Dirichlet series
| L(s) = 1 | − 30·3-s − 56·5-s + 464·7-s − 2.09e3·9-s − 3.64e3·11-s − 7.14e3·13-s + 1.68e3·15-s + 1.13e3·17-s + 2.11e3·19-s − 1.39e4·21-s + 832·23-s + 6.58e4·25-s + 7.18e4·27-s − 1.09e4·29-s + 1.09e5·33-s − 2.59e4·35-s + 2.14e5·39-s + 1.09e5·41-s + 1.10e5·43-s + 1.17e5·45-s + 1.07e5·47-s − 1.00e6·49-s − 3.39e4·51-s + 2.54e5·53-s + 2.04e5·55-s − 6.33e4·57-s − 6.10e5·59-s + ⋯ |
| L(s) = 1 | − 1.11·3-s − 0.447·5-s + 1.35·7-s − 2.87·9-s − 2.73·11-s − 3.24·13-s + 0.497·15-s + 0.230·17-s + 0.307·19-s − 1.50·21-s + 0.0683·23-s + 4.21·25-s + 3.65·27-s − 0.447·29-s + 3.04·33-s − 0.606·35-s + 3.61·39-s + 1.58·41-s + 1.39·43-s + 1.28·45-s + 1.03·47-s − 8.50·49-s − 0.256·51-s + 1.71·53-s + 1.22·55-s − 0.341·57-s − 2.97·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+3)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{40} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.12640\times 10^{24}\) |
| Root analytic conductor: | \(4.18140\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{40} \cdot 19^{20} ,\ ( \ : [3]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(13.96975936\) |
| \(L(\frac12)\) | \(\approx\) | \(13.96975936\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 2110 T + 6622231 p T^{2} - 846350042 p^{2} T^{3} + 1304265667148 p^{3} T^{4} - 11952690083474 p^{5} T^{5} + 7561004339847467 p^{6} T^{6} - 47831148373766037504 p^{6} T^{7} + \)\(54\!\cdots\!40\)\( p^{8} T^{8} - \)\(11\!\cdots\!48\)\( p^{11} T^{9} + \)\(51\!\cdots\!40\)\( p^{13} T^{10} - \)\(11\!\cdots\!48\)\( p^{17} T^{11} + \)\(54\!\cdots\!40\)\( p^{20} T^{12} - 47831148373766037504 p^{24} T^{13} + 7561004339847467 p^{30} T^{14} - 11952690083474 p^{35} T^{15} + 1304265667148 p^{39} T^{16} - 846350042 p^{44} T^{17} + 6622231 p^{49} T^{18} - 2110 p^{54} T^{19} + p^{60} T^{20} \) | |
| good | 3 | \( 1 + 10 p T + 2995 T^{2} + 26950 p T^{3} + 4280098 T^{4} + 3622774 p^{3} T^{5} + 1240733527 p T^{6} + 1918121830 p^{3} T^{7} + 188354116718 p^{2} T^{8} - 77839229342 p^{5} T^{9} - 12131318109209 p^{3} T^{10} - 250103921881930 p^{5} T^{11} - 14167043544527308 p^{4} T^{12} - 78935577076766510 p^{6} T^{13} - 2575487831351559569 p^{5} T^{14} - 11717242485072220406 p^{7} T^{15} + \)\(57\!\cdots\!33\)\( p^{6} T^{16} + \)\(37\!\cdots\!48\)\( p^{9} T^{17} + \)\(16\!\cdots\!10\)\( p^{8} T^{18} + \)\(12\!\cdots\!96\)\( p^{11} T^{19} + \)\(17\!\cdots\!28\)\( p^{10} T^{20} + \)\(12\!\cdots\!96\)\( p^{17} T^{21} + \)\(16\!\cdots\!10\)\( p^{20} T^{22} + \)\(37\!\cdots\!48\)\( p^{27} T^{23} + \)\(57\!\cdots\!33\)\( p^{30} T^{24} - 11717242485072220406 p^{37} T^{25} - 2575487831351559569 p^{41} T^{26} - 78935577076766510 p^{48} T^{27} - 14167043544527308 p^{52} T^{28} - 250103921881930 p^{59} T^{29} - 12131318109209 p^{63} T^{30} - 77839229342 p^{71} T^{31} + 188354116718 p^{74} T^{32} + 1918121830 p^{81} T^{33} + 1240733527 p^{85} T^{34} + 3622774 p^{93} T^{35} + 4280098 p^{96} T^{36} + 26950 p^{103} T^{37} + 2995 p^{108} T^{38} + 10 p^{115} T^{39} + p^{120} T^{40} \) |
