Dirichlet series
| L(s) = 1 | − 5·2-s − 12·3-s − 34·4-s + 60·6-s − 31·7-s + 213·8-s − 90·9-s + 36·11-s + 408·12-s − 67·13-s + 155·14-s + 479·16-s − 425·17-s + 450·18-s − 251·19-s + 372·21-s − 180·22-s − 584·23-s − 2.55e3·24-s + 335·26-s + 1.60e3·27-s + 1.05e3·28-s + 252·29-s + 329·31-s − 4.46e3·32-s − 432·33-s + 2.12e3·34-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 2.30·3-s − 4.25·4-s + 4.08·6-s − 1.67·7-s + 9.41·8-s − 3.33·9-s + 0.986·11-s + 9.81·12-s − 1.42·13-s + 2.95·14-s + 7.48·16-s − 6.06·17-s + 5.89·18-s − 3.03·19-s + 3.86·21-s − 1.74·22-s − 5.29·23-s − 21.7·24-s + 2.52·26-s + 11.4·27-s + 7.11·28-s + 1.61·29-s + 1.90·31-s − 24.6·32-s − 2.27·33-s + 10.7·34-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{40} \cdot 83^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{40} \cdot 83^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(5^{40} \cdot 83^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.72378\times 10^{41}\) |
| Root analytic conductor: | \(11.0647\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(20\) |
| Selberg data: | \((40,\ 5^{40} \cdot 83^{20} ,\ ( \ : [3/2]^{20} ),\ 1 )\) |
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 | \( 1 \) |
| 83 | \( ( 1 - p T )^{20} \) | |
| good | 2 | \( 1 + 5 T + 59 T^{2} + 63 p^{2} T^{3} + 861 p T^{4} + 6683 T^{5} + 34723 T^{6} + 125277 T^{7} + 552547 T^{8} + 1870819 T^{9} + 7431953 T^{10} + 11876027 p T^{11} + 21925635 p^{2} T^{12} + 266133005 T^{13} + 928557905 T^{14} + 2696780713 T^{15} + 4486794207 p T^{16} + 3138890233 p^{3} T^{17} + 10013383019 p^{3} T^{18} + 13531630899 p^{4} T^{19} + 5190626223 p^{7} T^{20} + 13531630899 p^{7} T^{21} + 10013383019 p^{9} T^{22} + 3138890233 p^{12} T^{23} + 4486794207 p^{13} T^{24} + 2696780713 p^{15} T^{25} + 928557905 p^{18} T^{26} + 266133005 p^{21} T^{27} + 21925635 p^{26} T^{28} + 11876027 p^{28} T^{29} + 7431953 p^{30} T^{30} + 1870819 p^{33} T^{31} + 552547 p^{36} T^{32} + 125277 p^{39} T^{33} + 34723 p^{42} T^{34} + 6683 p^{45} T^{35} + 861 p^{49} T^{36} + 63 p^{53} T^{37} + 59 p^{54} T^{38} + 5 p^{57} T^{39} + p^{60} T^{40} \) |
| 3 | \( 1 + 4 p T + 26 p^{2} T^{2} + 761 p T^{3} + 27131 T^{4} + 227131 T^{5} + 78107 p^{3} T^{6} + 15746033 T^{7} + 125223205 T^{8} + 31829536 p^{3} T^{9} + 6145082660 T^{10} + 39560557219 T^{11} + 3226865950 p^{4} T^{12} + 1596503957498 T^{13} + 9889485447262 T^{14} + 19197991917445 p T^{15} + 337154535226802 T^{16} + 1875603626966272 T^{17} + 10432899889258211 T^{18} + 55505257552617215 T^{19} + 294576487320857662 T^{20} + 55505257552617215 p^{3} T^{21} + 10432899889258211 p^{6} T^{22} + 1875603626966272 p^{9} T^{23} + 337154535226802 p^{12} T^{24} + 19197991917445 p^{16} T^{25} + 9889485447262 p^{18} T^{26} + 1596503957498 p^{21} T^{27} + 3226865950 p^{28} T^{28} + 39560557219 p^{27} T^{29} + 6145082660 p^{30} T^{30} + 31829536 p^{36} T^{31} + 125223205 p^{36} T^{32} + 15746033 p^{39} T^{33} + 78107 p^{45} T^{34} + 227131 p^{45} T^{35} + 27131 p^{48} T^{36} + 761 p^{52} T^{37} + 26 p^{56} T^{38} + 4 p^{58} T^{39} + p^{60} T^{40} \) | |
| 7 | \( 1 + 31 T + 3145 T^{2} + 87868 T^{3} + 5088208 T^{4} + 131512890 T^{5} + 5626669003 T^{6} + 135764609177 T^{7} + 4754671748682 T^{8} + 107774671221859 T^{9} + 3263448298055887 T^{10} + 69841655502545694 T^{11} + 270143665393007446 p T^{12} + 38374928830673819498 T^{13} + 19394249087606869479 p^{2} T^{14} + \)\(18\!\cdots\!49\)\( T^{15} + \)\(42\!\cdots\!33\)\( T^{16} + \)\(77\!\cdots\!48\)\( T^{17} + \)\(16\!\cdots\!02\)\( T^{18} + \)\(29\!\cdots\!06\)\( T^{19} + \)\(60\!\cdots\!52\)\( T^{20} + \)\(29\!\cdots\!06\)\( p^{3} T^{21} + \)\(16\!\cdots\!02\)\( p^{6} T^{22} + \)\(77\!\cdots\!48\)\( p^{9} T^{23} + \)\(42\!\cdots\!33\)\( p^{12} T^{24} + \)\(18\!\cdots\!49\)\( p^{15} T^{25} + 19394249087606869479 p^{20} T^{26} + 38374928830673819498 p^{21} T^{27} + 270143665393007446 p^{25} T^{28} + 69841655502545694 p^{27} T^{29} + 3263448298055887 p^{30} T^{30} + 107774671221859 p^{33} T^{31} + 4754671748682 p^{36} T^{32} + 135764609177 p^{39} T^{33} + 5626669003 p^{42} T^{34} + 131512890 p^{45} T^{35} + 5088208 p^{48} T^{36} + 87868 p^{51} T^{37} + 3145 p^{54} T^{38} + 31 p^{57} T^{39} + p^{60} T^{40} \) | |
