Dirichlet series
| L(s) = 1 | + 9·2-s + 22·3-s + 13·4-s + 69·5-s + 198·6-s + 55·7-s − 130·8-s + 108·9-s + 621·10-s + 130·11-s + 286·12-s + 131·13-s + 495·14-s + 1.51e3·15-s − 570·16-s + 510·17-s + 972·18-s + 142·19-s + 897·20-s + 1.21e3·21-s + 1.17e3·22-s + 509·23-s − 2.86e3·24-s + 1.71e3·25-s + 1.17e3·26-s − 1.17e3·27-s + 715·28-s + ⋯ |
| L(s) = 1 | + 3.18·2-s + 4.23·3-s + 13/8·4-s + 6.17·5-s + 13.4·6-s + 2.96·7-s − 5.74·8-s + 4·9-s + 19.6·10-s + 3.56·11-s + 6.88·12-s + 2.79·13-s + 9.44·14-s + 26.1·15-s − 8.90·16-s + 7.27·17-s + 12.7·18-s + 1.71·19-s + 10.0·20-s + 12.5·21-s + 11.3·22-s + 4.61·23-s − 24.3·24-s + 13.6·25-s + 8.89·26-s − 8.35·27-s + 4.82·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(127^{17}\right)^{s/2} \, \Gamma_{\C}(s)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(127^{17}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{17} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(34\) |
| Conductor: | \(127^{17}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.40263\times 10^{14}\) |
| Root analytic conductor: | \(2.73737\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((34,\ 127^{17} ,\ ( \ : [3/2]^{17} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(24106.99295\) |
| \(L(\frac12)\) | \(\approx\) | \(24106.99295\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 127 | \( ( 1 + p T )^{17} \) |
| good | 2 | \( 1 - 9 T + 17 p^{2} T^{2} - 365 T^{3} + 1801 T^{4} - 961 p^{3} T^{5} + 31611 T^{6} - 119707 T^{7} + 439321 T^{8} - 1518137 T^{9} + 5082275 T^{10} - 8145925 p T^{11} + 50899715 T^{12} - 38717913 p^{2} T^{13} + 57689049 p^{3} T^{14} - 42297929 p^{5} T^{15} + 243604995 p^{4} T^{16} - 347271833 p^{5} T^{17} + 243604995 p^{7} T^{18} - 42297929 p^{11} T^{19} + 57689049 p^{12} T^{20} - 38717913 p^{14} T^{21} + 50899715 p^{15} T^{22} - 8145925 p^{19} T^{23} + 5082275 p^{21} T^{24} - 1518137 p^{24} T^{25} + 439321 p^{27} T^{26} - 119707 p^{30} T^{27} + 31611 p^{33} T^{28} - 961 p^{39} T^{29} + 1801 p^{39} T^{30} - 365 p^{42} T^{31} + 17 p^{47} T^{32} - 9 p^{48} T^{33} + p^{51} T^{34} \) |
| 3 | \( 1 - 22 T + 376 T^{2} - 4724 T^{3} + 51233 T^{4} - 479348 T^{5} + 1355720 p T^{6} - 3498110 p^{2} T^{7} + 25305742 p^{2} T^{8} - 1551702844 T^{9} + 10105519174 T^{10} - 63294261982 T^{11} + 384682808972 T^{12} - 2271569871740 T^{13} + 13077957484721 T^{14} - 8130170616304 p^{2} T^{15} + 132754458976469 p T^{16} - 700039462822612 p T^{17} + 132754458976469 p^{4} T^{18} - 8130170616304 p^{8} T^{19} + 13077957484721 p^{9} T^{20} - 2271569871740 p^{12} T^{21} + 384682808972 p^{15} T^{22} - 63294261982 p^{18} T^{23} + 10105519174 p^{21} T^{24} - 1551702844 p^{24} T^{25} + 25305742 p^{29} T^{26} - 3498110 p^{32} T^{27} + 1355720 p^{34} T^{28} - 479348 p^{36} T^{29} + 51233 p^{39} T^{30} - 4724 p^{42} T^{31} + 376 p^{45} T^{32} - 22 p^{48} T^{33} + p^{51} T^{34} \) | |
| 5 | \( 1 - 69 T + 3051 T^{2} - 20234 p T^{3} + 2797833 T^{4} - 13403293 p T^{5} + 1436905966 T^{6} - 28032799927 T^{7} + 505419450389 T^{8} - 8497663317148 T^{9} + 26881514059323 p T^{10} - 2010593694939529 T^{11} + 28591976299288906 T^{12} - 387656403219095411 T^{13} + 5025591939003498979 T^{14} - 62378238846367632358 T^{15} + \)\(14\!\cdots\!24\)\( p T^{16} - \)\(84\!\cdots\!94\)\( T^{17} + \)\(14\!\cdots\!24\)\( p^{4} T^{18} - 62378238846367632358 p^{6} T^{19} + 5025591939003498979 p^{9} T^{20} - 387656403219095411 p^{12} T^{21} + 28591976299288906 p^{15} T^{22} - 2010593694939529 p^{18} T^{23} + 26881514059323 p^{22} T^{24} - 8497663317148 p^{24} T^{25} + 505419450389 p^{27} T^{26} - 28032799927 p^{30} T^{27} + 1436905966 p^{33} T^{28} - 13403293 p^{37} T^{29} + 2797833 p^{39} T^{30} - 20234 p^{43} T^{31} + 3051 p^{45} T^{32} - 69 p^{48} T^{33} + p^{51} T^{34} \) | |