| 5 | \( 1 + 56 T - 62708 T^{2} - 5374192 T^{3} + 1529957114 T^{4} + 188242959724 T^{5} - 15896786681932 T^{6} - 2697636731001008 T^{7} + 101516280306358371 T^{8} + 3309550096175668952 p T^{9} - \)\(19\!\cdots\!44\)\( p^{2} T^{10} - \)\(52\!\cdots\!84\)\( p^{3} T^{11} + \)\(18\!\cdots\!24\)\( p^{4} T^{12} + \)\(80\!\cdots\!28\)\( p^{5} T^{13} + \)\(13\!\cdots\!08\)\( p^{6} T^{14} - \)\(38\!\cdots\!64\)\( p^{7} T^{15} - \)\(89\!\cdots\!11\)\( p^{8} T^{16} - \)\(13\!\cdots\!04\)\( p^{10} T^{17} + \)\(26\!\cdots\!44\)\( p^{10} T^{18} + \)\(69\!\cdots\!76\)\( p^{11} T^{19} + \)\(58\!\cdots\!94\)\( p^{12} T^{20} + \)\(69\!\cdots\!76\)\( p^{17} T^{21} + \)\(26\!\cdots\!44\)\( p^{22} T^{22} - \)\(13\!\cdots\!04\)\( p^{28} T^{23} - \)\(89\!\cdots\!11\)\( p^{32} T^{24} - \)\(38\!\cdots\!64\)\( p^{37} T^{25} + \)\(13\!\cdots\!08\)\( p^{42} T^{26} + \)\(80\!\cdots\!28\)\( p^{47} T^{27} + \)\(18\!\cdots\!24\)\( p^{52} T^{28} - \)\(52\!\cdots\!84\)\( p^{57} T^{29} - \)\(19\!\cdots\!44\)\( p^{62} T^{30} + 3309550096175668952 p^{67} T^{31} + 101516280306358371 p^{72} T^{32} - 2697636731001008 p^{78} T^{33} - 15896786681932 p^{84} T^{34} + 188242959724 p^{90} T^{35} + 1529957114 p^{96} T^{36} - 5374192 p^{102} T^{37} - 62708 p^{108} T^{38} + 56 p^{114} T^{39} + p^{120} T^{40} \) | |
| 7 | \( ( 1 - 232 T + 581134 T^{2} - 188388740 T^{3} + 183362500385 T^{4} - 1315158698300 p^{2} T^{5} + 41782068410093156 T^{6} - 13974583183510863044 T^{7} + \)\(71\!\cdots\!22\)\( T^{8} - \)\(22\!\cdots\!64\)\( T^{9} + \)\(94\!\cdots\!64\)\( T^{10} - \)\(22\!\cdots\!64\)\( p^{6} T^{11} + \)\(71\!\cdots\!22\)\( p^{12} T^{12} - 13974583183510863044 p^{18} T^{13} + 41782068410093156 p^{24} T^{14} - 1315158698300 p^{32} T^{15} + 183362500385 p^{36} T^{16} - 188388740 p^{42} T^{17} + 581134 p^{48} T^{18} - 232 p^{54} T^{19} + p^{60} T^{20} )^{2} \) | |
| 11 | \( ( 1 + 1822 T + 9609381 T^{2} + 16756894966 T^{3} + 4504437483588 p T^{4} + 77303554955485850 T^{5} + \)\(16\!\cdots\!83\)\( T^{6} + \)\(23\!\cdots\!78\)\( T^{7} + \)\(42\!\cdots\!55\)\( T^{8} + \)\(54\!\cdots\!52\)\( T^{9} + \)\(84\!\cdots\!36\)\( T^{10} + \)\(54\!\cdots\!52\)\( p^{6} T^{11} + \)\(42\!\cdots\!55\)\( p^{12} T^{12} + \)\(23\!\cdots\!78\)\( p^{18} T^{13} + \)\(16\!\cdots\!83\)\( p^{24} T^{14} + 77303554955485850 p^{30} T^{15} + 4504437483588 p^{37} T^{16} + 16756894966 p^{42} T^{17} + 9609381 p^{48} T^{18} + 1822 p^{54} T^{19} + p^{60} T^{20} )^{2} \) | |
| 13 | \( 1 + 7140 T + 49417360 T^{2} + 231508502400 T^{3} + 990044348590234 T^{4} + 3586624711383929220 T^{5} + \)\(11\!\cdots\!60\)\( T^{6} + \)\(36\!\cdots\!44\)\( T^{7} + \)\(80\!\cdots\!71\)\( p T^{8} + \)\(27\!\cdots\!00\)\( T^{9} + \)\(71\!\cdots\!92\)\( T^{10} + \)\(17\!\cdots\!48\)\( T^{11} + \)\(39\!\cdots\!08\)\( T^{12} + \)\(87\!\cdots\!48\)\( T^{13} + \)\(18\!\cdots\!72\)\( T^{14} + \)\(28\!\cdots\!00\)\( p T^{15} + \)\(73\!\cdots\!17\)\( T^{16} + \)\(13\!\cdots\!96\)\( T^{17} + \)\(26\!\cdots\!88\)\( T^{18} + \)\(52\!\cdots\!28\)\( T^{19} + \)\(10\!\cdots\!18\)\( T^{20} + \)\(52\!\cdots\!28\)\( p^{6} T^{21} + \)\(26\!\cdots\!88\)\( p^{12} T^{22} + \)\(13\!\cdots\!96\)\( p^{18} T^{23} + \)\(73\!\cdots\!17\)\( p^{24} T^{24} + \)\(28\!\cdots\!00\)\( p^{31} T^{25} + \)\(18\!