| 11 | \( 1 - 36 T + 13998 T^{2} - 439129 T^{3} + 96454441 T^{4} - 2666660949 T^{5} + 437679303225 T^{6} - 10663343284151 T^{7} + 1472961104741851 T^{8} - 31382480208521008 T^{9} + 3928867491698925478 T^{10} - 6569779413246679967 p T^{11} + \)\(86\!\cdots\!10\)\( T^{12} - \)\(13\!\cdots\!42\)\( T^{13} + \)\(16\!\cdots\!50\)\( T^{14} - \)\(21\!\cdots\!53\)\( T^{15} + \)\(27\!\cdots\!20\)\( T^{16} - \)\(31\!\cdots\!56\)\( T^{17} + \)\(41\!\cdots\!73\)\( T^{18} - \)\(41\!\cdots\!73\)\( T^{19} + \)\(57\!\cdots\!18\)\( T^{20} - \)\(41\!\cdots\!73\)\( p^{3} T^{21} + \)\(41\!\cdots\!73\)\( p^{6} T^{22} - \)\(31\!\cdots\!56\)\( p^{9} T^{23} + \)\(27\!\cdots\!20\)\( p^{12} T^{24} - \)\(21\!\cdots\!53\)\( p^{15} T^{25} + \)\(16\!\cdots\!50\)\( p^{18} T^{26} - \)\(13\!\cdots\!42\)\( p^{21} T^{27} + \)\(86\!\cdots\!10\)\( p^{24} T^{28} - 6569779413246679967 p^{28} T^{29} + 3928867491698925478 p^{30} T^{30} - 31382480208521008 p^{33} T^{31} + 1472961104741851 p^{36} T^{32} - 10663343284151 p^{39} T^{33} + 437679303225 p^{42} T^{34} - 2666660949 p^{45} T^{35} + 96454441 p^{48} T^{36} - 439129 p^{51} T^{37} + 13998 p^{54} T^{38} - 36 p^{57} T^{39} + p^{60} T^{40} \) | |
| 13 | \( 1 + 67 T + 24091 T^{2} + 1335845 T^{3} + 282711525 T^{4} + 13181172438 T^{5} + 2158645452564 T^{6} + 84013796067902 T^{7} + 12027000204294618 T^{8} + 379746615000335860 T^{9} + 51983769461010892952 T^{10} + \)\(12\!\cdots\!18\)\( T^{11} + \)\(13\!\cdots\!36\)\( p T^{12} + \)\(29\!\cdots\!68\)\( T^{13} + \)\(52\!\cdots\!10\)\( T^{14} + \)\(42\!\cdots\!30\)\( T^{15} + \)\(13\!\cdots\!61\)\( T^{16} + \)\(72\!\cdots\!33\)\( T^{17} + \)\(30\!\cdots\!67\)\( T^{18} - \)\(13\!\cdots\!53\)\( T^{19} + \)\(68\!\cdots\!14\)\( T^{20} - \)\(13\!\cdots\!53\)\( p^{3} T^{21} + \)\(30\!\cdots\!67\)\( p^{6} T^{22} + \)\(72\!\cdots\!33\)\( p^{9} T^{23} + \)\(13\!\cdots\!61\)\( p^{12} T^{24} + \)\(42\!\cdots\!30\)\( p^{15} T^{25} + \)\(52\!\cdots\!10\)\( p^{18} T^{26} + \)\(29\!\cdots\!68\)\( p^{21} T^{27} + \)\(13\!\cdots\!36\)\( p^{25} T^{28} + \)\(12\!\cdots\!18\)\( p^{27} T^{29} + 51983769461010892952 p^{30} T^{30} + 379746615000335860 p^{33} T^{31} + 12027000204294618 p^{36} T^{32} + 84013796067902 p^{39} T^{33} + 2158645452564 p^{42} T^{34} + 13181172438 p^{45} T^{35} + 282711525 p^{48} T^{36} + 1335845 p^{51} T^{37} + 24091 p^{54} T^{38} + 67 p^{57} T^{39} + p^{60} T^{40} \) | |
| 17 | \( 1 + 25 p T + 140036 T^{2} + 32763397 T^{3} + 387989575 p T^{4} + 1115825240334 T^{5} + 169665366170301 T^{6} + 22947401223081439 T^{7} + 2858102316027931407 T^{8} + \)\(32\!\cdots\!42\)\( T^{9} + \)\(34\!\cdots\!50\)\( T^{10} + \)\(34\!\cdots\!30\)\( T^{11} + \)\(32\!\cdots\!56\)\( T^{12} + \)\(28\!\cdots\!84\)\( T^{13} + \)\(24\!\cdots\!40\)\( T^{14} + \)\(20\!\cdots\!90\)\( T^{15} + \)\(15\!\cdots\!28\)\( T^{16} + \)\(12\!\cdots\!96\)\( T^{17} + \)\(52\!\cdots\!17\)\( p T^{18} + \)\(65\!\cdots\!55\)\( T^{19} + \)\(46\!\cdots\!14\)\( T^{20} + \)\(65\!\cdots\!55\)\( p^{3} T^{21} + \)\(52\!\cdots\!17\)\( p^{7} T^{22} + \)\(12\!\cdots\!96\)\( p^{9} T^{23} + \)\(15\!\cdots\!28\)\( p^{12} T^{24} + \)\(20\!\cdots\!90\)\( p^{15} T^{25} + \)\(24\!\cdots\!40\)\( p^{18} T^{26} + \)\(28\!\cdots\!84\)\( p^{21} T^{27} + \)\(32\!\cdots\!56\)\( p^{24} T^{28} + \)\(34\!\cdots\!30\)\( p^{27} T^{29} + \)\(34\!\cdots\!50\)\( p^{30} T^{30} + \)\(32\!\cdots\!42\)\( p^{33} T^{31} + 2858102316027931407 p^{36} T^{32} + 22947401223081439 p^{39} T^{33} + 169665366170301 p^{42} T^{34} + 1115825240334 p^{45} T^{35} + 387989575 p^{49} T^{36} + 32763397 p^{51} T^{37} + 140036 p^{54} T^{38} + 25 p^{58} T^{39} + p^{60} T^{40} \) | |
