} + \)\(13\!\cdots\!97\)\( T^{13} + \)\(61\!\cdots\!85\)\( p T^{14} + \)\(46\!\cdots\!32\)\( T^{15} + \)\(25\!\cdots\!67\)\( T^{16} + \)\(13\!\cdots\!87\)\( T^{17} + \)\(71\!\cdots\!59\)\( T^{18} + \)\(34\!\cdots\!64\)\( T^{19} + \)\(16\!\cdots\!08\)\( T^{20} + \)\(34\!\cdots\!64\)\( p^{3} T^{21} + \)\(71\!\cdots\!59\)\( p^{6} T^{22} + \)\(13\!\cdots\!87\)\( p^{9} T^{23} + \)\(25\!\cdots\!67\)\( p^{12} T^{24} + \)\(46\!\cdots\!32\)\( p^{15} T^{25} + \)\(61\!\cdots\!85\)\( p^{19} T^{26} + \)\(13\!\cdots\!97\)\( p^{21} T^{27} + \)\(21\!\cdots\!22\)\( p^{24} T^{28} + \)\(31\!\cdots\!69\)\( p^{27} T^{29} + 35133624850540503902 p^{31} T^{30} + 6102742171517867051 p^{33} T^{31} + 79109487200748062 p^{36} T^{32} + 922368198843693 p^{39} T^{33} + 10372256055435 p^{42} T^{34} + 100853923880 p^{45} T^{35} + 939027110 p^{48} T^{36} + 6974376 p^{51} T^{37} + 49443 p^{54} T^{38} + 223 p^{57} T^{39} + p^{60} T^{40} \) | |
| 17 | \( 1 + 471 T + 153874 T^{2} + 36580853 T^{3} + 7305598954 T^{4} + 1244858696039 T^{5} + 189923663633044 T^{6} + 26147648699732747 T^{7} + 3325825781655648346 T^{8} + \)\(39\!\cdots\!93\)\( T^{9} + \)\(43\!\cdots\!79\)\( T^{10} + \)\(45\!\cdots\!64\)\( T^{11} + \)\(45\!\cdots\!77\)\( T^{12} + \)\(42\!\cdots\!49\)\( T^{13} + \)\(38\!\cdots\!79\)\( T^{14} + \)\(33\!\cdots\!00\)\( T^{15} + \)\(27\!\cdots\!29\)\( T^{16} + \)\(22\!\cdots\!89\)\( T^{17} + \)\(10\!\cdots\!56\)\( p T^{18} + \)\(12\!\cdots\!63\)\( T^{19} + \)\(91\!\cdots\!42\)\( T^{20} + \)\(12\!\cdots\!63\)\( p^{3} T^{21} + \)\(10\!\cdots\!56\)\( p^{7} T^{22} + \)\(22\!\cdots\!89\)\( p^{9} T^{23} + \)\(27\!\cdots\!29\)\( p^{12} T^{24} + \)\(33\!\cdots\!00\)\( p^{15} T^{25} + \)\(38\!\cdots\!79\)\( p^{18} T^{26} + \)\(42\!\cdots\!49\)\( p^{21} T^{27} + \)\(45\!\cdots\!77\)\( p^{24} T^{28} + \)\(45\!\cdots\!64\)\( p^{27} T^{29} + \)\(43\!\cdots\!79\)\( p^{30} T^{30} + \)\(39\!\cdots\!93\)\( p^{33} T^{31} + 3325825781655648346 p^{36} T^{32} + 26147648699732747 p^{39} T^{33} + 189923663633044 p^{42} T^{34} + 1244858696039 p^{45} T^{35} + 7305598954 p^{48} T^{36} + 36580853 p^{51} T^{37} + 153874 p^{54} T^{38} + 471 p^{57} T^{39} + p^{60} T^{40} \) | |
| 23 | \( 1 + 653 T + 13212 p T^{2} + 106638575 T^{3} + 31794180255 T^{4} + 8247481221174 T^{5} + 3645366715427 p^{2} T^{6} + 411420721358096459 T^{7} + 81371207965605002609 T^{8} + \)\(15\!\cdots\!44\)\( T^{9} + \)\(26\!\cdots\!21\)\( T^{10} + \)\(42\!\cdots\!53\)\( T^{11} + \)\(67\!\cdots\!76\)\( T^{12} + \)\(10\!\cdots\!19\)\( T^{13} + \)\(14\!\cdots\!54\)\( T^{14} + \)\(19\!\cdots\!27\)\( T^{15} + \)\(25\!\cdots\!58\)\( T^{16} + \)\(32\!\cdots\!87\)\( T^{17} + \)\(38\!\cdots\!70\)\( T^{18} + \)\(45\!\cdots\!43\)\( T^{19} + \)\(51\!\cdots\!38\)\( T^{20} + \)\(45\!\cdots\!43\)\( p^{3} T^{21} + \)\(38\!\cdots\!70\)\( p^{6} T^{22} + \)\(32\!\cdots\!87\)\( p^{9} T^{23} + \)\(25\!\cdots\!58\)\( p^{12} T^{24} + \)\(19\!\cdots\!27\)\( p^{15} T^{25} + \)\(14\!\cdots\!54\)\( p^{18} T^{26} + \)\(10\!\cdots\!19\)\( p^{21} T^{27} + \)\(67\!\cdots\!76\)\( p^{24} T^{28} + \)\(42\!\cdots\!53\)\( p^{27} T^{29} + \)\(26\!\cdots\!21\)\( p^{30} T^{30} + \)\(15\!\cdots\!44\)\( p^{33} T^{31} + 81371207965605002609 p^{36} T^{32} + 411420721358096459 p^{39} T^{33} + 3645366715427 p^{44} T^{34} + 8247481221174 p^{45} T^{35} + 31794180255 p^{48} T^{36} + 106638575 p^{51} T^{37} + 13212 p^{55} T^{38} + 653 p^{57} T^{39} + p^{60} T^{40} \) | |