\cdots\!72\)\( p^{36} T^{26} + \)\(87\!\cdots\!48\)\( p^{42} T^{27} + \)\(39\!\cdots\!08\)\( p^{48} T^{28} + \)\(17\!\cdots\!48\)\( p^{54} T^{29} + \)\(71\!\cdots\!92\)\( p^{60} T^{30} + \)\(27\!\cdots\!00\)\( p^{66} T^{31} + \)\(80\!\cdots\!71\)\( p^{73} T^{32} + \)\(36\!\cdots\!44\)\( p^{78} T^{33} + \)\(11\!\cdots\!60\)\( p^{84} T^{34} + 3586624711383929220 p^{90} T^{35} + 990044348590234 p^{96} T^{36} + 231508502400 p^{102} T^{37} + 49417360 p^{108} T^{38} + 7140 p^{114} T^{39} + p^{120} T^{40} \) | |
| 17 | \( 1 - 1132 T - 92232716 T^{2} + 237371390888 T^{3} + 3158834980187666 T^{4} - 15056157150780313724 T^{5} - \)\(28\!\cdots\!36\)\( T^{6} + \)\(43\!\cdots\!40\)\( T^{7} - \)\(90\!\cdots\!61\)\( T^{8} - \)\(49\!\cdots\!52\)\( T^{9} + \)\(22\!\cdots\!52\)\( T^{10} - \)\(53\!\cdots\!52\)\( T^{11} + \)\(28\!\cdots\!76\)\( T^{12} + \)\(28\!\cdots\!08\)\( T^{13} - \)\(31\!\cdots\!52\)\( T^{14} - \)\(26\!\cdots\!80\)\( T^{15} + \)\(10\!\cdots\!93\)\( T^{16} - \)\(10\!\cdots\!12\)\( T^{17} - \)\(15\!\cdots\!28\)\( T^{18} + \)\(74\!\cdots\!20\)\( p^{2} T^{19} + \)\(16\!\cdots\!82\)\( T^{20} + \)\(74\!\cdots\!20\)\( p^{8} T^{21} - \)\(15\!\cdots\!28\)\( p^{12} T^{22} - \)\(10\!\cdots\!12\)\( p^{18} T^{23} + \)\(10\!\cdots\!93\)\( p^{24} T^{24} - \)\(26\!\cdots\!80\)\( p^{30} T^{25} - \)\(31\!\cdots\!52\)\( p^{36} T^{26} + \)\(28\!\cdots\!08\)\( p^{42} T^{27} + \)\(28\!\cdots\!76\)\( p^{48} T^{28} - \)\(53\!\cdots\!52\)\( p^{54} T^{29} + \)\(22\!\cdots\!52\)\( p^{60} T^{30} - \)\(49\!\cdots\!52\)\( p^{66} T^{31} - \)\(90\!\cdots\!61\)\( p^{72} T^{32} + \)\(43\!\cdots\!40\)\( p^{78} T^{33} - \)\(28\!\cdots\!36\)\( p^{84} T^{34} - 15056157150780313724 p^{90} T^{35} + 3158834980187666 p^{96} T^{36} + 237371390888 p^{102} T^{37} - 92232716 p^{108} T^{38} - 1132 p^{114} T^{39} + p^{120} T^{40} \) | |
| 23 | \( 1 - 832 T - 716990246 T^{2} + 5006892125600 T^{3} + 248346305322584984 T^{4} - \)\(31\!\cdots\!00\)\( T^{5} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!68\)\( T^{7} + \)\(21\!\cdots\!65\)\( T^{8} - \)\(19\!\cdots\!28\)\( T^{9} + \)\(12\!\cdots\!08\)\( T^{10} + \)\(17\!\cdots\!48\)\( T^{11} - \)\(39\!\cdots\!18\)\( T^{12} + \)\(19\!\cdots\!00\)\( T^{13} + \)\(51\!\cdots\!46\)\( T^{14} - \)\(98\!\cdots\!00\)\( T^{15} + \)\(91\!\cdots\!33\)\( T^{16} + \)\(16\!\cdots\!20\)\( T^{17} - \)\(17\!\cdots\!20\)\( T^{18} - \)\(10\!\cdots\!64\)\( T^{19} + \)\(35\!\cdots\!18\)\( T^{20} - \)\(10\!\cdots\!64\)\( p^{6} T^{21} - \)\(17\!\cdots\!20\)\( p^{12} T^{22} + \)\(16\!\cdots\!20\)\( p^{18} T^{23} + \)\(91\!\cdots\!33\)\( p^{24} T^{24} - \)\(98\!\cdots\!00\)\( p^{30} T^{25} + \)\(51\!\cdots\!46\)\( p^{36} T^{26} + \)\(19\!\cdots\!00\)\( p^{42} T^{27} - \)\(39\!\cdots\!18\)\( p^{48} T^{28} + \)\(17\!\cdots\!48\)\( p^{54} T^{29} + \)\(12\!\cdots\!08\)\( p^{60} T^{30} - \)\(19\!\cdots\!28\)\( p^{66} T^{31} + \)\(21\!\cdots\!65\)\( p^{72} T^{32} + \)\(10\!\cdots\!68\)\( p^{78} T^{33} - \)\(44\!\cdots\!20\)\( p^{84} T^{34} - \)\(31\!\cdots\!00\)\( p^{90} T^{35} + 248346305322584984 p^{96} T^{36} + 5006892125600 p^{102} T^{37} - 716990246 p^{108} T^{38} - 832 p^{114} T^{39} + p^{120} T^{40} \) | |