| 19 | \( 1 + 251 T + 97783 T^{2} + 18823419 T^{3} + 4317182411 T^{4} + 690578966000 T^{5} + 119894028704628 T^{6} + 16688907177313850 T^{7} + 2413052182016183403 T^{8} + \)\(30\!\cdots\!05\)\( T^{9} + \)\(38\!\cdots\!19\)\( T^{10} + \)\(43\!\cdots\!91\)\( T^{11} + \)\(49\!\cdots\!76\)\( T^{12} + \)\(51\!\cdots\!31\)\( T^{13} + \)\(54\!\cdots\!97\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{15} + \)\(51\!\cdots\!36\)\( T^{16} + \)\(46\!\cdots\!11\)\( T^{17} + \)\(42\!\cdots\!49\)\( T^{18} + \)\(36\!\cdots\!93\)\( T^{19} + \)\(30\!\cdots\!54\)\( T^{20} + \)\(36\!\cdots\!93\)\( p^{3} T^{21} + \)\(42\!\cdots\!49\)\( p^{6} T^{22} + \)\(46\!\cdots\!11\)\( p^{9} T^{23} + \)\(51\!\cdots\!36\)\( p^{12} T^{24} + \)\(52\!\cdots\!41\)\( p^{15} T^{25} + \)\(54\!\cdots\!97\)\( p^{18} T^{26} + \)\(51\!\cdots\!31\)\( p^{21} T^{27} + \)\(49\!\cdots\!76\)\( p^{24} T^{28} + \)\(43\!\cdots\!91\)\( p^{27} T^{29} + \)\(38\!\cdots\!19\)\( p^{30} T^{30} + \)\(30\!\cdots\!05\)\( p^{33} T^{31} + 2413052182016183403 p^{36} T^{32} + 16688907177313850 p^{39} T^{33} + 119894028704628 p^{42} T^{34} + 690578966000 p^{45} T^{35} + 4317182411 p^{48} T^{36} + 18823419 p^{51} T^{37} + 97783 p^{54} T^{38} + 251 p^{57} T^{39} + p^{60} T^{40} \) | |
| 23 | \( 1 + 584 T + 280220 T^{2} + 96355946 T^{3} + 29106333253 T^{4} + 7488729142468 T^{5} + 1751299507600348 T^{6} + 368934125865680006 T^{7} + 72057248441721270789 T^{8} + \)\(13\!\cdots\!86\)\( T^{9} + \)\(22\!\cdots\!36\)\( T^{10} + \)\(35\!\cdots\!78\)\( T^{11} + \)\(53\!\cdots\!52\)\( T^{12} + \)\(33\!\cdots\!06\)\( p T^{13} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(13\!\cdots\!22\)\( T^{15} + \)\(17\!\cdots\!50\)\( T^{16} + \)\(21\!\cdots\!54\)\( T^{17} + \)\(11\!\cdots\!44\)\( p T^{18} + \)\(29\!\cdots\!58\)\( T^{19} + \)\(32\!\cdots\!30\)\( T^{20} + \)\(29\!\cdots\!58\)\( p^{3} T^{21} + \)\(11\!\cdots\!44\)\( p^{7} T^{22} + \)\(21\!\cdots\!54\)\( p^{9} T^{23} + \)\(17\!\cdots\!50\)\( p^{12} T^{24} + \)\(13\!\cdots\!22\)\( p^{15} T^{25} + \)\(10\!\cdots\!40\)\( p^{18} T^{26} + \)\(33\!\cdots\!06\)\( p^{22} T^{27} + \)\(53\!\cdots\!52\)\( p^{24} T^{28} + \)\(35\!\cdots\!78\)\( p^{27} T^{29} + \)\(22\!\cdots\!36\)\( p^{30} T^{30} + \)\(13\!\cdots\!86\)\( p^{33} T^{31} + 72057248441721270789 p^{36} T^{32} + 368934125865680006 p^{39} T^{33} + 1751299507600348 p^{42} T^{34} + 7488729142468 p^{45} T^{35} + 29106333253 p^{48} T^{36} + 96355946 p^{51} T^{37} + 280220 p^{54} T^{38} + 584 p^{57} T^{39} + p^{60} T^{40} \) | |
| 29 | \( 1 - 252 T + 283780 T^{2} - 58846239 T^{3} + 37296539278 T^{4} - 6477759967911 T^{5} + 3084921833700211 T^{6} - 453915235771491526 T^{7} + \)\(18\!\cdots\!28\)\( T^{8} - \)\(23\!\cdots\!07\)\( T^{9} + \)\(85\!\cdots\!85\)\( T^{10} - \)\(91\!\cdots\!74\)\( T^{11} + \)\(33\!\cdots\!80\)\( T^{12} - \)\(29\!\cdots\!32\)\( T^{13} + \)\(10\!\cdots\!70\)\( T^{14} - \)\(78\!\cdots\!57\)\( T^{15} + \)\(31\!\cdots\!65\)\( T^{16} - \)\(18\!\cdots\!55\)\( T^{17} + \)\(84\!\cdots\!16\)\( T^{18} - \)\(43\!\cdots\!23\)\( T^{19} + \)\(21\!\cdots\!52\)\( T^{20} - \)\(43\!\cdots\!23\)\( p^{3} T^{21} + \)\(84\!\cdots\!16\)\( p^{6} T^{22} - \)\(18\!\cdots\!55\)\( p^{9} T^{23} + \)\(31\!\cdots\!65\)\( p^{12} T^{24} - \)\(78\!\cdots\!57\)\( p^{15} T^{25} + \)\(10\!\cdots\!70\)\( p^{18} T^{26} - \)\(29\!\cdots\!32\)\( p^{21} T^{27} + \)\(33\!\cdots\!80\)\( p^{24} T^{28} - \)\(91\!\cdots\!74\)\( p^{27} T^{29} + \)\(85\!\cdots\!85\)\( p^{30} T^{30} - \)\(23\!\cdots\!07\)\( p^{33} T^{31} + \)\(18\!\cdots\!28\)\( p^{36} T^{32} - 453915235771491526 p^{39} T^{33} + 3084921833700211 p^{42} T^{34} - 6477759967911 p^{45} T^{35} + 37296539278 p^{48} T^{36} - 58846239 p^{51} T^{37} + 283780 p^{54} T^{38} - 252 p^{57} T^{39} + p^{60} T^{40} \) | |