| 7 | \( 1 - 55 T + 4224 T^{2} - 24389 p T^{3} + 7892393 T^{4} - 261456256 T^{5} + 9249733356 T^{6} - 265517596228 T^{7} + 7862721531936 T^{8} - 202065870846934 T^{9} + 5240540658912378 T^{10} - 17567115215648078 p T^{11} + 2866978576110162382 T^{12} - 8877145910951909960 p T^{13} + \)\(13\!\cdots\!19\)\( T^{14} - \)\(26\!\cdots\!59\)\( T^{15} + \)\(52\!\cdots\!09\)\( T^{16} - \)\(98\!\cdots\!54\)\( T^{17} + \)\(52\!\cdots\!09\)\( p^{3} T^{18} - \)\(26\!\cdots\!59\)\( p^{6} T^{19} + \)\(13\!\cdots\!19\)\( p^{9} T^{20} - 8877145910951909960 p^{13} T^{21} + 2866978576110162382 p^{15} T^{22} - 17567115215648078 p^{19} T^{23} + 5240540658912378 p^{21} T^{24} - 202065870846934 p^{24} T^{25} + 7862721531936 p^{27} T^{26} - 265517596228 p^{30} T^{27} + 9249733356 p^{33} T^{28} - 261456256 p^{36} T^{29} + 7892393 p^{39} T^{30} - 24389 p^{43} T^{31} + 4224 p^{45} T^{32} - 55 p^{48} T^{33} + p^{51} T^{34} \) | |
| 11 | \( 1 - 130 T + 18222 T^{2} - 1408287 T^{3} + 115037367 T^{4} - 6360244185 T^{5} + 390365088131 T^{6} - 16889952424393 T^{7} + 81231590632721 p T^{8} - 32518176469000086 T^{9} + 1657863288784268014 T^{10} - 4906567922840877745 p T^{11} + \)\(27\!\cdots\!12\)\( T^{12} - \)\(83\!\cdots\!28\)\( T^{13} + \)\(43\!\cdots\!36\)\( T^{14} - \)\(10\!\cdots\!47\)\( p T^{15} + \)\(60\!\cdots\!52\)\( T^{16} - \)\(15\!\cdots\!78\)\( T^{17} + \)\(60\!\cdots\!52\)\( p^{3} T^{18} - \)\(10\!\cdots\!47\)\( p^{7} T^{19} + \)\(43\!\cdots\!36\)\( p^{9} T^{20} - \)\(83\!\cdots\!28\)\( p^{12} T^{21} + \)\(27\!\cdots\!12\)\( p^{15} T^{22} - 4906567922840877745 p^{19} T^{23} + 1657863288784268014 p^{21} T^{24} - 32518176469000086 p^{24} T^{25} + 81231590632721 p^{28} T^{26} - 16889952424393 p^{30} T^{27} + 390365088131 p^{33} T^{28} - 6360244185 p^{36} T^{29} + 115037367 p^{39} T^{30} - 1408287 p^{42} T^{31} + 18222 p^{45} T^{32} - 130 p^{48} T^{33} + p^{51} T^{34} \) | |
| 13 | \( 1 - 131 T + 29496 T^{2} - 2952556 T^{3} + 393078099 T^{4} - 32499161040 T^{5} + 3273903304629 T^{6} - 233400352332974 T^{7} + 19505770528052542 T^{8} - 1231508041501303261 T^{9} + 89461289374882532086 T^{10} - \)\(50\!\cdots\!59\)\( T^{11} + \)\(33\!\cdots\!84\)\( T^{12} - \)\(10\!\cdots\!30\)\( p^{2} T^{13} + \)\(10\!\cdots\!54\)\( T^{14} - \)\(48\!\cdots\!43\)\( T^{15} + \)\(26\!\cdots\!33\)\( T^{16} - \)\(88\!\cdots\!20\)\( p T^{17} + \)\(26\!\cdots\!33\)\( p^{3} T^{18} - \)\(48\!\cdots\!43\)\( p^{6} T^{19} + \)\(10\!\cdots\!54\)\( p^{9} T^{20} - \)\(10\!\cdots\!30\)\( p^{14} T^{21} + \)\(33\!\cdots\!84\)\( p^{15} T^{22} - \)\(50\!\cdots\!59\)\( p^{18} T^{23} + 89461289374882532086 p^{21} T^{24} - 1231508041501303261 p^{24} T^{25} + 19505770528052542 p^{27} T^{26} - 233400352332974 p^{30} T^{27} + 3273903304629 p^{33} T^{28} - 32499161040 p^{36} T^{29} + 393078099 p^{39} T^{30} - 2952556 p^{42} T^{31} + 29496 p^{45} T^{32} - 131 p^{48} T^{33} + p^{51} T^{34} \) | |