| 29 | \( 1 + 10920 T + 2635751980 T^{2} + 28348354725600 T^{3} + 3171525990934176802 T^{4} + \)\(91\!\cdots\!20\)\( p T^{5} + \)\(25\!\cdots\!32\)\( T^{6} + \)\(18\!\cdots\!76\)\( T^{7} + \)\(17\!\cdots\!39\)\( T^{8} - \)\(21\!\cdots\!60\)\( T^{9} + \)\(97\!\cdots\!96\)\( T^{10} - \)\(28\!\cdots\!04\)\( T^{11} + \)\(53\!\cdots\!36\)\( T^{12} - \)\(80\!\cdots\!68\)\( p T^{13} + \)\(41\!\cdots\!12\)\( T^{14} - \)\(15\!\cdots\!08\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} - \)\(78\!\cdots\!04\)\( T^{17} + \)\(29\!\cdots\!76\)\( T^{18} - \)\(36\!\cdots\!72\)\( T^{19} + \)\(19\!\cdots\!82\)\( T^{20} - \)\(36\!\cdots\!72\)\( p^{6} T^{21} + \)\(29\!\cdots\!76\)\( p^{12} T^{22} - \)\(78\!\cdots\!04\)\( p^{18} T^{23} + \)\(37\!\cdots\!41\)\( p^{24} T^{24} - \)\(15\!\cdots\!08\)\( p^{30} T^{25} + \)\(41\!\cdots\!12\)\( p^{36} T^{26} - \)\(80\!\cdots\!68\)\( p^{43} T^{27} + \)\(53\!\cdots\!36\)\( p^{48} T^{28} - \)\(28\!\cdots\!04\)\( p^{54} T^{29} + \)\(97\!\cdots\!96\)\( p^{60} T^{30} - \)\(21\!\cdots\!60\)\( p^{66} T^{31} + \)\(17\!\cdots\!39\)\( p^{72} T^{32} + \)\(18\!\cdots\!76\)\( p^{78} T^{33} + \)\(25\!\cdots\!32\)\( p^{84} T^{34} + \)\(91\!\cdots\!20\)\( p^{91} T^{35} + 3171525990934176802 p^{96} T^{36} + 28348354725600 p^{102} T^{37} + 2635751980 p^{108} T^{38} + 10920 p^{114} T^{39} + p^{120} T^{40} \) | |
| 31 | \( 1 - 9005172860 T^{2} + 1326535283941524394 p T^{4} - \)\(12\!\cdots\!76\)\( T^{6} + \)\(30\!\cdots\!57\)\( T^{8} - \)\(58\!\cdots\!68\)\( T^{10} + \)\(94\!\cdots\!40\)\( T^{12} - \)\(13\!\cdots\!96\)\( T^{14} + \)\(15\!\cdots\!54\)\( T^{16} - \)\(17\!\cdots\!48\)\( T^{18} + \)\(16\!\cdots\!00\)\( T^{20} - \)\(17\!\cdots\!48\)\( p^{12} T^{22} + \)\(15\!\cdots\!54\)\( p^{24} T^{24} - \)\(13\!\cdots\!96\)\( p^{36} T^{26} + \)\(94\!\cdots\!40\)\( p^{48} T^{28} - \)\(58\!\cdots\!68\)\( p^{60} T^{30} + \)\(30\!\cdots\!57\)\( p^{72} T^{32} - \)\(12\!\cdots\!76\)\( p^{84} T^{34} + 1326535283941524394 p^{97} T^{36} - 9005172860 p^{108} T^{38} + p^{120} T^{40} \) | |
| 37 | \( 1 - 13220908844 T^{2} + \)\(11\!\cdots\!34\)\( T^{4} - \)\(79\!\cdots\!96\)\( T^{6} + \)\(44\!\cdots\!49\)\( T^{8} - \)\(20\!\cdots\!24\)\( T^{10} + \)\(85\!\cdots\!28\)\( T^{12} - \)\(31\!\cdots\!00\)\( T^{14} + \)\(10\!\cdots\!38\)\( T^{16} - \)\(30\!\cdots\!40\)\( T^{18} + \)\(82\!\cdots\!32\)\( T^{20} - \)\(30\!\cdots\!40\)\( p^{12} T^{22} + \)\(10\!\cdots\!38\)\( p^{24} T^{24} - \)\(31\!\cdots\!00\)\( p^{36} T^{26} + \)\(85\!\cdots\!28\)\( p^{48} T^{28} - \)\(20\!\cdots\!24\)\( p^{60} T^{30} + \)\(44\!\cdots\!49\)\( p^{72} T^{32} - \)\(79\!\cdots\!96\)\( p^{84} T^{34} + \)\(11\!\cdots\!34\)\( p^{96} T^{36} - 13220908844 p^{108} T^{38} + p^{120} T^{40} \) | |