| 31 | \( 1 - 329 T + 283840 T^{2} - 86996051 T^{3} + 42280563532 T^{4} - 11834454533697 T^{5} + 4314397372496163 T^{6} - 1106508532182978798 T^{7} + \)\(33\!\cdots\!99\)\( T^{8} - \)\(79\!\cdots\!33\)\( T^{9} + \)\(21\!\cdots\!33\)\( T^{10} - \)\(46\!\cdots\!06\)\( T^{11} + \)\(11\!\cdots\!47\)\( T^{12} - \)\(74\!\cdots\!43\)\( p T^{13} + \)\(51\!\cdots\!79\)\( T^{14} - \)\(32\!\cdots\!42\)\( p T^{15} + \)\(20\!\cdots\!86\)\( T^{16} - \)\(37\!\cdots\!66\)\( T^{17} + \)\(72\!\cdots\!90\)\( T^{18} - \)\(12\!\cdots\!78\)\( T^{19} + \)\(22\!\cdots\!20\)\( T^{20} - \)\(12\!\cdots\!78\)\( p^{3} T^{21} + \)\(72\!\cdots\!90\)\( p^{6} T^{22} - \)\(37\!\cdots\!66\)\( p^{9} T^{23} + \)\(20\!\cdots\!86\)\( p^{12} T^{24} - \)\(32\!\cdots\!42\)\( p^{16} T^{25} + \)\(51\!\cdots\!79\)\( p^{18} T^{26} - \)\(74\!\cdots\!43\)\( p^{22} T^{27} + \)\(11\!\cdots\!47\)\( p^{24} T^{28} - \)\(46\!\cdots\!06\)\( p^{27} T^{29} + \)\(21\!\cdots\!33\)\( p^{30} T^{30} - \)\(79\!\cdots\!33\)\( p^{33} T^{31} + \)\(33\!\cdots\!99\)\( p^{36} T^{32} - 1106508532182978798 p^{39} T^{33} + 4314397372496163 p^{42} T^{34} - 11834454533697 p^{45} T^{35} + 42280563532 p^{48} T^{36} - 86996051 p^{51} T^{37} + 283840 p^{54} T^{38} - 329 p^{57} T^{39} + p^{60} T^{40} \) | |
| 37 | \( 1 + 260 T + 588252 T^{2} + 115651759 T^{3} + 166097872111 T^{4} + 24532558573873 T^{5} + 30655954727962413 T^{6} + 3301097592810465245 T^{7} + \)\(42\!\cdots\!67\)\( T^{8} + \)\(31\!\cdots\!38\)\( T^{9} + \)\(46\!\cdots\!94\)\( T^{10} + \)\(22\!\cdots\!89\)\( T^{11} + \)\(43\!\cdots\!96\)\( T^{12} + \)\(12\!\cdots\!14\)\( T^{13} + \)\(34\!\cdots\!52\)\( T^{14} + \)\(48\!\cdots\!43\)\( T^{15} + \)\(23\!\cdots\!08\)\( T^{16} + \)\(13\!\cdots\!98\)\( T^{17} + \)\(14\!\cdots\!53\)\( T^{18} + \)\(24\!\cdots\!33\)\( T^{19} + \)\(78\!\cdots\!74\)\( T^{20} + \)\(24\!\cdots\!33\)\( p^{3} T^{21} + \)\(14\!\cdots\!53\)\( p^{6} T^{22} + \)\(13\!\cdots\!98\)\( p^{9} T^{23} + \)\(23\!\cdots\!08\)\( p^{12} T^{24} + \)\(48\!\cdots\!43\)\( p^{15} T^{25} + \)\(34\!\cdots\!52\)\( p^{18} T^{26} + \)\(12\!\cdots\!14\)\( p^{21} T^{27} + \)\(43\!\cdots\!96\)\( p^{24} T^{28} + \)\(22\!\cdots\!89\)\( p^{27} T^{29} + \)\(46\!\cdots\!94\)\( p^{30} T^{30} + \)\(31\!\cdots\!38\)\( p^{33} T^{31} + \)\(42\!\cdots\!67\)\( p^{36} T^{32} + 3301097592810465245 p^{39} T^{33} + 30655954727962413 p^{42} T^{34} + 24532558573873 p^{45} T^{35} + 166097872111 p^{48} T^{36} + 115651759 p^{51} T^{37} + 588252 p^{54} T^{38} + 260 p^{57} T^{39} + p^{60} T^{40} \) | |
| 41 | \( 1 - 1830 T + 2301088 T^{2} - 2126736014 T^{3} + 1644993260527 T^{4} - 1089043119103482 T^{5} + 642962482061043544 T^{6} - \)\(34\!\cdots\!24\)\( T^{7} + \)\(16\!\cdots\!58\)\( T^{8} - \)\(76\!\cdots\!26\)\( T^{9} + \)\(32\!\cdots\!24\)\( T^{10} - \)\(12\!\cdots\!64\)\( T^{11} + \)\(49\!\cdots\!34\)\( T^{12} - \)\(17\!\cdots\!56\)\( T^{13} + \)\(60\!\cdots\!72\)\( T^{14} - \)\(19\!\cdots\!74\)\( T^{15} + \)\(62\!\cdots\!89\)\( T^{16} - \)\(18\!\cdots\!10\)\( T^{17} + \)\(54\!\cdots\!64\)\( T^{18} - \)\(15\!\cdots\!68\)\( T^{19} + \)\(40\!\cdots\!42\)\( T^{20} - \)\(15\!\cdots\!68\)\( p^{3} T^{21} + \)\(54\!\cdots\!64\)\( p^{6} T^{22} - \)\(18\!\cdots\!10\)\( p^{9} T^{23} + \)\(62\!\cdots\!89\)\( p^{12} T^{24} - \)\(19\!\cdots\!74\)\( p^{15} T^{25} + \)\(60\!\cdots\!72\)\( p^{18} T^{26} - \)\(17\!\cdots\!56\)\( p^{21} T^{27} + \)\(49\!\cdots\!34\)\( p^{24} T^{28} - \)\(12\!\cdots\!64\)\( p^{27} T^{29} + \)\(32\!\cdots\!24\)\( p^{30} T^{30} - \)\(76\!\cdots\!26\)\( p^{33} T^{31} + \)\(16\!\cdots\!58\)\( p^{36} T^{32} - \)\(34\!\cdots\!24\)\( p^{39} T^{33} + 642962482061043544 p^{42} T^{34} - 1089043119103482 p^{45} T^{35} + 1644993260527 p^{48} T^{36} - 2126736014 p^{51} T^{37} + 2301088 p^{54} T^{38} - 1830 p^{57} T^{39} + p^{60} T^{40} \) | |