| 17 | \( 1 - 30 p T + 169952 T^{2} - 42057213 T^{3} + 8601631989 T^{4} - 1507267585893 T^{5} + 13750005602125 p T^{6} - 32618896486211615 T^{7} + 4156772284397472269 T^{8} - \)\(48\!\cdots\!48\)\( T^{9} + \)\(53\!\cdots\!04\)\( T^{10} - \)\(54\!\cdots\!17\)\( T^{11} + \)\(51\!\cdots\!60\)\( T^{12} - \)\(46\!\cdots\!52\)\( T^{13} + \)\(39\!\cdots\!46\)\( T^{14} - \)\(31\!\cdots\!75\)\( T^{15} + \)\(14\!\cdots\!54\)\( p T^{16} - \)\(17\!\cdots\!26\)\( T^{17} + \)\(14\!\cdots\!54\)\( p^{4} T^{18} - \)\(31\!\cdots\!75\)\( p^{6} T^{19} + \)\(39\!\cdots\!46\)\( p^{9} T^{20} - \)\(46\!\cdots\!52\)\( p^{12} T^{21} + \)\(51\!\cdots\!60\)\( p^{15} T^{22} - \)\(54\!\cdots\!17\)\( p^{18} T^{23} + \)\(53\!\cdots\!04\)\( p^{21} T^{24} - \)\(48\!\cdots\!48\)\( p^{24} T^{25} + 4156772284397472269 p^{27} T^{26} - 32618896486211615 p^{30} T^{27} + 13750005602125 p^{34} T^{28} - 1507267585893 p^{36} T^{29} + 8601631989 p^{39} T^{30} - 42057213 p^{42} T^{31} + 169952 p^{45} T^{32} - 30 p^{49} T^{33} + p^{51} T^{34} \) | |
| 19 | \( 1 - 142 T + 68329 T^{2} - 7571737 T^{3} + 2125643490 T^{4} - 547001422 p^{2} T^{5} + 42483280089414 T^{6} - 3491166582714047 T^{7} + 635404295347969129 T^{8} - 47819397644700924268 T^{9} + \)\(76\!\cdots\!16\)\( T^{10} - \)\(28\!\cdots\!58\)\( p T^{11} + \)\(78\!\cdots\!91\)\( T^{12} - \)\(51\!\cdots\!27\)\( T^{13} + \)\(69\!\cdots\!91\)\( T^{14} - \)\(43\!\cdots\!66\)\( T^{15} + \)\(53\!\cdots\!91\)\( T^{16} - \)\(31\!\cdots\!06\)\( T^{17} + \)\(53\!\cdots\!91\)\( p^{3} T^{18} - \)\(43\!\cdots\!66\)\( p^{6} T^{19} + \)\(69\!\cdots\!91\)\( p^{9} T^{20} - \)\(51\!\cdots\!27\)\( p^{12} T^{21} + \)\(78\!\cdots\!91\)\( p^{15} T^{22} - \)\(28\!\cdots\!58\)\( p^{19} T^{23} + \)\(76\!\cdots\!16\)\( p^{21} T^{24} - 47819397644700924268 p^{24} T^{25} + 635404295347969129 p^{27} T^{26} - 3491166582714047 p^{30} T^{27} + 42483280089414 p^{33} T^{28} - 547001422 p^{38} T^{29} + 2125643490 p^{39} T^{30} - 7571737 p^{42} T^{31} + 68329 p^{45} T^{32} - 142 p^{48} T^{33} + p^{51} T^{34} \) | |
| 23 | \( 1 - 509 T + 212016 T^{2} - 62658069 T^{3} + 16380510211 T^{4} - 3649434013454 T^{5} + 746262419184806 T^{6} - 138080503606139926 T^{7} + 23907687208256032884 T^{8} - \)\(38\!\cdots\!42\)\( T^{9} + \)\(58\!\cdots\!10\)\( T^{10} - \)\(84\!\cdots\!60\)\( T^{11} + \)\(11\!\cdots\!28\)\( T^{12} - \)\(15\!\cdots\!06\)\( T^{13} + \)\(19\!\cdots\!05\)\( T^{14} - \)\(23\!\cdots\!97\)\( T^{15} + \)\(11\!\cdots\!91\)\( p T^{16} - \)\(30\!\cdots\!50\)\( T^{17} + \)\(11\!\cdots\!91\)\( p^{4} T^{18} - \)\(23\!\cdots\!97\)\( p^{6} T^{19} + \)\(19\!\cdots\!05\)\( p^{9} T^{20} - \)\(15\!\cdots\!06\)\( p^{12} T^{21} + \)\(11\!\cdots\!28\)\( p^{15} T^{22} - \)\(84\!\cdots\!60\)\( p^{18} T^{23} + \)\(58\!\cdots\!10\)\( p^{21} T^{24} - \)\(38\!\cdots\!42\)\( p^{24} T^{25} + 23907687208256032884 p^{27} T^{26} - 138080503606139926 p^{30} T^{27} + 746262419184806 p^{33} T^{28} - 3649434013454 p^{36} T^{29} + 16380510211 p^{39} T^{30} - 62658069 p^{42} T^{31} + 212016 p^{45} T^{32} - 509 p^{48} T^{33} + p^{51} T^{34} \) | |
| 29 | \( 1 - 416 T + 267118 T^{2} - 85202410 T^{3} + 33774495675 T^{4} - 9156847060760 T^{5} + 2822897106789796 T^{6} - 676434784361533210 T^{7} + \)\(17\!\cdots\!10\)\( T^{8} - \)\(38\!\cdots\!44\)\( T^{9} + \)\(86\!\cdots\!34\)\( T^{10} - \)\(17\!\cdots\!98\)\( T^{11} + \)\(35\!\cdots\!00\)\( T^{12} - \)\(63\!\cdots\!76\)\( T^{13} + \)\(11\!\cdots\!09\)\( T^{14} - \)\(19\!\cdots\!10\)\( T^{15} + \)\(34\!\cdots\!85\)\( T^{16} - \)\(52\!\cdots\!84\)\( T^{17} + \)\(34\!\cdots\!85\)\( p^{3} T^{18} - \)\(19\!\cdots\!10\)\( p^{6} T^{19} + \)\(11\!\cdots\!09\)\( p^{9} T^{20} - \)\(63\!\cdots\!76\)\( p^{12} T^{21} + \)\(35\!\cdots\!00\)\( p^{15} T^{22} - \)\(17\!\cdots\!98\)\( p^{18} T^{23} + \)\(86\!\cdots\!34\)\( p^{21} T^{24} - \)\(38\!\cdots\!44\)\( p^{24} T^{25} + \)\(17\!\cdots\!10\)\( p^{27} T^{26} - 676434784361533210 p^{30} T^{27} + 2822897106789796 p^{33} T^{28} - 9156847060760 p^{36} T^{29} + 33774495675 p^{39} T^{30} - 85202410 p^{42} T^{31} + 267118 p^{45} T^{32} - 416 p^{48} T^{33} + p^{51} T^{34} \) | |