| 29 | \( 1 - 51 T + 272954 T^{2} - 9147183 T^{3} + 36388732382 T^{4} - 579081280549 T^{5} + 3188677351871038 T^{6} + 2498590670994077 T^{7} + \)\(20\!\cdots\!74\)\( T^{8} + \)\(32\!\cdots\!65\)\( T^{9} + \)\(10\!\cdots\!87\)\( T^{10} + \)\(29\!\cdots\!26\)\( T^{11} + \)\(48\!\cdots\!55\)\( T^{12} + \)\(17\!\cdots\!29\)\( T^{13} + \)\(18\!\cdots\!53\)\( T^{14} + \)\(73\!\cdots\!50\)\( T^{15} + \)\(60\!\cdots\!89\)\( T^{16} + \)\(25\!\cdots\!27\)\( T^{17} + \)\(17\!\cdots\!20\)\( T^{18} + \)\(74\!\cdots\!09\)\( T^{19} + \)\(46\!\cdots\!74\)\( T^{20} + \)\(74\!\cdots\!09\)\( p^{3} T^{21} + \)\(17\!\cdots\!20\)\( p^{6} T^{22} + \)\(25\!\cdots\!27\)\( p^{9} T^{23} + \)\(60\!\cdots\!89\)\( p^{12} T^{24} + \)\(73\!\cdots\!50\)\( p^{15} T^{25} + \)\(18\!\cdots\!53\)\( p^{18} T^{26} + \)\(17\!\cdots\!29\)\( p^{21} T^{27} + \)\(48\!\cdots\!55\)\( p^{24} T^{28} + \)\(29\!\cdots\!26\)\( p^{27} T^{29} + \)\(10\!\cdots\!87\)\( p^{30} T^{30} + \)\(32\!\cdots\!65\)\( p^{33} T^{31} + \)\(20\!\cdots\!74\)\( p^{36} T^{32} + 2498590670994077 p^{39} T^{33} + 3188677351871038 p^{42} T^{34} - 579081280549 p^{45} T^{35} + 36388732382 p^{48} T^{36} - 9147183 p^{51} T^{37} + 272954 p^{54} T^{38} - 51 p^{57} T^{39} + p^{60} T^{40} \) | |
| 31 | \( 1 + 90 T + 231560 T^{2} + 26497490 T^{3} + 28867729842 T^{4} + 3656745505614 T^{5} + 2539038575878738 T^{6} + 328360225285040044 T^{7} + \)\(17\!\cdots\!74\)\( T^{8} + \)\(21\!\cdots\!64\)\( T^{9} + \)\(98\!\cdots\!79\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{11} + \)\(47\!\cdots\!67\)\( T^{12} + \)\(53\!\cdots\!94\)\( T^{13} + \)\(19\!\cdots\!47\)\( T^{14} + \)\(20\!\cdots\!86\)\( T^{15} + \)\(23\!\cdots\!39\)\( p T^{16} + \)\(73\!\cdots\!64\)\( T^{17} + \)\(24\!\cdots\!20\)\( T^{18} + \)\(23\!\cdots\!02\)\( T^{19} + \)\(77\!\cdots\!82\)\( T^{20} + \)\(23\!\cdots\!02\)\( p^{3} T^{21} + \)\(24\!\cdots\!20\)\( p^{6} T^{22} + \)\(73\!\cdots\!64\)\( p^{9} T^{23} + \)\(23\!\cdots\!39\)\( p^{13} T^{24} + \)\(20\!\cdots\!86\)\( p^{15} T^{25} + \)\(19\!\cdots\!47\)\( p^{18} T^{26} + \)\(53\!\cdots\!94\)\( p^{21} T^{27} + \)\(47\!\cdots\!67\)\( p^{24} T^{28} + \)\(11\!\cdots\!00\)\( p^{27} T^{29} + \)\(98\!\cdots\!79\)\( p^{30} T^{30} + \)\(21\!\cdots\!64\)\( p^{33} T^{31} + \)\(17\!\cdots\!74\)\( p^{36} T^{32} + 328360225285040044 p^{39} T^{33} + 2539038575878738 p^{42} T^{34} + 3656745505614 p^{45} T^{35} + 28867729842 p^{48} T^{36} + 26497490 p^{51} T^{37} + 231560 p^{54} T^{38} + 90 p^{57} T^{39} + p^{60} T^{40} \) | |
| 37 | \( 1 + 96 T + 353975 T^{2} + 155332 p T^{3} + 63718911506 T^{4} - 1505290677868 T^{5} + 8490782217527005 T^{6} - 279281590397141912 T^{7} + \)\(90\!\cdots\!54\)\( T^{8} - \)\(28\!\cdots\!08\)\( T^{9} + \)\(80\!\cdots\!65\)\( T^{10} - \)\(19\!\cdots\!40\)\( T^{11} + \)\(61\!\cdots\!48\)\( T^{12} - \)\(77\!\cdots\!40\)\( T^{13} + \)\(41\!\cdots\!57\)\( T^{14} + \)\(33\!\cdots\!72\)\( T^{15} + \)\(25\!\cdots\!17\)\( T^{16} + \)\(27\!\cdots\!64\)\( T^{17} + \)\(14\!\cdots\!66\)\( T^{18} + \)\(23\!