| 43 | \( 1 - 87 T + 838989 T^{2} - 116495265 T^{3} + 347692011589 T^{4} - 64523305285704 T^{5} + 95765595477779716 T^{6} - 21448528229300726276 T^{7} + \)\(19\!\cdots\!86\)\( T^{8} - \)\(49\!\cdots\!86\)\( T^{9} + \)\(33\!\cdots\!00\)\( T^{10} - \)\(88\!\cdots\!48\)\( T^{11} + \)\(46\!\cdots\!36\)\( T^{12} - \)\(12\!\cdots\!02\)\( T^{13} + \)\(56\!\cdots\!18\)\( T^{14} - \)\(15\!\cdots\!44\)\( T^{15} + \)\(59\!\cdots\!01\)\( T^{16} - \)\(15\!\cdots\!15\)\( T^{17} + \)\(55\!\cdots\!89\)\( T^{18} - \)\(13\!\cdots\!81\)\( T^{19} + \)\(46\!\cdots\!42\)\( T^{20} - \)\(13\!\cdots\!81\)\( p^{3} T^{21} + \)\(55\!\cdots\!89\)\( p^{6} T^{22} - \)\(15\!\cdots\!15\)\( p^{9} T^{23} + \)\(59\!\cdots\!01\)\( p^{12} T^{24} - \)\(15\!\cdots\!44\)\( p^{15} T^{25} + \)\(56\!\cdots\!18\)\( p^{18} T^{26} - \)\(12\!\cdots\!02\)\( p^{21} T^{27} + \)\(46\!\cdots\!36\)\( p^{24} T^{28} - \)\(88\!\cdots\!48\)\( p^{27} T^{29} + \)\(33\!\cdots\!00\)\( p^{30} T^{30} - \)\(49\!\cdots\!86\)\( p^{33} T^{31} + \)\(19\!\cdots\!86\)\( p^{36} T^{32} - 21448528229300726276 p^{39} T^{33} + 95765595477779716 p^{42} T^{34} - 64523305285704 p^{45} T^{35} + 347692011589 p^{48} T^{36} - 116495265 p^{51} T^{37} + 838989 p^{54} T^{38} - 87 p^{57} T^{39} + p^{60} T^{40} \) | |
| 47 | \( 1 + 1028 T + 1868652 T^{2} + 1472410928 T^{3} + 1553228363511 T^{4} + 1007299364982438 T^{5} + 792417249379037636 T^{6} + \)\(44\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!22\)\( T^{8} + \)\(13\!\cdots\!90\)\( T^{9} + \)\(77\!\cdots\!00\)\( T^{10} + \)\(33\!\cdots\!60\)\( T^{11} + \)\(16\!\cdots\!90\)\( T^{12} + \)\(66\!\cdots\!48\)\( T^{13} + \)\(29\!\cdots\!04\)\( T^{14} + \)\(10\!\cdots\!82\)\( T^{15} + \)\(44\!\cdots\!65\)\( T^{16} + \)\(14\!\cdots\!20\)\( T^{17} + \)\(56\!\cdots\!28\)\( T^{18} + \)\(17\!\cdots\!10\)\( T^{19} + \)\(63\!\cdots\!74\)\( T^{20} + \)\(17\!\cdots\!10\)\( p^{3} T^{21} + \)\(56\!\cdots\!28\)\( p^{6} T^{22} + \)\(14\!\cdots\!20\)\( p^{9} T^{23} + \)\(44\!\cdots\!65\)\( p^{12} T^{24} + \)\(10\!\cdots\!82\)\( p^{15} T^{25} + \)\(29\!\cdots\!04\)\( p^{18} T^{26} + \)\(66\!\cdots\!48\)\( p^{21} T^{27} + \)\(16\!\cdots\!90\)\( p^{24} T^{28} + \)\(33\!\cdots\!60\)\( p^{27} T^{29} + \)\(77\!\cdots\!00\)\( p^{30} T^{30} + \)\(13\!\cdots\!90\)\( p^{33} T^{31} + \)\(28\!\cdots\!22\)\( p^{36} T^{32} + \)\(44\!\cdots\!00\)\( p^{39} T^{33} + 792417249379037636 p^{42} T^{34} + 1007299364982438 p^{45} T^{35} + 1553228363511 p^{48} T^{36} + 1472410928 p^{51} T^{37} + 1868652 p^{54} T^{38} + 1028 p^{57} T^{39} + p^{60} T^{40} \) | |
| 53 | \( 1 + 1491 T + 2672789 T^{2} + 2850213315 T^{3} + 3135149691329 T^{4} + 2686131407456752 T^{5} + 2282631829413123748 T^{6} + \)\(16\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!98\)\( T^{8} + \)\(75\!\cdots\!54\)\( T^{9} + \)\(47\!\cdots\!72\)\( T^{10} + \)\(27\!\cdots\!56\)\( T^{11} + \)\(15\!\cdots\!96\)\( T^{12} + \)\(78\!\cdots\!82\)\( T^{13} + \)\(40\!\cdots\!90\)\( T^{14} + \)\(19\!\cdots\!88\)\( T^{15} + \)\(88\!\cdots\!57\)\( T^{16} + \)\(39\!\cdots\!91\)\( T^{17} + \)\(16\!\cdots\!13\)\( T^{18} + \)\(67\!\cdots\!39\)\( T^{19} + \)\(26\!\cdots\!06\)\( T^{20} + \)\(67\!\cdots\!39\)\( p^{3} T^{21} + \)\(16\!\cdots\!13\)\( p^{6} T^{22} + \)\(39\!\cdots\!91\)\( p^{9} T^{23} + \)\(88\!\cdots\!57\)\( p^{12} T^{24} + \)\(19\!\cdots\!88\)\( p^{15} T^{25} + \)\(40\!\cdots\!90\)\( p^{18} T^{26} + \)\(78\!\cdots\!82\)\( p^{21} T^{27} + \)\(15\!\cdots\!96\)\( p^{24} T^{28} + \)\(27\!\cdots\!56\)\( p^{27} T^{29} + \)\(47\!\cdots\!72\)\( p^{30} T^{30} + \)\(75\!\cdots\!54\)\( p^{33} T^{31} + \)\(11\!\cdots\!98\)\( p^{36} T^{32} + \)\(16\!\cdots\!68\)\( p^{39} T^{33} + 2282631829413123748 p^{42} T^{34} + 2686131407456752 p^{45} T^{35} + 3135149691329 p^{48} T^{36} + 2850213315 p^{51} T^{37} + 2672789 p^{54} T^{38} + 1491 p^{57} T^{39} + p^{60} T^{40} \) | |