| 31 | \( 1 - 55 T + 206019 T^{2} - 10504913 T^{3} + 21947518521 T^{4} - 1028525029014 T^{5} + 1636036593315269 T^{6} - 71159784155909727 T^{7} + 96013320958078115934 T^{8} - \)\(39\!\cdots\!18\)\( T^{9} + \)\(47\!\cdots\!31\)\( T^{10} - \)\(17\!\cdots\!12\)\( T^{11} + \)\(19\!\cdots\!57\)\( T^{12} - \)\(70\!\cdots\!44\)\( T^{13} + \)\(74\!\cdots\!99\)\( T^{14} - \)\(24\!\cdots\!32\)\( T^{15} + \)\(24\!\cdots\!29\)\( T^{16} - \)\(77\!\cdots\!34\)\( T^{17} + \)\(24\!\cdots\!29\)\( p^{3} T^{18} - \)\(24\!\cdots\!32\)\( p^{6} T^{19} + \)\(74\!\cdots\!99\)\( p^{9} T^{20} - \)\(70\!\cdots\!44\)\( p^{12} T^{21} + \)\(19\!\cdots\!57\)\( p^{15} T^{22} - \)\(17\!\cdots\!12\)\( p^{18} T^{23} + \)\(47\!\cdots\!31\)\( p^{21} T^{24} - \)\(39\!\cdots\!18\)\( p^{24} T^{25} + 96013320958078115934 p^{27} T^{26} - 71159784155909727 p^{30} T^{27} + 1636036593315269 p^{33} T^{28} - 1028525029014 p^{36} T^{29} + 21947518521 p^{39} T^{30} - 10504913 p^{42} T^{31} + 206019 p^{45} T^{32} - 55 p^{48} T^{33} + p^{51} T^{34} \) | |
| 37 | \( 1 - 345 T + 557759 T^{2} - 182115247 T^{3} + 153788682664 T^{4} - 47233971696177 T^{5} + 753580741886313 p T^{6} - 8026325517525410111 T^{7} + \)\(37\!\cdots\!05\)\( T^{8} - \)\(10\!\cdots\!34\)\( T^{9} + \)\(39\!\cdots\!33\)\( T^{10} - \)\(98\!\cdots\!45\)\( T^{11} + \)\(33\!\cdots\!41\)\( T^{12} - \)\(78\!\cdots\!47\)\( T^{13} + \)\(23\!\cdots\!02\)\( T^{14} - \)\(51\!\cdots\!85\)\( T^{15} + \)\(38\!\cdots\!30\)\( p T^{16} - \)\(28\!\cdots\!46\)\( T^{17} + \)\(38\!\cdots\!30\)\( p^{4} T^{18} - \)\(51\!\cdots\!85\)\( p^{6} T^{19} + \)\(23\!\cdots\!02\)\( p^{9} T^{20} - \)\(78\!\cdots\!47\)\( p^{12} T^{21} + \)\(33\!\cdots\!41\)\( p^{15} T^{22} - \)\(98\!\cdots\!45\)\( p^{18} T^{23} + \)\(39\!\cdots\!33\)\( p^{21} T^{24} - \)\(10\!\cdots\!34\)\( p^{24} T^{25} + \)\(37\!\cdots\!05\)\( p^{27} T^{26} - 8026325517525410111 p^{30} T^{27} + 753580741886313 p^{34} T^{28} - 47233971696177 p^{36} T^{29} + 153788682664 p^{39} T^{30} - 182115247 p^{42} T^{31} + 557759 p^{45} T^{32} - 345 p^{48} T^{33} + p^{51} T^{34} \) | |
| 41 | \( 1 - 1010 T + 1045710 T^{2} - 726904589 T^{3} + 476793378244 T^{4} - 262131978804739 T^{5} + 135162888960748114 T^{6} - 62619771592281574378 T^{7} + \)\(27\!\cdots\!69\)\( T^{8} - \)\(11\!\cdots\!48\)\( T^{9} + \)\(42\!\cdots\!51\)\( T^{10} - \)\(15\!\cdots\!88\)\( T^{11} + \)\(52\!\cdots\!66\)\( T^{12} - \)\(16\!\cdots\!33\)\( T^{13} + \)\(52\!\cdots\!68\)\( T^{14} - \)\(15\!\cdots\!41\)\( T^{15} + \)\(43\!\cdots\!57\)\( T^{16} - \)\(11\!\cdots\!00\)\( T^{17} + \)\(43\!\cdots\!57\)\( p^{3} T^{18} - \)\(15\!\cdots\!41\)\( p^{6} T^{19} + \)\(52\!\cdots\!68\)\( p^{9} T^{20} - \)\(16\!\cdots\!33\)\( p^{12} T^{21} + \)\(52\!\cdots\!66\)\( p^{15} T^{22} - \)\(15\!\cdots\!88\)\( p^{18} T^{23} + \)\(42\!\cdots\!51\)\( p^{21} T^{24} - \)\(11\!\cdots\!48\)\( p^{24} T^{25} + \)\(27\!\cdots\!69\)\( p^{27} T^{26} - 62619771592281574378 p^{30} T^{27} + 135162888960748114 p^{33} T^{28} - 262131978804739 p^{36} T^{29} + 476793378244 p^{39} T^{30} - 726904589 p^{42} T^{31} + 1045710 p^{45} T^{32} - 1010 p^{48} T^{33} + p^{51} T^{34} \) | |