\cdots\!40\)\( T^{19} + \)\(19\!\cdots\!28\)\( p T^{20} + \)\(23\!\cdots\!40\)\( p^{3} T^{21} + \)\(14\!\cdots\!66\)\( p^{6} T^{22} + \)\(27\!\cdots\!64\)\( p^{9} T^{23} + \)\(25\!\cdots\!17\)\( p^{12} T^{24} + \)\(33\!\cdots\!72\)\( p^{15} T^{25} + \)\(41\!\cdots\!57\)\( p^{18} T^{26} - \)\(77\!\cdots\!40\)\( p^{21} T^{27} + \)\(61\!\cdots\!48\)\( p^{24} T^{28} - \)\(19\!\cdots\!40\)\( p^{27} T^{29} + \)\(80\!\cdots\!65\)\( p^{30} T^{30} - \)\(28\!\cdots\!08\)\( p^{33} T^{31} + \)\(90\!\cdots\!54\)\( p^{36} T^{32} - 279281590397141912 p^{39} T^{33} + 8490782217527005 p^{42} T^{34} - 1505290677868 p^{45} T^{35} + 63718911506 p^{48} T^{36} + 155332 p^{52} T^{37} + 353975 p^{54} T^{38} + 96 p^{57} T^{39} + p^{60} T^{40} \) | |
| 41 | \( 1 + 1284 T + 1700556 T^{2} + 1439105366 T^{3} + 1161574008797 T^{4} + 755961122483408 T^{5} + 465490872335580551 T^{6} + \)\(25\!\cdots\!08\)\( T^{7} + \)\(12\!\cdots\!69\)\( T^{8} + \)\(59\!\cdots\!10\)\( T^{9} + \)\(26\!\cdots\!85\)\( T^{10} + \)\(10\!\cdots\!70\)\( T^{11} + \)\(41\!\cdots\!73\)\( T^{12} + \)\(15\!\cdots\!90\)\( T^{13} + \)\(53\!\cdots\!71\)\( T^{14} + \)\(43\!\cdots\!46\)\( p T^{15} + \)\(56\!\cdots\!78\)\( T^{16} + \)\(17\!\cdots\!92\)\( T^{17} + \)\(50\!\cdots\!49\)\( T^{18} + \)\(13\!\cdots\!50\)\( T^{19} + \)\(37\!\cdots\!48\)\( T^{20} + \)\(13\!\cdots\!50\)\( p^{3} T^{21} + \)\(50\!\cdots\!49\)\( p^{6} T^{22} + \)\(17\!\cdots\!92\)\( p^{9} T^{23} + \)\(56\!\cdots\!78\)\( p^{12} T^{24} + \)\(43\!\cdots\!46\)\( p^{16} T^{25} + \)\(53\!\cdots\!71\)\( p^{18} T^{26} + \)\(15\!\cdots\!90\)\( p^{21} T^{27} + \)\(41\!\cdots\!73\)\( p^{24} T^{28} + \)\(10\!\cdots\!70\)\( p^{27} T^{29} + \)\(26\!\cdots\!85\)\( p^{30} T^{30} + \)\(59\!\cdots\!10\)\( p^{33} T^{31} + \)\(12\!\cdots\!69\)\( p^{36} T^{32} + \)\(25\!\cdots\!08\)\( p^{39} T^{33} + 465490872335580551 p^{42} T^{34} + 755961122483408 p^{45} T^{35} + 1161574008797 p^{48} T^{36} + 1439105366 p^{51} T^{37} + 1700556 p^{54} T^{38} + 1284 p^{57} T^{39} + p^{60} T^{40} \) | |
| 43 | \( 1 + 1592 T + 2102278 T^{2} + 1971485358 T^{3} + 1630527707458 T^{4} + 1145307742036204 T^{5} + 733571121541748085 T^{6} + \)\(42\!\cdots\!34\)\( T^{7} + \)\(22\!\cdots\!76\)\( T^{8} + \)\(11\!\cdots\!64\)\( T^{9} + \)\(52\!\cdots\!65\)\( T^{10} + \)\(23\!\cdots\!74\)\( T^{11} + \)\(95\!\cdots\!45\)\( T^{12} + \)\(37\!\cdots\!80\)\( T^{13} + \)\(14\!\cdots\!93\)\( T^{14} + \)\(50\!\cdots\!26\)\( T^{15} + \)\(17\!\cdots\!99\)\( T^{16} + \)\(56\!\cdots\!16\)\( T^{17} + \)\(17\!\cdots\!07\)\( T^{18} + \)\(53\!\cdots\!04\)\( T^{19} + \)\(15\!\cdots\!74\)\( T^{20} + \)\(53\!\cdots\!04\)\( p^{3} T^{21} + \)\(17\!\cdots\!07\)\( p^{6} T^{22} + \)\(56\!\cdots\!16\)\( p^{9} T^{23} + \)\(17\!\cdots\!99\)\( p^{12} T^{24} + \)\(50\!\cdots\!26\)\( p^{15} T^{25} + \)\(14\!\cdots\!93\)\( p^{18} T^{26} + \)\(37\!\cdots\!80\)\( p^{21} T^{27} + \)\(95\!\cdots\!45\)\( p^{24} T^{28} + \)\(23\!\cdots\!74\)\( p^{27} T^{29} + \)\(52\!\cdots\!65\)\( p^{30} T^{30} + \)\(11\!\cdots\!64\)\( p^{33} T^{31} + \)\(22\!\cdots\!76\)\( p^{36} T^{32} + \)\(42\!\cdots\!34\)\( p^{39} T^{33} + 733571121541748085 p^{42} T^{34} + 1145307742036204 p^{45} T^{35} + 1630527707458 p^{48} T^{36} + 1971485358 p^{51} T^{37} + 2102278 p^{54} T^{38} + 1592 p^{57} T^{39} + p^{60} T^{40} \) | |