| 59 | \( 1 - 782 T + 2310806 T^{2} - 1812073397 T^{3} + 2778210480960 T^{4} - 2098034979609803 T^{5} + 2279691544997374549 T^{6} - \)\(16\!\cdots\!58\)\( T^{7} + \)\(14\!\cdots\!70\)\( T^{8} - \)\(93\!\cdots\!87\)\( T^{9} + \)\(70\!\cdots\!61\)\( T^{10} - \)\(42\!\cdots\!58\)\( T^{11} + \)\(28\!\cdots\!58\)\( T^{12} - \)\(16\!\cdots\!28\)\( T^{13} + \)\(98\!\cdots\!10\)\( T^{14} - \)\(52\!\cdots\!01\)\( T^{15} + \)\(28\!\cdots\!97\)\( T^{16} - \)\(14\!\cdots\!19\)\( T^{17} + \)\(73\!\cdots\!86\)\( T^{18} - \)\(33\!\cdots\!17\)\( T^{19} + \)\(16\!\cdots\!44\)\( T^{20} - \)\(33\!\cdots\!17\)\( p^{3} T^{21} + \)\(73\!\cdots\!86\)\( p^{6} T^{22} - \)\(14\!\cdots\!19\)\( p^{9} T^{23} + \)\(28\!\cdots\!97\)\( p^{12} T^{24} - \)\(52\!\cdots\!01\)\( p^{15} T^{25} + \)\(98\!\cdots\!10\)\( p^{18} T^{26} - \)\(16\!\cdots\!28\)\( p^{21} T^{27} + \)\(28\!\cdots\!58\)\( p^{24} T^{28} - \)\(42\!\cdots\!58\)\( p^{27} T^{29} + \)\(70\!\cdots\!61\)\( p^{30} T^{30} - \)\(93\!\cdots\!87\)\( p^{33} T^{31} + \)\(14\!\cdots\!70\)\( p^{36} T^{32} - \)\(16\!\cdots\!58\)\( p^{39} T^{33} + 2279691544997374549 p^{42} T^{34} - 2098034979609803 p^{45} T^{35} + 2778210480960 p^{48} T^{36} - 1812073397 p^{51} T^{37} + 2310806 p^{54} T^{38} - 782 p^{57} T^{39} + p^{60} T^{40} \) | |
| 61 | \( 1 + 686 T + 2443481 T^{2} + 1526808353 T^{3} + 2948217102836 T^{4} + 1686293181815472 T^{5} + 2331660081340700489 T^{6} + \)\(12\!\cdots\!31\)\( T^{7} + \)\(13\!\cdots\!67\)\( T^{8} + \)\(65\!\cdots\!01\)\( T^{9} + \)\(61\!\cdots\!76\)\( T^{10} + \)\(27\!\cdots\!97\)\( T^{11} + \)\(23\!\cdots\!57\)\( T^{12} + \)\(93\!\cdots\!69\)\( T^{13} + \)\(73\!\cdots\!69\)\( T^{14} + \)\(27\!\cdots\!84\)\( T^{15} + \)\(20\!\cdots\!66\)\( T^{16} + \)\(71\!\cdots\!75\)\( T^{17} + \)\(52\!\cdots\!43\)\( T^{18} + \)\(17\!\cdots\!56\)\( T^{19} + \)\(12\!\cdots\!82\)\( T^{20} + \)\(17\!\cdots\!56\)\( p^{3} T^{21} + \)\(52\!\cdots\!43\)\( p^{6} T^{22} + \)\(71\!\cdots\!75\)\( p^{9} T^{23} + \)\(20\!\cdots\!66\)\( p^{12} T^{24} + \)\(27\!\cdots\!84\)\( p^{15} T^{25} + \)\(73\!\cdots\!69\)\( p^{18} T^{26} + \)\(93\!\cdots\!69\)\( p^{21} T^{27} + \)\(23\!\cdots\!57\)\( p^{24} T^{28} + \)\(27\!\cdots\!97\)\( p^{27} T^{29} + \)\(61\!\cdots\!76\)\( p^{30} T^{30} + \)\(65\!\cdots\!01\)\( p^{33} T^{31} + \)\(13\!\cdots\!67\)\( p^{36} T^{32} + \)\(12\!\cdots\!31\)\( p^{39} T^{33} + 2331660081340700489 p^{42} T^{34} + 1686293181815472 p^{45} T^{35} + 2948217102836 p^{48} T^{36} + 1526808353 p^{51} T^{37} + 2443481 p^{54} T^{38} + 686 p^{57} T^{39} + p^{60} T^{40} \) | |
| 67 | \( 1 + 661 T + 1920916 T^{2} + 1691609208 T^{3} + 2466486904133 T^{4} + 2167865459962972 T^{5} + 2403299815302165779 T^{6} + \)\(19\!\cdots\!18\)\( T^{7} + \)\(18\!\cdots\!75\)\( T^{8} + \)\(14\!\cdots\!66\)\( T^{9} + \)\(11\!\cdots\!74\)\( T^{10} + \)\(12\!\cdots\!01\)\( p T^{11} + \)\(65\!\cdots\!98\)\( T^{12} + \)\(44\!\cdots\!41\)\( T^{13} + \)\(30\!\cdots\!12\)\( T^{14} + \)\(19\!\cdots\!84\)\( T^{15} + \)\(12\!\cdots\!88\)\( T^{16} + \)\(11\!\cdots\!21\)\( p T^{17} + \)\(46\!\cdots\!15\)\( T^{18} + \)\(25\!\cdots\!54\)\( T^{19} + \)\(14\!\cdots\!70\)\( T^{20} + \)\(25\!\cdots\!54\)\( p^{3} T^{21} + \)\(46\!\cdots\!15\)\( p^{6} T^{22} + \)\(11\!\cdots\!21\)\( p^{10} T^{23} + \)\(12\!\cdots\!88\)\( p^{12} T^{24} + \)\(19\!\cdots\!84\)\( p^{15} T^{25} + \)\(30\!\cdots\!12\)\( p^{18} T^{26} + \)\(44\!\cdots\!41\)\( p^{21} T^{27} + \)\(65\!\cdots\!98\)\( p^{24} T^{28} + \)\(12\!\cdots\!01\)\( p^{28} T^{29} + \)\(11\!\cdots\!74\)\( p^{30} T^{30} + \)\(14\!\cdots\!66\)\( p^{33} T^{31} + \)\(18\!\cdots\!75\)\( p^{36} T^{32} + \)\(19\!\cdots\!18\)\( p^{39} T^{33} + 2403299815302165779 p^{42} T^{34} + 2167865459962972 p^{45} T^{35} + 2466486904133 p^{48} T^{36} + 1691609208 p^{51} T^{37} + 1920916 p^{54} T^{38} + 661 p^{57} T^{39} + p^{60} T^{40} \) | |