| 43 | \( 1 + 964 T + 1087404 T^{2} + 693289666 T^{3} + 462708135188 T^{4} + 223110621416952 T^{5} + 110233825537159935 T^{6} + 42368630770923834264 T^{7} + \)\(16\!\cdots\!20\)\( T^{8} + \)\(51\!\cdots\!88\)\( T^{9} + \)\(16\!\cdots\!36\)\( T^{10} + \)\(37\!\cdots\!92\)\( T^{11} + \)\(90\!\cdots\!43\)\( T^{12} + \)\(86\!\cdots\!32\)\( T^{13} - \)\(32\!\cdots\!36\)\( T^{14} - \)\(15\!\cdots\!90\)\( T^{15} - \)\(54\!\cdots\!05\)\( T^{16} - \)\(21\!\cdots\!28\)\( T^{17} - \)\(54\!\cdots\!05\)\( p^{3} T^{18} - \)\(15\!\cdots\!90\)\( p^{6} T^{19} - \)\(32\!\cdots\!36\)\( p^{9} T^{20} + \)\(86\!\cdots\!32\)\( p^{12} T^{21} + \)\(90\!\cdots\!43\)\( p^{15} T^{22} + \)\(37\!\cdots\!92\)\( p^{18} T^{23} + \)\(16\!\cdots\!36\)\( p^{21} T^{24} + \)\(51\!\cdots\!88\)\( p^{24} T^{25} + \)\(16\!\cdots\!20\)\( p^{27} T^{26} + 42368630770923834264 p^{30} T^{27} + 110233825537159935 p^{33} T^{28} + 223110621416952 p^{36} T^{29} + 462708135188 p^{39} T^{30} + 693289666 p^{42} T^{31} + 1087404 p^{45} T^{32} + 964 p^{48} T^{33} + p^{51} T^{34} \) | |
| 47 | \( 1 - 719 T + 1323228 T^{2} - 869458254 T^{3} + 866550859630 T^{4} - 515965610106861 T^{5} + 370514207539714960 T^{6} - \)\(19\!\cdots\!69\)\( T^{7} + \)\(11\!\cdots\!22\)\( T^{8} - \)\(55\!\cdots\!02\)\( T^{9} + \)\(27\!\cdots\!79\)\( T^{10} - \)\(12\!\cdots\!48\)\( T^{11} + \)\(52\!\cdots\!34\)\( T^{12} - \)\(20\!\cdots\!87\)\( T^{13} + \)\(80\!\cdots\!50\)\( T^{14} - \)\(28\!\cdots\!81\)\( T^{15} + \)\(10\!\cdots\!34\)\( T^{16} - \)\(33\!\cdots\!90\)\( T^{17} + \)\(10\!\cdots\!34\)\( p^{3} T^{18} - \)\(28\!\cdots\!81\)\( p^{6} T^{19} + \)\(80\!\cdots\!50\)\( p^{9} T^{20} - \)\(20\!\cdots\!87\)\( p^{12} T^{21} + \)\(52\!\cdots\!34\)\( p^{15} T^{22} - \)\(12\!\cdots\!48\)\( p^{18} T^{23} + \)\(27\!\cdots\!79\)\( p^{21} T^{24} - \)\(55\!\cdots\!02\)\( p^{24} T^{25} + \)\(11\!\cdots\!22\)\( p^{27} T^{26} - \)\(19\!\cdots\!69\)\( p^{30} T^{27} + 370514207539714960 p^{33} T^{28} - 515965610106861 p^{36} T^{29} + 866550859630 p^{39} T^{30} - 869458254 p^{42} T^{31} + 1323228 p^{45} T^{32} - 719 p^{48} T^{33} + p^{51} T^{34} \) | |
| 53 | \( 1 - 729 T + 1774036 T^{2} - 1041544739 T^{3} + 1454120145511 T^{4} - 716926651368382 T^{5} + 754375528356801150 T^{6} - \)\(32\!\cdots\!38\)\( T^{7} + \)\(28\!\cdots\!58\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{9} + \)\(84\!\cdots\!50\)\( T^{10} - \)\(28\!\cdots\!90\)\( T^{11} + \)\(20\!\cdots\!82\)\( T^{12} - \)\(64\!\cdots\!50\)\( T^{13} + \)\(41\!\cdots\!05\)\( T^{14} - \)\(12\!\cdots\!57\)\( T^{15} + \)\(72\!\cdots\!59\)\( T^{16} - \)\(19\!\cdots\!58\)\( T^{17} + \)\(72\!\cdots\!59\)\( p^{3} T^{18} - \)\(12\!\cdots\!57\)\( p^{6} T^{19} + \)\(41\!\cdots\!05\)\( p^{9} T^{20} - \)\(64\!\cdots\!50\)\( p^{12} T^{21} + \)\(20\!\cdots\!82\)\( p^{15} T^{22} - \)\(28\!\cdots\!90\)\( p^{18} T^{23} + \)\(84\!\cdots\!50\)\( p^{21} T^{24} - \)\(10\!\cdots\!40\)\( p^{24} T^{25} + \)\(28\!\cdots\!58\)\( p^{27} T^{26} - \)\(32\!\cdots\!38\)\( p^{30} T^{27} + 754375528356801150 p^{33} T^{28} - 716926651368382 p^{36} T^{29} + 1454120145511 p^{39} T^{30} - 1041544739 p^{42} T^{31} + 1774036 p^{45} T^{32} - 729 p^{48} T^{33} + p^{51} T^{34} \) | |
| 59 | \( 1 - 1358 T + 2555641 T^{2} - 2569102984 T^{3} + 2916165433393 T^{4} - 2380615740546836 T^{5} + 2068292643806280246 T^{6} - \)\(14\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!67\)\( T^{8} - \)\(64\!\cdots\!72\)\( T^{9} + \)\(41\!\cdots\!59\)\( T^{10} - \)\(23\!\cdots\!56\)\( T^{11} + \)\(13\!\cdots\!34\)\( T^{12} - \)\(69\!\cdots\!16\)\( T^{13} + \)\(37\!\cdots\!21\)\( T^{14} - \)\(17\!\cdots\!24\)\( T^{15} + \)\(87\!\cdots\!36\)\( T^{16} - \)\(39\!\cdots\!48\)\( T^{17} + \)\(87\!\cdots\!36\)\( p^{3} T^{18} - \)\(17\!\cdots\!24\)\( p^{6} T^{19} + \)\(37\!\cdots\!21\)\( p^{9} T^{20} - \)\(69\!\cdots\!16\)\( p^{12} T^{21} + \)\(13\!\cdots\!34\)\( p^{15} T^{22} - \)\(23\!\cdots\!56\)\( p^{18} T^{23} + \)\(41\!\cdots\!59\)\( p^{21} T^{24} - \)\(64\!\cdots\!72\)\( p^{24} T^{25} + \)\(10\!\cdots\!67\)\( p^{27} T^{26} - \)\(14\!\cdots\!16\)\( p^{30} T^{27} + 2068292643806280246 p^{33} T^{28} - 2380615740546836 p^{36} T^{29} + 2916165433393 p^{39} T^{30} - 2569102984 p^{42} T^{31} + 2555641 p^{45} T^{32} - 1358 p^{48} T^{33} + p^{51} T^{34} \) | |