| 41 | \( 1 - 109206 T + 30888917557 T^{2} - 2939126682958470 T^{3} + \)\(43\!\cdots\!74\)\( T^{4} - \)\(35\!\cdots\!66\)\( T^{5} + \)\(38\!\cdots\!11\)\( T^{6} - \)\(26\!\cdots\!94\)\( T^{7} + \)\(25\!\cdots\!54\)\( T^{8} - \)\(15\!\cdots\!94\)\( T^{9} + \)\(15\!\cdots\!59\)\( T^{10} - \)\(96\!\cdots\!34\)\( T^{11} + \)\(93\!\cdots\!32\)\( T^{12} - \)\(57\!\cdots\!90\)\( T^{13} + \)\(53\!\cdots\!71\)\( T^{14} - \)\(31\!\cdots\!14\)\( T^{15} + \)\(28\!\cdots\!53\)\( T^{16} - \)\(17\!\cdots\!44\)\( T^{17} + \)\(15\!\cdots\!90\)\( T^{18} - \)\(91\!\cdots\!40\)\( T^{19} + \)\(76\!\cdots\!76\)\( T^{20} - \)\(91\!\cdots\!40\)\( p^{6} T^{21} + \)\(15\!\cdots\!90\)\( p^{12} T^{22} - \)\(17\!\cdots\!44\)\( p^{18} T^{23} + \)\(28\!\cdots\!53\)\( p^{24} T^{24} - \)\(31\!\cdots\!14\)\( p^{30} T^{25} + \)\(53\!\cdots\!71\)\( p^{36} T^{26} - \)\(57\!\cdots\!90\)\( p^{42} T^{27} + \)\(93\!\cdots\!32\)\( p^{48} T^{28} - \)\(96\!\cdots\!34\)\( p^{54} T^{29} + \)\(15\!\cdots\!59\)\( p^{60} T^{30} - \)\(15\!\cdots\!94\)\( p^{66} T^{31} + \)\(25\!\cdots\!54\)\( p^{72} T^{32} - \)\(26\!\cdots\!94\)\( p^{78} T^{33} + \)\(38\!\cdots\!11\)\( p^{84} T^{34} - \)\(35\!\cdots\!66\)\( p^{90} T^{35} + \)\(43\!\cdots\!74\)\( p^{96} T^{36} - 2939126682958470 p^{102} T^{37} + 30888917557 p^{108} T^{38} - 109206 p^{114} T^{39} + p^{120} T^{40} \) | |
| 43 | \( 1 - 110740 T - 17106204822 T^{2} + 2237380127564168 T^{3} + \)\(20\!\cdots\!00\)\( T^{4} - \)\(31\!\cdots\!08\)\( T^{5} - \)\(52\!\cdots\!76\)\( T^{6} + \)\(23\!\cdots\!48\)\( T^{7} - \)\(11\!\cdots\!19\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{9} + \)\(23\!\cdots\!88\)\( T^{10} - \)\(55\!\cdots\!44\)\( T^{11} - \)\(19\!\cdots\!50\)\( T^{12} + \)\(11\!\cdots\!52\)\( T^{13} + \)\(95\!\cdots\!78\)\( T^{14} - \)\(11\!\cdots\!00\)\( T^{15} + \)\(17\!\cdots\!45\)\( T^{16} + \)\(60\!\cdots\!92\)\( T^{17} - \)\(61\!\cdots\!08\)\( T^{18} - \)\(16\!\cdots\!88\)\( T^{19} + \)\(55\!\cdots\!26\)\( T^{20} - \)\(16\!\cdots\!88\)\( p^{6} T^{21} - \)\(61\!\cdots\!08\)\( p^{12} T^{22} + \)\(60\!\cdots\!92\)\( p^{18} T^{23} + \)\(17\!\cdots\!45\)\( p^{24} T^{24} - \)\(11\!\cdots\!00\)\( p^{30} T^{25} + \)\(95\!\cdots\!78\)\( p^{36} T^{26} + \)\(11\!\cdots\!52\)\( p^{42} T^{27} - \)\(19\!\cdots\!50\)\( p^{48} T^{28} - \)\(55\!\cdots\!44\)\( p^{54} T^{29} + \)\(23\!\cdots\!88\)\( p^{60} T^{30} - \)\(10\!\cdots\!96\)\( p^{66} T^{31} - \)\(11\!\cdots\!19\)\( p^{72} T^{32} + \)\(23\!\cdots\!48\)\( p^{78} T^{33} - \)\(52\!\cdots\!76\)\( p^{84} T^{34} - \)\(31\!\cdots\!08\)\( p^{90} T^{35} + \)\(20\!\cdots\!00\)\( p^{96} T^{36} + 2237380127564168 p^{102} T^{37} - 17106204822 p^{108} T^{38} - 110740 p^{114} T^{39} + p^{120} T^{40} \) | |
| 47 | \( 1 - 107080 T - 53747106506 T^{2} + 2233149505966232 T^{3} + \)\(18\!\cdots\!76\)\( T^{4} + \)\(13\!\cdots\!12\)\( T^{5} - \)\(39\!\cdots\!96\)\( T^{6} - \)\(22\!\cdots\!68\)\( T^{7} + \)\(11\!\cdots\!27\)\( p T^{8} + \)\(57\!\cdots\!76\)\( T^{9} - \)\(31\!\cdots\!96\)\( T^{10} - \)\(78\!\cdots\!48\)\( T^{11} - \)\(20\!\cdots\!46\)\( T^{12} + \)\(35\!\cdots\!16\)\( T^{13} + \)\(53\!\cdots\!30\)\( T^{14} + \)\(56\!\cdots\!92\)\( T^{15} + \)\(41\!\cdots\!13\)\( T^{16} - \)\(11\!\cdots\!84\)\( T^{17} - \)\(12\!\cdots\!48\)\( T^{18} + \)\(62\!\cdots\!32\)\( T^{19} + \)\(20\!\cdots\!18\)\( T^{20} + \)\(62\!\cdots\!32\)\( p^{6} T^{21} - \)\(12\!\cdots\!48\)\( p^{12} T^{22} - \)\(11\!\cdots\!84\)\( p^{18} T^{23} + \)\(41\!\cdots\!13\)\( p^{24} T^{24} + \)\(56\!\cdots\!92\)\( p^{30} T^{25} + \)\(53\!