| 47 | \( 1 + 2030 T + 2770219 T^{2} + 2753356366 T^{3} + 2257148722586 T^{4} + 1567047199239076 T^{5} + 959380512503947968 T^{6} + 11177238406341780658 p T^{7} + \)\(55\!\cdots\!34\)\( p T^{8} + \)\(12\!\cdots\!68\)\( T^{9} + \)\(52\!\cdots\!52\)\( T^{10} + \)\(21\!\cdots\!28\)\( T^{11} + \)\(81\!\cdots\!19\)\( T^{12} + \)\(63\!\cdots\!22\)\( p T^{13} + \)\(10\!\cdots\!74\)\( T^{14} + \)\(35\!\cdots\!66\)\( T^{15} + \)\(11\!\cdots\!29\)\( T^{16} + \)\(38\!\cdots\!94\)\( T^{17} + \)\(12\!\cdots\!71\)\( T^{18} + \)\(38\!\cdots\!52\)\( T^{19} + \)\(12\!\cdots\!26\)\( T^{20} + \)\(38\!\cdots\!52\)\( p^{3} T^{21} + \)\(12\!\cdots\!71\)\( p^{6} T^{22} + \)\(38\!\cdots\!94\)\( p^{9} T^{23} + \)\(11\!\cdots\!29\)\( p^{12} T^{24} + \)\(35\!\cdots\!66\)\( p^{15} T^{25} + \)\(10\!\cdots\!74\)\( p^{18} T^{26} + \)\(63\!\cdots\!22\)\( p^{22} T^{27} + \)\(81\!\cdots\!19\)\( p^{24} T^{28} + \)\(21\!\cdots\!28\)\( p^{27} T^{29} + \)\(52\!\cdots\!52\)\( p^{30} T^{30} + \)\(12\!\cdots\!68\)\( p^{33} T^{31} + \)\(55\!\cdots\!34\)\( p^{37} T^{32} + 11177238406341780658 p^{40} T^{33} + 959380512503947968 p^{42} T^{34} + 1567047199239076 p^{45} T^{35} + 2257148722586 p^{48} T^{36} + 2753356366 p^{51} T^{37} + 2770219 p^{54} T^{38} + 2030 p^{57} T^{39} + p^{60} T^{40} \) | |
| 53 | \( 1 + 943 T + 1621962 T^{2} + 1107883133 T^{3} + 1151403047368 T^{4} + 625546193894031 T^{5} + 504285245900658308 T^{6} + \)\(22\!\cdots\!15\)\( T^{7} + \)\(29\!\cdots\!82\)\( p T^{8} + \)\(58\!\cdots\!07\)\( T^{9} + \)\(37\!\cdots\!73\)\( T^{10} + \)\(11\!\cdots\!66\)\( T^{11} + \)\(72\!\cdots\!95\)\( T^{12} + \)\(17\!\cdots\!97\)\( T^{13} + \)\(12\!\cdots\!39\)\( T^{14} + \)\(22\!\cdots\!48\)\( T^{15} + \)\(18\!\cdots\!13\)\( T^{16} + \)\(23\!\cdots\!07\)\( T^{17} + \)\(26\!\cdots\!14\)\( T^{18} + \)\(26\!\cdots\!49\)\( T^{19} + \)\(39\!\cdots\!26\)\( T^{20} + \)\(26\!\cdots\!49\)\( p^{3} T^{21} + \)\(26\!\cdots\!14\)\( p^{6} T^{22} + \)\(23\!\cdots\!07\)\( p^{9} T^{23} + \)\(18\!\cdots\!13\)\( p^{12} T^{24} + \)\(22\!\cdots\!48\)\( p^{15} T^{25} + \)\(12\!\cdots\!39\)\( p^{18} T^{26} + \)\(17\!\cdots\!97\)\( p^{21} T^{27} + \)\(72\!\cdots\!95\)\( p^{24} T^{28} + \)\(11\!\cdots\!66\)\( p^{27} T^{29} + \)\(37\!\cdots\!73\)\( p^{30} T^{30} + \)\(58\!\cdots\!07\)\( p^{33} T^{31} + \)\(29\!\cdots\!82\)\( p^{37} T^{32} + \)\(22\!\cdots\!15\)\( p^{39} T^{33} + 504285245900658308 p^{42} T^{34} + 625546193894031 p^{45} T^{35} + 1151403047368 p^{48} T^{36} + 1107883133 p^{51} T^{37} + 1621962 p^{54} T^{38} + 943 p^{57} T^{39} + p^{60} T^{40} \) | |
| 59 | \( 1 + 515 T + 2365452 T^{2} + 1208751701 T^{3} + 2875682291388 T^{4} + 1430170195584121 T^{5} + 2363726826843982116 T^{6} + \)\(11\!\cdots\!47\)\( T^{7} + \)\(14\!\cdots\!96\)\( T^{8} + \)\(66\!\cdots\!71\)\( T^{9} + \)\(72\!\cdots\!11\)\( T^{10} + \)\(31\!\cdots\!72\)\( T^{11} + \)\(29\!\cdots\!75\)\( T^{12} + \)\(12\!\cdots\!91\)\( T^{13} + \)\(10\!\cdots\!55\)\( T^{14} + \)\(39\!\cdots\!84\)\( T^{15} + \)\(29\!\cdots\!87\)\( T^{16} + \)\(10\!\cdots\!69\)\( T^{17} + \)\(75\!\cdots\!98\)\( T^{18} + \)\(25\!\cdots\!87\)\( T^{19} + \)\(16\!\cdots\!42\)\( T^{20} + \)\(25\!