| 61 | \( 1 + 339 T + 1990092 T^{2} + 711368626 T^{3} + 2037817887838 T^{4} + 738529327253477 T^{5} + 1413446279572950662 T^{6} + \)\(50\!\cdots\!77\)\( T^{7} + \)\(74\!\cdots\!86\)\( T^{8} + \)\(25\!\cdots\!38\)\( T^{9} + \)\(31\!\cdots\!99\)\( T^{10} + \)\(10\!\cdots\!16\)\( T^{11} + \)\(10\!\cdots\!64\)\( T^{12} + \)\(34\!\cdots\!55\)\( T^{13} + \)\(32\!\cdots\!82\)\( T^{14} + \)\(96\!\cdots\!97\)\( T^{15} + \)\(84\!\cdots\!20\)\( T^{16} + \)\(23\!\cdots\!66\)\( T^{17} + \)\(84\!\cdots\!20\)\( p^{3} T^{18} + \)\(96\!\cdots\!97\)\( p^{6} T^{19} + \)\(32\!\cdots\!82\)\( p^{9} T^{20} + \)\(34\!\cdots\!55\)\( p^{12} T^{21} + \)\(10\!\cdots\!64\)\( p^{15} T^{22} + \)\(10\!\cdots\!16\)\( p^{18} T^{23} + \)\(31\!\cdots\!99\)\( p^{21} T^{24} + \)\(25\!\cdots\!38\)\( p^{24} T^{25} + \)\(74\!\cdots\!86\)\( p^{27} T^{26} + \)\(50\!\cdots\!77\)\( p^{30} T^{27} + 1413446279572950662 p^{33} T^{28} + 738529327253477 p^{36} T^{29} + 2037817887838 p^{39} T^{30} + 711368626 p^{42} T^{31} + 1990092 p^{45} T^{32} + 339 p^{48} T^{33} + p^{51} T^{34} \) | |
| 67 | \( 1 - 1768 T + 5420866 T^{2} - 7388308412 T^{3} + 13055600776175 T^{4} - 14629279449412942 T^{5} + 19129931269326547238 T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(19\!\cdots\!20\)\( T^{8} - \)\(16\!\cdots\!96\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} - \)\(10\!\cdots\!72\)\( T^{11} + \)\(85\!\cdots\!94\)\( T^{12} - \)\(56\!\cdots\!42\)\( T^{13} + \)\(39\!\cdots\!39\)\( T^{14} - \)\(23\!\cdots\!04\)\( T^{15} + \)\(14\!\cdots\!65\)\( T^{16} - \)\(77\!\cdots\!44\)\( T^{17} + \)\(14\!\cdots\!65\)\( p^{3} T^{18} - \)\(23\!\cdots\!04\)\( p^{6} T^{19} + \)\(39\!\cdots\!39\)\( p^{9} T^{20} - \)\(56\!\cdots\!42\)\( p^{12} T^{21} + \)\(85\!\cdots\!94\)\( p^{15} T^{22} - \)\(10\!\cdots\!72\)\( p^{18} T^{23} + \)\(14\!\cdots\!08\)\( p^{21} T^{24} - \)\(16\!\cdots\!96\)\( p^{24} T^{25} + \)\(19\!\cdots\!20\)\( p^{27} T^{26} - \)\(18\!\cdots\!00\)\( p^{30} T^{27} + 19129931269326547238 p^{33} T^{28} - 14629279449412942 p^{36} T^{29} + 13055600776175 p^{39} T^{30} - 7388308412 p^{42} T^{31} + 5420866 p^{45} T^{32} - 1768 p^{48} T^{33} + p^{51} T^{34} \) | |
| 71 | \( 1 + 897 T + 3494108 T^{2} + 2553532472 T^{3} + 5631585288762 T^{4} + 3387498516095327 T^{5} + 5671174779544430752 T^{6} + \)\(28\!\cdots\!67\)\( T^{7} + \)\(40\!\cdots\!62\)\( T^{8} + \)\(16\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!99\)\( T^{10} + \)\(74\!\cdots\!12\)\( T^{11} + \)\(10\!\cdots\!38\)\( T^{12} + \)\(27\!\cdots\!95\)\( T^{13} + \)\(44\!\cdots\!50\)\( T^{14} + \)\(87\!\cdots\!13\)\( T^{15} + \)\(16\!\cdots\!10\)\( T^{16} + \)\(29\!\cdots\!42\)\( T^{17} + \)\(16\!\cdots\!10\)\( p^{3} T^{18} + \)\(87\!\cdots\!13\)\( p^{6} T^{19} + \)\(44\!\cdots\!50\)\( p^{9} T^{20} + \)\(27\!\cdots\!95\)\( p^{12} T^{21} + \)\(10\!\cdots\!38\)\( p^{15} T^{22} + \)\(74\!\cdots\!12\)\( p^{18} T^{23} + \)\(23\!\cdots\!99\)\( p^{21} T^{24} + \)\(16\!\cdots\!00\)\( p^{24} T^{25} + \)\(40\!\cdots\!62\)\( p^{27} T^{26} + \)\(28\!\cdots\!67\)\( p^{30} T^{27} + 5671174779544430752 p^{33} T^{28} + 3387498516095327 p^{36} T^{29} + 5631585288762 p^{39} T^{30} + 2553532472 p^{42} T^{31} + 3494108 p^{45} T^{32} + 897 p^{48} T^{33} + p^{51} T^{34} \) | |