\cdots\!30\)\( p^{36} T^{26} + \)\(35\!\cdots\!16\)\( p^{42} T^{27} - \)\(20\!\cdots\!46\)\( p^{48} T^{28} - \)\(78\!\cdots\!48\)\( p^{54} T^{29} - \)\(31\!\cdots\!96\)\( p^{60} T^{30} + \)\(57\!\cdots\!76\)\( p^{66} T^{31} + \)\(11\!\cdots\!27\)\( p^{73} T^{32} - \)\(22\!\cdots\!68\)\( p^{78} T^{33} - \)\(39\!\cdots\!96\)\( p^{84} T^{34} + \)\(13\!\cdots\!12\)\( p^{90} T^{35} + \)\(18\!\cdots\!76\)\( p^{96} T^{36} + 2233149505966232 p^{102} T^{37} - 53747106506 p^{108} T^{38} - 107080 p^{114} T^{39} + p^{120} T^{40} \) | |
| 53 | \( 1 - 254796 T + 140839223812 T^{2} - 30371400361152240 T^{3} + \)\(94\!\cdots\!38\)\( T^{4} - \)\(16\!\cdots\!36\)\( T^{5} + \)\(37\!\cdots\!60\)\( T^{6} - \)\(50\!\cdots\!92\)\( T^{7} + \)\(17\!\cdots\!83\)\( p T^{8} - \)\(88\!\cdots\!96\)\( T^{9} + \)\(13\!\cdots\!96\)\( T^{10} - \)\(44\!\cdots\!56\)\( T^{11} + \)\(72\!\cdots\!16\)\( T^{12} + \)\(19\!\cdots\!04\)\( T^{13} - \)\(14\!\cdots\!68\)\( T^{14} + \)\(87\!\cdots\!56\)\( T^{15} - \)\(85\!\cdots\!83\)\( T^{16} + \)\(30\!\cdots\!24\)\( T^{17} - \)\(34\!\cdots\!40\)\( T^{18} + \)\(92\!\cdots\!60\)\( T^{19} - \)\(96\!\cdots\!94\)\( T^{20} + \)\(92\!\cdots\!60\)\( p^{6} T^{21} - \)\(34\!\cdots\!40\)\( p^{12} T^{22} + \)\(30\!\cdots\!24\)\( p^{18} T^{23} - \)\(85\!\cdots\!83\)\( p^{24} T^{24} + \)\(87\!\cdots\!56\)\( p^{30} T^{25} - \)\(14\!\cdots\!68\)\( p^{36} T^{26} + \)\(19\!\cdots\!04\)\( p^{42} T^{27} + \)\(72\!\cdots\!16\)\( p^{48} T^{28} - \)\(44\!\cdots\!56\)\( p^{54} T^{29} + \)\(13\!\cdots\!96\)\( p^{60} T^{30} - \)\(88\!\cdots\!96\)\( p^{66} T^{31} + \)\(17\!\cdots\!83\)\( p^{73} T^{32} - \)\(50\!\cdots\!92\)\( p^{78} T^{33} + \)\(37\!\cdots\!60\)\( p^{84} T^{34} - \)\(16\!\cdots\!36\)\( p^{90} T^{35} + \)\(94\!\cdots\!38\)\( p^{96} T^{36} - 30371400361152240 p^{102} T^{37} + 140839223812 p^{108} T^{38} - 254796 p^{114} T^{39} + p^{120} T^{40} \) | |
| 59 | \( 1 + 610638 T + 430037064151 T^{2} + 186698991262300314 T^{3} + \)\(81\!\cdots\!66\)\( T^{4} + \)\(28\!\cdots\!18\)\( T^{5} + \)\(96\!\cdots\!01\)\( T^{6} + \)\(28\!\cdots\!10\)\( T^{7} + \)\(79\!\cdots\!38\)\( T^{8} + \)\(20\!\cdots\!18\)\( T^{9} + \)\(47\!\cdots\!01\)\( T^{10} + \)\(10\!\cdots\!30\)\( T^{11} + \)\(19\!\cdots\!80\)\( T^{12} + \)\(31\!\cdots\!50\)\( T^{13} + \)\(39\!\cdots\!49\)\( T^{14} + \)\(27\!\cdots\!42\)\( T^{15} - \)\(16\!\cdots\!55\)\( T^{16} - \)\(71\!\cdots\!92\)\( T^{17} - \)\(20\!\cdots\!30\)\( T^{18} - \)\(51\!\cdots\!04\)\( T^{19} - \)\(11\!\cdots\!56\)\( T^{20} - \)\(51\!\cdots\!04\)\( p^{6} T^{21} - \)\(20\!\cdots\!30\)\( p^{12} T^{22} - \)\(71\!\cdots\!92\)\( p^{18} T^{23} - \)\(16\!\cdots\!55\)\( p^{24} T^{24} + \)\(27\!\cdots\!42\)\( p^{30} T^{25} + \)\(39\!\cdots\!49\)\( p^{36} T^{26} + \)\(31\!\cdots\!50\)\( p^{42} T^{27} + \)\(19\!\cdots\!80\)\( p^{48} T^{28} + \)\(10\!\cdots\!30\)\( p^{54} T^{29} + \)\(47\!\cdots\!01\)\( p^{60} T^{30} + \)\(20\!\cdots\!18\)\( p^{66} T^{31} + \)\(79\!\cdots\!38\)\( p^{72} T^{32} + \)\(28\!\cdots\!10\)\( p^{78} T^{33} + \)\(96\!\cdots\!01\)\( p^{84} T^{34} + \)\(28\!\cdots\!18\)\( p^{90} T^{35} + \)\(81\!\cdots\!66\)\( p^{96} T^{36} + 186698991262300314 p^{102} T^{37} + 430037064151 p^{108} T^{38} + 610638 p^{114} T^{39} + p^{120} T^{40} \) | |