| 71 | \( 1 - 298 T + 3468105 T^{2} - 873918996 T^{3} + 5900140159609 T^{4} - 1246891360545770 T^{5} + 6640723102306818101 T^{6} - \)\(11\!\cdots\!56\)\( T^{7} + \)\(56\!\cdots\!93\)\( T^{8} - \)\(77\!\cdots\!26\)\( T^{9} + \)\(38\!\cdots\!08\)\( T^{10} - \)\(41\!\cdots\!62\)\( T^{11} + \)\(22\!\cdots\!84\)\( T^{12} - \)\(18\!\cdots\!50\)\( T^{13} + \)\(11\!\cdots\!92\)\( T^{14} - \)\(73\!\cdots\!34\)\( T^{15} + \)\(52\!\cdots\!10\)\( T^{16} - \)\(27\!\cdots\!26\)\( T^{17} + \)\(21\!\cdots\!14\)\( T^{18} - \)\(10\!\cdots\!62\)\( T^{19} + \)\(81\!\cdots\!86\)\( T^{20} - \)\(10\!\cdots\!62\)\( p^{3} T^{21} + \)\(21\!\cdots\!14\)\( p^{6} T^{22} - \)\(27\!\cdots\!26\)\( p^{9} T^{23} + \)\(52\!\cdots\!10\)\( p^{12} T^{24} - \)\(73\!\cdots\!34\)\( p^{15} T^{25} + \)\(11\!\cdots\!92\)\( p^{18} T^{26} - \)\(18\!\cdots\!50\)\( p^{21} T^{27} + \)\(22\!\cdots\!84\)\( p^{24} T^{28} - \)\(41\!\cdots\!62\)\( p^{27} T^{29} + \)\(38\!\cdots\!08\)\( p^{30} T^{30} - \)\(77\!\cdots\!26\)\( p^{33} T^{31} + \)\(56\!\cdots\!93\)\( p^{36} T^{32} - \)\(11\!\cdots\!56\)\( p^{39} T^{33} + 6640723102306818101 p^{42} T^{34} - 1246891360545770 p^{45} T^{35} + 5900140159609 p^{48} T^{36} - 873918996 p^{51} T^{37} + 3468105 p^{54} T^{38} - 298 p^{57} T^{39} + p^{60} T^{40} \) | |
| 73 | \( 1 + 656 T + 2215105 T^{2} + 1266409002 T^{3} + 2744493972132 T^{4} + 1666085376302670 T^{5} + 2625147183377014158 T^{6} + \)\(16\!\cdots\!50\)\( T^{7} + \)\(20\!\cdots\!19\)\( T^{8} + \)\(13\!\cdots\!86\)\( T^{9} + \)\(19\!\cdots\!76\)\( p T^{10} + \)\(93\!\cdots\!78\)\( T^{11} + \)\(87\!\cdots\!77\)\( T^{12} + \)\(55\!\cdots\!90\)\( T^{13} + \)\(48\!\cdots\!58\)\( T^{14} + \)\(29\!\cdots\!50\)\( T^{15} + \)\(23\!\cdots\!80\)\( T^{16} + \)\(14\!\cdots\!22\)\( T^{17} + \)\(10\!\cdots\!79\)\( T^{18} + \)\(61\!\cdots\!36\)\( T^{19} + \)\(43\!\cdots\!86\)\( T^{20} + \)\(61\!\cdots\!36\)\( p^{3} T^{21} + \)\(10\!\cdots\!79\)\( p^{6} T^{22} + \)\(14\!\cdots\!22\)\( p^{9} T^{23} + \)\(23\!\cdots\!80\)\( p^{12} T^{24} + \)\(29\!\cdots\!50\)\( p^{15} T^{25} + \)\(48\!\cdots\!58\)\( p^{18} T^{26} + \)\(55\!\cdots\!90\)\( p^{21} T^{27} + \)\(87\!\cdots\!77\)\( p^{24} T^{28} + \)\(93\!\cdots\!78\)\( p^{27} T^{29} + \)\(19\!\cdots\!76\)\( p^{31} T^{30} + \)\(13\!\cdots\!86\)\( p^{33} T^{31} + \)\(20\!\cdots\!19\)\( p^{36} T^{32} + \)\(16\!\cdots\!50\)\( p^{39} T^{33} + 2625147183377014158 p^{42} T^{34} + 1666085376302670 p^{45} T^{35} + 2744493972132 p^{48} T^{36} + 1266409002 p^{51} T^{37} + 2215105 p^{54} T^{38} + 656 p^{57} T^{39} + p^{60} T^{40} \) | |
| 79 | \( 1 + 748 T + 6603998 T^{2} + 4492033364 T^{3} + 21235317516387 T^{4} + 12987971960672784 T^{5} + 44168987738321462434 T^{6} + \)\(24\!\cdots\!12\)\( T^{7} + \)\(66\!\cdots\!33\)\( T^{8} + \)\(31\!\cdots\!48\)\( T^{9} + \)\(78\!\cdots\!80\)\( T^{10} + \)\(32\!\cdots\!04\)\( T^{11} + \)\(73\!\cdots\!12\)\( T^{12} + \)\(26\!\cdots\!52\)\( T^{13} + \)\(58\!\cdots\!08\)\( T^{14} + \)\(17\!\cdots\!28\)\( T^{15} + \)\(39\!\cdots\!54\)\( T^{16} + \)\(10\!\cdots\!84\)\( T^{17} + \)\(23\!\cdots\!08\)\( T^{18} + \)\(55\!\cdots\!68\)\( T^{19} + \)\(12\!\cdots\!62\)\( T^{20} + \)\(55\!\cdots\!68\)\( p^{3} T^{21} + \)\(23\!\cdots\!08\)\( p^{6} T^{22} + \)\(10\!\cdots\!84\)\( p^{9} T^{23} + \)\(39\!\cdots\!54\)\( p^{12} T^{24} + \)\(17\!\cdots\!28\)\( p^{15} T^{25} + \)\(58\!\cdots\!08\)\( p^{18} T^{26} + \)\(26\!\cdots\!52\)\( p^{21} T^{27} + \)\(73\!\cdots\!12\)\( p^{24} T^{28} + \)\(32\!\cdots\!04\)\( p^{27} T^{29} + \)\(78\!\cdots\!80\)\( p^{30} T^{30} + \)\(31\!\cdots\!48\)\( p^{33} T^{31} + \)\(66\!\cdots\!33\)\( p^{36} T^{32} + \)\(24\!\cdots\!12\)\( p^{39} T^{33} + 44168987738321462434 p^{42} T^{34} + 12987971960672784 p^{45} T^{35} + 21235317516387 p^{48} T^{36} + 4492033364 p^{51} T^{37} + 6603998 p^{54} T^{38} + 748 p^{57} T^{39} + p^{60} T^{40} \) | |