| 73 | \( 1 - 1167 T + 3616082 T^{2} - 3195094254 T^{3} + 6132345749969 T^{4} - 4399366623755496 T^{5} + 6757822833654712947 T^{6} - \)\(40\!\cdots\!24\)\( T^{7} + \)\(55\!\cdots\!60\)\( T^{8} - \)\(28\!\cdots\!01\)\( T^{9} + \)\(36\!\cdots\!58\)\( T^{10} - \)\(16\!\cdots\!43\)\( T^{11} + \)\(20\!\cdots\!42\)\( T^{12} - \)\(82\!\cdots\!20\)\( T^{13} + \)\(10\!\cdots\!36\)\( T^{14} - \)\(36\!\cdots\!83\)\( T^{15} + \)\(44\!\cdots\!17\)\( T^{16} - \)\(14\!\cdots\!20\)\( T^{17} + \)\(44\!\cdots\!17\)\( p^{3} T^{18} - \)\(36\!\cdots\!83\)\( p^{6} T^{19} + \)\(10\!\cdots\!36\)\( p^{9} T^{20} - \)\(82\!\cdots\!20\)\( p^{12} T^{21} + \)\(20\!\cdots\!42\)\( p^{15} T^{22} - \)\(16\!\cdots\!43\)\( p^{18} T^{23} + \)\(36\!\cdots\!58\)\( p^{21} T^{24} - \)\(28\!\cdots\!01\)\( p^{24} T^{25} + \)\(55\!\cdots\!60\)\( p^{27} T^{26} - \)\(40\!\cdots\!24\)\( p^{30} T^{27} + 6757822833654712947 p^{33} T^{28} - 4399366623755496 p^{36} T^{29} + 6132345749969 p^{39} T^{30} - 3195094254 p^{42} T^{31} + 3616082 p^{45} T^{32} - 1167 p^{48} T^{33} + p^{51} T^{34} \) | |
| 79 | \( 1 - 743 T + 4426866 T^{2} - 2872774568 T^{3} + 9768681829361 T^{4} - 5705460756341624 T^{5} + 14492300439053579031 T^{6} - \)\(77\!\cdots\!76\)\( T^{7} + \)\(16\!\cdots\!52\)\( T^{8} - \)\(80\!\cdots\!49\)\( T^{9} + \)\(14\!\cdots\!82\)\( T^{10} - \)\(67\!\cdots\!29\)\( T^{11} + \)\(11\!\cdots\!94\)\( T^{12} - \)\(47\!\cdots\!40\)\( T^{13} + \)\(72\!\cdots\!70\)\( T^{14} - \)\(28\!\cdots\!99\)\( T^{15} + \)\(40\!\cdots\!49\)\( T^{16} - \)\(15\!\cdots\!12\)\( T^{17} + \)\(40\!\cdots\!49\)\( p^{3} T^{18} - \)\(28\!\cdots\!99\)\( p^{6} T^{19} + \)\(72\!\cdots\!70\)\( p^{9} T^{20} - \)\(47\!\cdots\!40\)\( p^{12} T^{21} + \)\(11\!\cdots\!94\)\( p^{15} T^{22} - \)\(67\!\cdots\!29\)\( p^{18} T^{23} + \)\(14\!\cdots\!82\)\( p^{21} T^{24} - \)\(80\!\cdots\!49\)\( p^{24} T^{25} + \)\(16\!\cdots\!52\)\( p^{27} T^{26} - \)\(77\!\cdots\!76\)\( p^{30} T^{27} + 14492300439053579031 p^{33} T^{28} - 5705460756341624 p^{36} T^{29} + 9768681829361 p^{39} T^{30} - 2872774568 p^{42} T^{31} + 4426866 p^{45} T^{32} - 743 p^{48} T^{33} + p^{51} T^{34} \) | |
| 83 | \( 1 - 2046 T + 7727033 T^{2} - 13018731476 T^{3} + 28572742647837 T^{4} - 41010681879671388 T^{5} + 67479899387162065454 T^{6} - \)\(84\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!31\)\( T^{8} - \)\(12\!\cdots\!40\)\( T^{9} + \)\(14\!\cdots\!71\)\( T^{10} - \)\(14\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!94\)\( T^{12} - \)\(13\!\cdots\!24\)\( T^{13} + \)\(12\!\cdots\!81\)\( T^{14} - \)\(10\!\cdots\!60\)\( T^{15} + \)\(88\!\cdots\!96\)\( T^{16} - \)\(66\!\cdots\!60\)\( T^{17} + \)\(88\!\cdots\!96\)\( p^{3} T^{18} - \)\(10\!\cdots\!60\)\( p^{6} T^{19} + \)\(12\!\cdots\!81\)\( p^{9} T^{20} - \)\(13\!\cdots\!24\)\( p^{12} T^{21} + \)\(15\!\cdots\!94\)\( p^{15} T^{22} - \)\(14\!\cdots\!96\)\( p^{18} T^{23} + \)\(14\!\cdots\!71\)\( p^{21} T^{24} - \)\(12\!\cdots\!40\)\( p^{24} T^{25} + \)\(11\!\cdots\!31\)\( p^{27} T^{26} - \)\(84\!\cdots\!36\)\( p^{30} T^{27} + 67479899387162065454 p^{33} T^{28} - 41010681879671388 p^{36} T^{29} + 28572742647837 p^{39} T^{30} - 13018731476 p^{42} T^{31} + 7727033 p^{45} T^{32} - 2046 p^{48} T^{33} + p^{51} T^{34} \) | |