| 61 | \( 1 - 47864 T - 179309477916 T^{2} - 3995567212486080 T^{3} + \)\(17\!\cdots\!10\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} - \)\(84\!\cdots\!48\)\( T^{6} - \)\(72\!\cdots\!68\)\( T^{7} + \)\(37\!\cdots\!79\)\( T^{8} + \)\(65\!\cdots\!36\)\( T^{9} + \)\(23\!\cdots\!08\)\( T^{10} + \)\(16\!\cdots\!12\)\( T^{11} - \)\(16\!\cdots\!24\)\( T^{12} + \)\(43\!\cdots\!80\)\( T^{13} + \)\(47\!\cdots\!12\)\( T^{14} - \)\(14\!\cdots\!28\)\( T^{15} + \)\(36\!\cdots\!53\)\( T^{16} + \)\(11\!\cdots\!32\)\( T^{17} - \)\(10\!\cdots\!04\)\( T^{18} - \)\(27\!\cdots\!60\)\( T^{19} + \)\(66\!\cdots\!90\)\( T^{20} - \)\(27\!\cdots\!60\)\( p^{6} T^{21} - \)\(10\!\cdots\!04\)\( p^{12} T^{22} + \)\(11\!\cdots\!32\)\( p^{18} T^{23} + \)\(36\!\cdots\!53\)\( p^{24} T^{24} - \)\(14\!\cdots\!28\)\( p^{30} T^{25} + \)\(47\!\cdots\!12\)\( p^{36} T^{26} + \)\(43\!\cdots\!80\)\( p^{42} T^{27} - \)\(16\!\cdots\!24\)\( p^{48} T^{28} + \)\(16\!\cdots\!12\)\( p^{54} T^{29} + \)\(23\!\cdots\!08\)\( p^{60} T^{30} + \)\(65\!\cdots\!36\)\( p^{66} T^{31} + \)\(37\!\cdots\!79\)\( p^{72} T^{32} - \)\(72\!\cdots\!68\)\( p^{78} T^{33} - \)\(84\!\cdots\!48\)\( p^{84} T^{34} + \)\(11\!\cdots\!16\)\( p^{90} T^{35} + \)\(17\!\cdots\!10\)\( p^{96} T^{36} - 3995567212486080 p^{102} T^{37} - 179309477916 p^{108} T^{38} - 47864 p^{114} T^{39} + p^{120} T^{40} \) | |
| 67 | \( 1 + 839562 T + 845433768823 T^{2} + 512534957499624750 T^{3} + \)\(32\!\cdots\!10\)\( T^{4} + \)\(16\!\cdots\!82\)\( T^{5} + \)\(87\!\cdots\!89\)\( T^{6} + \)\(39\!\cdots\!98\)\( T^{7} + \)\(17\!\cdots\!38\)\( T^{8} + \)\(72\!\cdots\!54\)\( T^{9} + \)\(29\!\cdots\!81\)\( T^{10} + \)\(11\!\cdots\!10\)\( T^{11} + \)\(42\!\cdots\!56\)\( T^{12} + \)\(15\!\cdots\!02\)\( T^{13} + \)\(53\!\cdots\!77\)\( T^{14} + \)\(18\!\cdots\!98\)\( T^{15} + \)\(60\!\cdots\!21\)\( T^{16} + \)\(19\!\cdots\!12\)\( T^{17} + \)\(62\!\cdots\!74\)\( T^{18} + \)\(19\!\cdots\!44\)\( T^{19} + \)\(58\!\cdots\!88\)\( T^{20} + \)\(19\!\cdots\!44\)\( p^{6} T^{21} + \)\(62\!\cdots\!74\)\( p^{12} T^{22} + \)\(19\!\cdots\!12\)\( p^{18} T^{23} + \)\(60\!\cdots\!21\)\( p^{24} T^{24} + \)\(18\!\cdots\!98\)\( p^{30} T^{25} + \)\(53\!\cdots\!77\)\( p^{36} T^{26} + \)\(15\!\cdots\!02\)\( p^{42} T^{27} + \)\(42\!\cdots\!56\)\( p^{48} T^{28} + \)\(11\!\cdots\!10\)\( p^{54} T^{29} + \)\(29\!\cdots\!81\)\( p^{60} T^{30} + \)\(72\!\cdots\!54\)\( p^{66} T^{31} + \)\(17\!\cdots\!38\)\( p^{72} T^{32} + \)\(39\!\cdots\!98\)\( p^{78} T^{33} + \)\(87\!\cdots\!89\)\( p^{84} T^{34} + \)\(16\!\cdots\!82\)\( p^{90} T^{35} + \)\(32\!\cdots\!10\)\( p^{96} T^{36} + 512534957499624750 p^{102} T^{37} + 845433768823 p^{108} T^{38} + 839562 p^{114} T^{39} + p^{120} T^{40} \) | |