\cdots\!87\)\( p^{3} T^{21} + \)\(75\!\cdots\!98\)\( p^{6} T^{22} + \)\(10\!\cdots\!69\)\( p^{9} T^{23} + \)\(29\!\cdots\!87\)\( p^{12} T^{24} + \)\(39\!\cdots\!84\)\( p^{15} T^{25} + \)\(10\!\cdots\!55\)\( p^{18} T^{26} + \)\(12\!\cdots\!91\)\( p^{21} T^{27} + \)\(29\!\cdots\!75\)\( p^{24} T^{28} + \)\(31\!\cdots\!72\)\( p^{27} T^{29} + \)\(72\!\cdots\!11\)\( p^{30} T^{30} + \)\(66\!\cdots\!71\)\( p^{33} T^{31} + \)\(14\!\cdots\!96\)\( p^{36} T^{32} + \)\(11\!\cdots\!47\)\( p^{39} T^{33} + 2363726826843982116 p^{42} T^{34} + 1430170195584121 p^{45} T^{35} + 2875682291388 p^{48} T^{36} + 1208751701 p^{51} T^{37} + 2365452 p^{54} T^{38} + 515 p^{57} T^{39} + p^{60} T^{40} \) | |
| 61 | \( 1 - 446 T + 2547714 T^{2} - 1136462978 T^{3} + 3262048350066 T^{4} - 1421580430389922 T^{5} + 2790878543751703308 T^{6} - \)\(11\!\cdots\!14\)\( T^{7} + \)\(17\!\cdots\!48\)\( T^{8} - \)\(70\!\cdots\!30\)\( T^{9} + \)\(91\!\cdots\!77\)\( T^{10} - \)\(33\!\cdots\!50\)\( T^{11} + \)\(38\!\cdots\!25\)\( T^{12} - \)\(13\!\cdots\!56\)\( T^{13} + \)\(13\!\cdots\!49\)\( T^{14} - \)\(45\!\cdots\!12\)\( T^{15} + \)\(42\!\cdots\!19\)\( T^{16} - \)\(13\!\cdots\!14\)\( T^{17} + \)\(11\!\cdots\!08\)\( T^{18} - \)\(33\!\cdots\!06\)\( T^{19} + \)\(27\!\cdots\!54\)\( T^{20} - \)\(33\!\cdots\!06\)\( p^{3} T^{21} + \)\(11\!\cdots\!08\)\( p^{6} T^{22} - \)\(13\!\cdots\!14\)\( p^{9} T^{23} + \)\(42\!\cdots\!19\)\( p^{12} T^{24} - \)\(45\!\cdots\!12\)\( p^{15} T^{25} + \)\(13\!\cdots\!49\)\( p^{18} T^{26} - \)\(13\!\cdots\!56\)\( p^{21} T^{27} + \)\(38\!\cdots\!25\)\( p^{24} T^{28} - \)\(33\!\cdots\!50\)\( p^{27} T^{29} + \)\(91\!\cdots\!77\)\( p^{30} T^{30} - \)\(70\!\cdots\!30\)\( p^{33} T^{31} + \)\(17\!\cdots\!48\)\( p^{36} T^{32} - \)\(11\!\cdots\!14\)\( p^{39} T^{33} + 2790878543751703308 p^{42} T^{34} - 1421580430389922 p^{45} T^{35} + 3262048350066 p^{48} T^{36} - 1136462978 p^{51} T^{37} + 2547714 p^{54} T^{38} - 446 p^{57} T^{39} + p^{60} T^{40} \) | |
| 67 | \( 1 + 1719 T + 3696096 T^{2} + 4178703531 T^{3} + 5378971002386 T^{4} + 4636990131264511 T^{5} + 4533905616666898844 T^{6} + \)\(31\!\cdots\!93\)\( T^{7} + \)\(25\!\cdots\!18\)\( T^{8} + \)\(14\!\cdots\!87\)\( T^{9} + \)\(10\!\cdots\!63\)\( T^{10} + \)\(53\!\cdots\!62\)\( T^{11} + \)\(37\!\cdots\!06\)\( T^{12} + \)\(16\!\cdots\!32\)\( T^{13} + \)\(12\!\cdots\!90\)\( T^{14} + \)\(56\!\cdots\!26\)\( T^{15} + \)\(46\!\cdots\!45\)\( T^{16} + \)\(21\!\cdots\!75\)\( T^{17} + \)\(17\!\cdots\!19\)\( T^{18} + \)\(79\!\cdots\!52\)\( T^{19} + \)\(57\!\cdots\!36\)\( T^{20} + \)\(79\!\cdots\!52\)\( p^{3} T^{21} + \)\(17\!\cdots\!19\)\( p^{6} T^{22} + \)\(21\!\cdots\!75\)\( p^{9} T^{23} + \)\(46\!\cdots\!45\)\( p^{12} T^{24} + \)\(56\!\cdots\!26\)\( p^{15} T^{25} + \)\(12\!\cdots\!90\)\( p^{18} T^{26} + \)\(16\!\cdots\!32\)\( p^{21} T^{27} + \)\(37\!\cdots\!06\)\( p^{24} T^{28} + \)\(53\!\cdots\!62\)\( p^{27} T^{29} + \)\(10\!\cdots\!63\)\( p^{30} T^{30} + \)\(14\!\cdots\!87\)\( p^{33} T^{31} + \)\(25\!\cdots\!18\)\( p^{36} T^{32} + \)\(31\!\cdots\!93\)\( p^{39} T^{33} + 4533905616666898844 p^{42} T^{34} + 4636990131264511 p^{45} T^{35} + 5378971002386 p^{48} T^{36} + 4178703531 p^{51} T^{37} + 3696096 p^{54} T^{38} + 1719 p^{57} T^{39} + p^{60} T^{40} \) | |