| 89 | \( 1 - 4300 T + 12303414 T^{2} - 24591398520 T^{3} + 40684380750019 T^{4} - 55622434568862402 T^{5} + 67855708663039434194 T^{6} - \)\(73\!\cdots\!88\)\( T^{7} + \)\(73\!\cdots\!34\)\( T^{8} - \)\(68\!\cdots\!08\)\( T^{9} + \)\(62\!\cdots\!80\)\( T^{10} - \)\(53\!\cdots\!32\)\( T^{11} + \)\(44\!\cdots\!04\)\( T^{12} - \)\(34\!\cdots\!42\)\( T^{13} + \)\(27\!\cdots\!79\)\( T^{14} - \)\(19\!\cdots\!44\)\( T^{15} + \)\(15\!\cdots\!83\)\( T^{16} - \)\(11\!\cdots\!32\)\( T^{17} + \)\(15\!\cdots\!83\)\( p^{3} T^{18} - \)\(19\!\cdots\!44\)\( p^{6} T^{19} + \)\(27\!\cdots\!79\)\( p^{9} T^{20} - \)\(34\!\cdots\!42\)\( p^{12} T^{21} + \)\(44\!\cdots\!04\)\( p^{15} T^{22} - \)\(53\!\cdots\!32\)\( p^{18} T^{23} + \)\(62\!\cdots\!80\)\( p^{21} T^{24} - \)\(68\!\cdots\!08\)\( p^{24} T^{25} + \)\(73\!\cdots\!34\)\( p^{27} T^{26} - \)\(73\!\cdots\!88\)\( p^{30} T^{27} + 67855708663039434194 p^{33} T^{28} - 55622434568862402 p^{36} T^{29} + 40684380750019 p^{39} T^{30} - 24591398520 p^{42} T^{31} + 12303414 p^{45} T^{32} - 4300 p^{48} T^{33} + p^{51} T^{34} \) | |
| 97 | \( 1 - 3910 T + 14994474 T^{2} - 36013545306 T^{3} + 83247119043275 T^{4} - 151435045707048558 T^{5} + \)\(26\!\cdots\!34\)\( T^{6} - \)\(40\!\cdots\!64\)\( T^{7} + \)\(61\!\cdots\!94\)\( T^{8} - \)\(82\!\cdots\!28\)\( T^{9} + \)\(11\!\cdots\!04\)\( T^{10} - \)\(13\!\cdots\!56\)\( T^{11} + \)\(16\!\cdots\!64\)\( T^{12} - \)\(18\!\cdots\!02\)\( T^{13} + \)\(20\!\cdots\!39\)\( T^{14} - \)\(20\!\cdots\!98\)\( T^{15} + \)\(21\!\cdots\!27\)\( T^{16} - \)\(20\!\cdots\!48\)\( T^{17} + \)\(21\!\cdots\!27\)\( p^{3} T^{18} - \)\(20\!\cdots\!98\)\( p^{6} T^{19} + \)\(20\!\cdots\!39\)\( p^{9} T^{20} - \)\(18\!\cdots\!02\)\( p^{12} T^{21} + \)\(16\!\cdots\!64\)\( p^{15} T^{22} - \)\(13\!\cdots\!56\)\( p^{18} T^{23} + \)\(11\!\cdots\!04\)\( p^{21} T^{24} - \)\(82\!\cdots\!28\)\( p^{24} T^{25} + \)\(61\!\cdots\!94\)\( p^{27} T^{26} - \)\(40\!\cdots\!64\)\( p^{30} T^{27} + \)\(26\!\cdots\!34\)\( p^{33} T^{28} - 151435045707048558 p^{36} T^{29} + 83247119043275 p^{39} T^{30} - 36013545306 p^{42} T^{31} + 14994474 p^{45} T^{32} - 3910 p^{48} T^{33} + p^{51} T^{34} \) | |
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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
−3.37309811932280967534891413000, −3.19273207668009809099624479851, −3.12525934850234527795956415992, −3.12209905954949704512474634222, −3.07455569778211551209907813179, −2.75971788204891132617142417562, −2.64614816998065907293198014788, −2.48748476372029560535613239712, −2.39475470155882964940105630406, −2.36546212230062814293159746897, −2.31099405051971974834585298591, −2.30569400679089286235476320848, −1.99115937920199438932070022335, −1.98799015365310660462902793809, −1.80962458520370297199596129826, −1.58668414501646774822698024951, −1.48478464498748798594764560905, −1.30056728692860383472999661862, −1.26886165200197063729775714251, −1.19397272809351510601140270168, −0.989010276537050799677545965430, −0.936123513515804689358352059219, −0.791615180294638828658002356500, −0.65341590522690005313382644781, −0.60923106269564754868199216411, 0.60923106269564754868199216411, 0.65341590522690005313382644781, 0.791615180294638828658002356500, 0.936123513515804689358352059219, 0.989010276537050799677545965430, 1.19397272809351510601140270168, 1.26886165200197063729775714251, 1.30056728692860383472999661862, 1.48478464498748798594764560905, 1.58668414501646774822698024951, 1.80962458520370297199596129826, 1.98799015365310660462902793809, 1.99115937920199438932070022335, 2.30569400679089286235476320848, 2.31099405051971974834585298591, 2.36546212230062814293159746897, 2.39475470155882964940105630406, 2.48748476372029560535613239712, 2.64614816998065907293198014788, 2.75971788204891132617142417562, 3.07455569778211551209907813179, 3.12209905954949704512474634222, 3.12525934850234527795956415992, 3.19273207668009809099624479851, 3.37309811932280967534891413000
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.