| 89 | \( 1 - 3702 T + 16105467 T^{2} - 42123807962 T^{3} + 110863382601835 T^{4} - 228933319292101808 T^{5} + \)\(45\!\cdots\!79\)\( T^{6} - \)\(79\!\cdots\!76\)\( T^{7} + \)\(13\!\cdots\!65\)\( T^{8} - \)\(19\!\cdots\!30\)\( T^{9} + \)\(28\!\cdots\!44\)\( T^{10} - \)\(37\!\cdots\!10\)\( T^{11} + \)\(47\!\cdots\!76\)\( T^{12} - \)\(55\!\cdots\!50\)\( T^{13} + \)\(63\!\cdots\!20\)\( T^{14} - \)\(67\!\cdots\!34\)\( T^{15} + \)\(70\!\cdots\!34\)\( T^{16} - \)\(68\!\cdots\!98\)\( T^{17} + \)\(64\!\cdots\!54\)\( T^{18} - \)\(57\!\cdots\!42\)\( T^{19} + \)\(49\!\cdots\!10\)\( T^{20} - \)\(57\!\cdots\!42\)\( p^{3} T^{21} + \)\(64\!\cdots\!54\)\( p^{6} T^{22} - \)\(68\!\cdots\!98\)\( p^{9} T^{23} + \)\(70\!\cdots\!34\)\( p^{12} T^{24} - \)\(67\!\cdots\!34\)\( p^{15} T^{25} + \)\(63\!\cdots\!20\)\( p^{18} T^{26} - \)\(55\!\cdots\!50\)\( p^{21} T^{27} + \)\(47\!\cdots\!76\)\( p^{24} T^{28} - \)\(37\!\cdots\!10\)\( p^{27} T^{29} + \)\(28\!\cdots\!44\)\( p^{30} T^{30} - \)\(19\!\cdots\!30\)\( p^{33} T^{31} + \)\(13\!\cdots\!65\)\( p^{36} T^{32} - \)\(79\!\cdots\!76\)\( p^{39} T^{33} + \)\(45\!\cdots\!79\)\( p^{42} T^{34} - 228933319292101808 p^{45} T^{35} + 110863382601835 p^{48} T^{36} - 42123807962 p^{51} T^{37} + 16105467 p^{54} T^{38} - 3702 p^{57} T^{39} + p^{60} T^{40} \) | |
| 97 | \( 1 + 4506 T + 18053870 T^{2} + 49647519832 T^{3} + 124729703272072 T^{4} + 262042671852601338 T^{5} + \)\(51\!\cdots\!42\)\( T^{6} + \)\(90\!\cdots\!78\)\( T^{7} + \)\(15\!\cdots\!67\)\( T^{8} + \)\(23\!\cdots\!42\)\( T^{9} + \)\(33\!\cdots\!06\)\( T^{10} + \)\(46\!\cdots\!74\)\( T^{11} + \)\(62\!\cdots\!47\)\( T^{12} + \)\(78\!\cdots\!98\)\( T^{13} + \)\(95\!\cdots\!26\)\( T^{14} + \)\(11\!\cdots\!46\)\( T^{15} + \)\(12\!\cdots\!84\)\( T^{16} + \)\(13\!\cdots\!72\)\( T^{17} + \)\(14\!\cdots\!80\)\( T^{18} + \)\(14\!\cdots\!18\)\( T^{19} + \)\(13\!\cdots\!98\)\( T^{20} + \)\(14\!\cdots\!18\)\( p^{3} T^{21} + \)\(14\!\cdots\!80\)\( p^{6} T^{22} + \)\(13\!\cdots\!72\)\( p^{9} T^{23} + \)\(12\!\cdots\!84\)\( p^{12} T^{24} + \)\(11\!\cdots\!46\)\( p^{15} T^{25} + \)\(95\!\cdots\!26\)\( p^{18} T^{26} + \)\(78\!\cdots\!98\)\( p^{21} T^{27} + \)\(62\!\cdots\!47\)\( p^{24} T^{28} + \)\(46\!\cdots\!74\)\( p^{27} T^{29} + \)\(33\!\cdots\!06\)\( p^{30} T^{30} + \)\(23\!\cdots\!42\)\( p^{33} T^{31} + \)\(15\!\cdots\!67\)\( p^{36} T^{32} + \)\(90\!\cdots\!78\)\( p^{39} T^{33} + \)\(51\!\cdots\!42\)\( p^{42} T^{34} + 262042671852601338 p^{45} T^{35} + 124729703272072 p^{48} T^{36} + 49647519832 p^{51} T^{37} + 18053870 p^{54} T^{38} + 4506 p^{57} T^{39} + p^{60} T^{40} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.14884231015821405455347198414, −2.11675062827810649118273750459, −2.03799558780634595365660290386, −2.03375292270111182518773307124, −1.80460302932585183830585545012, −1.79520904510719416898773343214, −1.69919559802905016704578153611, −1.62404456201882577692062768117, −1.54393522163348982689398525504, −1.48069375883693280065552538593, −1.31946817315890105400330373076, −1.29672826819601840405071497886, −1.25231552200219556008678514565, −1.23567640842918360134842987616, −1.16359218571475920810279078990, −1.14190777538541651142049744031, −1.09757644777938081688952207611, −1.07483596683471005721807264752, −1.03591201786108866599699689205, −0.950262764033342380407198475406, −0.900702772175224917443129021307, −0.75559163444257945837940767310, −0.74927184645836745626566387602, −0.71984079132725234002119837509, −0.54881571685131542914550037299, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.54881571685131542914550037299, 0.71984079132725234002119837509, 0.74927184645836745626566387602, 0.75559163444257945837940767310, 0.900702772175224917443129021307, 0.950262764033342380407198475406, 1.03591201786108866599699689205, 1.07483596683471005721807264752, 1.09757644777938081688952207611, 1.14190777538541651142049744031, 1.16359218571475920810279078990, 1.23567640842918360134842987616, 1.25231552200219556008678514565, 1.29672826819601840405071497886, 1.31946817315890105400330373076, 1.48069375883693280065552538593, 1.54393522163348982689398525504, 1.62404456201882577692062768117, 1.69919559802905016704578153611, 1.79520904510719416898773343214, 1.80460302932585183830585545012, 2.03375292270111182518773307124, 2.03799558780634595365660290386, 2.11675062827810649118273750459, 2.14884231015821405455347198414
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.