| 71 | \( 1 + 90 T + 3685116 T^{2} - 36432056 T^{3} + 6792419176472 T^{4} - 667300493646884 T^{5} + 8420994374828947535 T^{6} - \)\(14\!\cdots\!08\)\( T^{7} + \)\(79\!\cdots\!79\)\( T^{8} - \)\(18\!\cdots\!90\)\( T^{9} + \)\(60\!\cdots\!14\)\( T^{10} - \)\(17\!\cdots\!16\)\( T^{11} + \)\(38\!\cdots\!28\)\( T^{12} - \)\(12\!\cdots\!64\)\( T^{13} + \)\(21\!\cdots\!24\)\( T^{14} - \)\(73\!\cdots\!78\)\( T^{15} + \)\(10\!\cdots\!80\)\( T^{16} - \)\(36\!\cdots\!66\)\( T^{17} + \)\(44\!\cdots\!27\)\( T^{18} - \)\(15\!\cdots\!00\)\( T^{19} + \)\(16\!\cdots\!72\)\( T^{20} - \)\(15\!\cdots\!00\)\( p^{3} T^{21} + \)\(44\!\cdots\!27\)\( p^{6} T^{22} - \)\(36\!\cdots\!66\)\( p^{9} T^{23} + \)\(10\!\cdots\!80\)\( p^{12} T^{24} - \)\(73\!\cdots\!78\)\( p^{15} T^{25} + \)\(21\!\cdots\!24\)\( p^{18} T^{26} - \)\(12\!\cdots\!64\)\( p^{21} T^{27} + \)\(38\!\cdots\!28\)\( p^{24} T^{28} - \)\(17\!\cdots\!16\)\( p^{27} T^{29} + \)\(60\!\cdots\!14\)\( p^{30} T^{30} - \)\(18\!\cdots\!90\)\( p^{33} T^{31} + \)\(79\!\cdots\!79\)\( p^{36} T^{32} - \)\(14\!\cdots\!08\)\( p^{39} T^{33} + 8420994374828947535 p^{42} T^{34} - 667300493646884 p^{45} T^{35} + 6792419176472 p^{48} T^{36} - 36432056 p^{51} T^{37} + 3685116 p^{54} T^{38} + 90 p^{57} T^{39} + p^{60} T^{40} \) | |
| 73 | \( 1 + 3763 T + 10842910 T^{2} + 22199341355 T^{3} + 39099182997618 T^{4} + 57981925544828355 T^{5} + 77566663513408877530 T^{6} + \)\(92\!\cdots\!05\)\( T^{7} + \)\(10\!\cdots\!86\)\( T^{8} + \)\(10\!\cdots\!67\)\( T^{9} + \)\(10\!\cdots\!93\)\( T^{10} + \)\(90\!\cdots\!94\)\( T^{11} + \)\(77\!\cdots\!30\)\( T^{12} + \)\(63\!\cdots\!70\)\( T^{13} + \)\(49\!\cdots\!62\)\( T^{14} + \)\(37\!\cdots\!96\)\( T^{15} + \)\(27\!\cdots\!27\)\( T^{16} + \)\(19\!\cdots\!43\)\( T^{17} + \)\(12\!\cdots\!09\)\( T^{18} + \)\(84\!\cdots\!12\)\( T^{19} + \)\(53\!\cdots\!16\)\( T^{20} + \)\(84\!\cdots\!12\)\( p^{3} T^{21} + \)\(12\!\cdots\!09\)\( p^{6} T^{22} + \)\(19\!\cdots\!43\)\( p^{9} T^{23} + \)\(27\!\cdots\!27\)\( p^{12} T^{24} + \)\(37\!\cdots\!96\)\( p^{15} T^{25} + \)\(49\!\cdots\!62\)\( p^{18} T^{26} + \)\(63\!\cdots\!70\)\( p^{21} T^{27} + \)\(77\!\cdots\!30\)\( p^{24} T^{28} + \)\(90\!\cdots\!94\)\( p^{27} T^{29} + \)\(10\!\cdots\!93\)\( p^{30} T^{30} + \)\(10\!\cdots\!67\)\( p^{33} T^{31} + \)\(10\!\cdots\!86\)\( p^{36} T^{32} + \)\(92\!\cdots\!05\)\( p^{39} T^{33} + 77566663513408877530 p^{42} T^{34} + 57981925544828355 p^{45} T^{35} + 39099182997618 p^{48} T^{36} + 22199341355 p^{51} T^{37} + 10842910 p^{54} T^{38} + 3763 p^{57} T^{39} + p^{60} T^{40} \) | |
| 79 | \( 1 - 2 T + 5896969 T^{2} - 230407352 T^{3} + 17019807099688 T^{4} - 1368004496833264 T^{5} + 32093286419356342587 T^{6} - \)\(40\!\cdots\!18\)\( T^{7} + \)\(44\!\cdots\!02\)\( T^{8} - \)\(76\!\cdots\!50\)\( T^{9} + \)\(48\!\cdots\!63\)\( T^{10} - \)\(10\!\cdots\!64\)\( T^{11} + \)\(44\!\cdots\!22\)\( T^{12} - \)\(11\!\cdots\!88\)\( T^{13} + \)\(33\!\cdots\!91\)\( T^{14} - \)\(96\!\cdots\!78\)\( T^{15} + \)\(22\!\cdots\!85\)\( T^{16} - \)\(67\!\cdots\!28\)\( T^{17} + \)\(13\!\cdots\!78\)\( T^{18} - \)\(39\!\cdots\!28\)\( T^{19} + \)\(69\!\cdots\!28\)\( T^{20} - \)\(39\!\cdots\!28\)\( p^{3} T^{21} + \)\(13\!\cdots\!78\)\( p^{6} T^{22} - \)\(67\!\cdots\!28\)\( p^{9} T^{23} + \)\(22\!\cdots\!85\)\( p^{12} T^{24} - \)\(96\!\cdots\!78\)\( p^{15} T^{25} + \)\(33\!\cdots\!91\)\( p^{18} T^{26} - \)\(11\!\cdots\!88\)\( p^{21} T^{27} + \)\(44\!\cdots\!22\)\( p^{24} T^{28} - \)\(10\!\cdots\!64\)\( p^{27} T^{29} + \)\(48\!\cdots\!63\)\( p^{30} T^{30} - \)\(76\!\cdots\!50\)\( p^{33} T^{31} + \)\(44\!\cdots\!02\)\( p^{36} T^{32} - \)\(40\!\cdots\!18\)\( p^{39} T^{33} + 32093286419356342587 p^{42} T^{34} - 1368004496833264 p^{45} T^{35} + 17019807099688 p^{48} T^{36} - 230407352 p^{51} T^{37} + 5896969 p^{54} T^{38} - 2 p^{57} T^{39} + p^{60} T^{40} \) | |
| 83 | \( 1 + 3436 T + 10103288 T^{2} + 21351605198 T^{3} + 40460228658182 T^{4} + 65802564430773590 T^{5} + 98875867653925624622 T^{6} + \)\(13\!\cdots\!68\)\( T^{7} + \)\(17\!\cdots\!86\)\( T^{8} + \)\(20\!\cdots\!30\)\( T^{9} + \)\(23\!\cdots\!63\)\( T^{10} + \)\(25\!\cdots\!86\)\( T^{11} + \)\(26\!\cdots\!17\)\( T^{12} + \)\(26\!\cdots\!44\)\( T^{13} + \)\(24\!\cdots\!65\)\( T^{14} + \)\(22\!\cdots\!60\)\( T^{15} + \)\(19\!\cdots\!49\)\( T^{16} + \)\(16\!\cdots\!02\)\( T^{17} + \)\(13\!\cdots\!42\)\( T^{18} + \)\(11\!\cdots\!78\)\( T^{19} + \)\(85\!\cdots\!58\)\( T^{20} + \)\(11\!\cdots\!78\)\( p^{3} T^{21} + \)\(13\!\cdots\!42\)\( p^{6} T^{22} + \)\(16\!\cdots\!02\)\( p^{9} T^{23} + \)\(19\!\cdots\!49\)\( p^{12} T^{24} + \)\(22\!\cdots\!60\)\( p^{15} T^{25} + \)\(24\!\cdots\!65\)\( p^{18} T^{26} + \)\(26\!\cdots\!44\)\( p^{21} T^{27} + \)\(26\!\cdots\!17\)\( p^{24} T^{28} + \)\(25\!\cdots\!86\)\( p^{27} T^{29} + \)\(23\!\cdots\!63\)\( p^{30} T^{30} + \)\(20\!\cdots\!30\)\( p^{33} T^{31} + \)\(17\!\cdots\!86\)\( p^{36} T^{32} + \)\(13\!\cdots\!68\)\( p^{39} T^{33} + 98875867653925624622 p^{42} T^{34} + 65802564430773590 p^{45} T^{35} + 40460228658182 p^{48} T^{36} + 21351605198 p^{51} T^{37} + 10103288 p^{54} T^{38} + 3436 p^{57} T^{39} + p^{60} T^{40} \) | |
| 89 | \( 1 - 1700 T + 7408080 T^{2} - 10741350784 T^{3} + 26461667775515 T^{4} - 33688258836001328 T^{5} + 61415370864426697506 T^{6} - \)\(70\!\cdots\!00\)\( T^{7} + \)\(10\!\cdots\!13\)\( T^{8} - \)\(10\!\cdots\!12\)\( T^{9} + \)\(14\!\cdots\!45\)\( T^{10} - \)\(13\!\cdots\!56\)\( T^{11} + \)\(16\!\cdots\!28\)\( T^{12} - \)\(14\!\cdots\!08\)\( T^{13} + \)\(15\!\cdots\!61\)\( T^{14} - \)\(13\!\cdots\!80\)\( T^{15} + \)\(13\!\cdots\!98\)\( T^{16} - \)\(11\!\cdots\!08\)\( T^{17} + \)\(11\!\cdots\!88\)\( T^{18} - \)\(86\!\cdots\!44\)\( T^{19} + \)\(81\!\cdots\!98\)\( T^{20} - \)\(86\!\cdots\!44\)\( p^{3} T^{21} + \)\(11\!\cdots\!88\)\( p^{6} T^{22} - \)\(11\!\cdots\!08\)\( p^{9} T^{23} + \)\(13\!\cdots\!98\)\( p^{12} T^{24} - \)\(13\!\cdots\!80\)\( p^{15} T^{25} + \)\(15\!\cdots\!61\)\( p^{18} T^{26} - \)\(14\!\cdots\!08\)\( p^{21} T^{27} + \)\(16\!\cdots\!28\)\( p^{24} T^{28} - \)\(13\!\cdots\!56\)\( p^{27} T^{29} + \)\(14\!\cdots\!45\)\( p^{30} T^{30} - \)\(10\!\cdots\!12\)\( p^{33} T^{31} + \)\(10\!\cdots\!13\)\( p^{36} T^{32} - \)\(70\!\cdots\!00\)\( p^{39} T^{33} + 61415370864426697506 p^{42} T^{34} - 33688258836001328 p^{45} T^{35} + 26461667775515 p^{48} T^{36} - 10741350784 p^{51} T^{37} + 7408080 p^{54} T^{38} - 1700 p^{57} T^{39} + p^{60} T^{40} \) | |
| 97 | \( 1 + 1956 T + 11422560 T^{2} + 18020633220 T^{3} + 60603375116175 T^{4} + 81930257486509620 T^{5} + \)\(20\!\cdots\!70\)\( T^{6} + \)\(24\!\cdots\!24\)\( T^{7} + \)\(52\!\cdots\!25\)\( T^{8} + \)\(57\!\cdots\!24\)\( T^{9} + \)\(10\!\cdots\!73\)\( T^{10} + \)\(10\!\cdots\!88\)\( T^{11} + \)\(17\!\cdots\!56\)\( T^{12} + \)\(17\!\cdots\!24\)\( T^{13} + \)\(25\!\cdots\!97\)\( T^{14} + \)\(23\!\cdots\!72\)\( T^{15} + \)\(32\!\cdots\!94\)\( T^{16} + \)\(27\!\cdots\!92\)\( T^{17} + \)\(35\!\cdots\!16\)\( T^{18} + \)\(28\!\cdots\!68\)\( T^{19} + \)\(34\!\cdots\!86\)\( T^{20} + \)\(28\!\cdots\!68\)\( p^{3} T^{21} + \)\(35\!\cdots\!16\)\( p^{6} T^{22} + \)\(27\!\cdots\!92\)\( p^{9} T^{23} + \)\(32\!\cdots\!94\)\( p^{12} T^{24} + \)\(23\!\cdots\!72\)\( p^{15} T^{25} + \)\(25\!\cdots\!97\)\( p^{18} T^{26} + \)\(17\!\cdots\!24\)\( p^{21} T^{27} + \)\(17\!\cdots\!56\)\( p^{24} T^{28} + \)\(10\!\cdots\!88\)\( p^{27} T^{29} + \)\(10\!\cdots\!73\)\( p^{30} T^{30} + \)\(57\!\cdots\!24\)\( p^{33} T^{31} + \)\(52\!\cdots\!25\)\( p^{36} T^{32} + \)\(24\!\cdots\!24\)\( p^{39} T^{33} + \)\(20\!\cdots\!70\)\( p^{42} T^{34} + 81930257486509620 p^{45} T^{35} + 60603375116175 p^{48} T^{36} + 18020633220 p^{51} T^{37} + 11422560 p^{54} T^{38} + 1956 p^{57} T^{39} + p^{60} T^{40} \) | |
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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
−2.26285939762291746931162964867, −2.25980411144149737964090791038, −2.24777128976828683707147338441, −2.08569901027069852791038690417, −2.05389592888045422565814685111, −1.95471585747135326387990086831, −1.95118628005982309430974229897, −1.84174494716756956305066249145, −1.76282225887518569748456943635, −1.61636906542104148984014410611, −1.57249998701406809563623814150, −1.56630800513363904968317720447, −1.48669648110549190136971835988, −1.35660072197402426252494194916, −1.35148713813861051964456066544, −1.33823175500369525244390804452, −1.31399443633970114171411257933, −1.18534316791990826882656179642, −1.18386540656461283252527306317, −1.15911253572117360550740419605, −1.03551601378853316931698247308, −1.02723008815609245747743557830, −0.939224681892638328765909120851, −0.860840580904576960409155344707, −0.78393263726872298081125107267, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.78393263726872298081125107267, 0.860840580904576960409155344707, 0.939224681892638328765909120851, 1.02723008815609245747743557830, 1.03551601378853316931698247308, 1.15911253572117360550740419605, 1.18386540656461283252527306317, 1.18534316791990826882656179642, 1.31399443633970114171411257933, 1.33823175500369525244390804452, 1.35148713813861051964456066544, 1.35660072197402426252494194916, 1.48669648110549190136971835988, 1.56630800513363904968317720447, 1.57249998701406809563623814150, 1.61636906542104148984014410611, 1.76282225887518569748456943635, 1.84174494716756956305066249145, 1.95118628005982309430974229897, 1.95471585747135326387990086831, 2.05389592888045422565814685111, 2.08569901027069852791038690417, 2.24777128976828683707147338441, 2.25980411144149737964090791038, 2.26285939762291746931162964867
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.