| 71 | \( 1 - 366660 T + 920338890550 T^{2} - 321020255123631000 T^{3} + \)\(43\!\cdots\!24\)\( T^{4} - \)\(15\!\cdots\!16\)\( T^{5} + \)\(14\!\cdots\!60\)\( T^{6} - \)\(52\!\cdots\!88\)\( T^{7} + \)\(36\!\cdots\!05\)\( T^{8} - \)\(14\!\cdots\!24\)\( T^{9} + \)\(80\!\cdots\!16\)\( T^{10} - \)\(31\!\cdots\!28\)\( T^{11} + \)\(15\!\cdots\!22\)\( T^{12} - \)\(60\!\cdots\!24\)\( T^{13} + \)\(26\!\cdots\!26\)\( T^{14} - \)\(10\!\cdots\!88\)\( T^{15} + \)\(42\!\cdots\!53\)\( T^{16} - \)\(15\!\cdots\!64\)\( T^{17} + \)\(62\!\cdots\!84\)\( T^{18} - \)\(22\!\cdots\!04\)\( T^{19} + \)\(83\!\cdots\!70\)\( T^{20} - \)\(22\!\cdots\!04\)\( p^{6} T^{21} + \)\(62\!\cdots\!84\)\( p^{12} T^{22} - \)\(15\!\cdots\!64\)\( p^{18} T^{23} + \)\(42\!\cdots\!53\)\( p^{24} T^{24} - \)\(10\!\cdots\!88\)\( p^{30} T^{25} + \)\(26\!\cdots\!26\)\( p^{36} T^{26} - \)\(60\!\cdots\!24\)\( p^{42} T^{27} + \)\(15\!\cdots\!22\)\( p^{48} T^{28} - \)\(31\!\cdots\!28\)\( p^{54} T^{29} + \)\(80\!\cdots\!16\)\( p^{60} T^{30} - \)\(14\!\cdots\!24\)\( p^{66} T^{31} + \)\(36\!\cdots\!05\)\( p^{72} T^{32} - \)\(52\!\cdots\!88\)\( p^{78} T^{33} + \)\(14\!\cdots\!60\)\( p^{84} T^{34} - \)\(15\!\cdots\!16\)\( p^{90} T^{35} + \)\(43\!\cdots\!24\)\( p^{96} T^{36} - 321020255123631000 p^{102} T^{37} + 920338890550 p^{108} T^{38} - 366660 p^{114} T^{39} + p^{120} T^{40} \) | |
| 73 | \( 1 - 854482 T - 243568471875 T^{2} + 256329443111397446 T^{3} + \)\(11\!\cdots\!98\)\( T^{4} - \)\(70\!\cdots\!62\)\( p T^{5} - \)\(41\!\cdots\!57\)\( T^{6} + \)\(37\!\cdots\!54\)\( T^{7} + \)\(10\!\cdots\!10\)\( T^{8} + \)\(15\!\cdots\!66\)\( T^{9} - \)\(18\!\cdots\!97\)\( T^{10} - \)\(72\!\cdots\!90\)\( T^{11} + \)\(19\!\cdots\!64\)\( T^{12} + \)\(18\!\cdots\!62\)\( T^{13} + \)\(27\!\cdots\!35\)\( T^{14} - \)\(31\!\cdots\!90\)\( T^{15} - \)\(69\!\cdots\!11\)\( T^{16} + \)\(53\!\cdots\!28\)\( p T^{17} + \)\(18\!\cdots\!30\)\( T^{18} - \)\(22\!\cdots\!52\)\( T^{19} - \)\(33\!\cdots\!72\)\( T^{20} - \)\(22\!\cdots\!52\)\( p^{6} T^{21} + \)\(18\!\cdots\!30\)\( p^{12} T^{22} + \)\(53\!\cdots\!28\)\( p^{19} T^{23} - \)\(69\!\cdots\!11\)\( p^{24} T^{24} - \)\(31\!\cdots\!90\)\( p^{30} T^{25} + \)\(27\!\cdots\!35\)\( p^{36} T^{26} + \)\(18\!\cdots\!62\)\( p^{42} T^{27} + \)\(19\!\cdots\!64\)\( p^{48} T^{28} - \)\(72\!\cdots\!90\)\( p^{54} T^{29} - \)\(18\!\cdots\!97\)\( p^{60} T^{30} + \)\(15\!\cdots\!66\)\( p^{66} T^{31} + \)\(10\!\cdots\!10\)\( p^{72} T^{32} + \)\(37\!\cdots\!54\)\( p^{78} T^{33} - \)\(41\!\cdots\!57\)\( p^{84} T^{34} - \)\(70\!\cdots\!62\)\( p^{91} T^{35} + \)\(11\!\cdots\!98\)\( p^{96} T^{36} + 256329443111397446 p^{102} T^{37} - 243568471875 p^{108} T^{38} - 854482 p^{114} T^{39} + p^{120} T^{40} \) | |
| 79 | \( 1 - 1718592 T + 3105179824546 T^{2} - 3644549889985182336 T^{3} + \)\(41\!\cdots\!32\)\( T^{4} - \)\(38\!\cdots\!96\)\( T^{5} + \)\(34\!\cdots\!76\)\( T^{6} - \)\(27\!\cdots\!52\)\( T^{7} + \)\(21\!\cdots\!05\)\( T^{8} - \)\(15\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!08\)\( T^{10} - \)\(69\!\cdots\!84\)\( T^{11} + \)\(44\!\cdots\!06\)\( T^{12} - \)\(27\!\cdots\!48\)\( T^{13} + \)\(16\!\cdots\!98\)\( T^{14} - \)\(92\!\cdots\!32\)\( T^{15} + \)\(51\!\cdots\!69\)\( T^{16} - \)\(27\!\cdots\!60\)\( T^{17} + \)\(14\!\cdots\!48\)\( T^{18} - \)\(75\!\cdots\!16\)\( T^{19} + \)\(37\!\cdots\!82\)\( T^{20} - \)\(75\!\cdots\!16\)\( p^{6} T^{21} + \)\(14\!\cdots\!48\)\( p^{12} T^{22} - \)\(27\!\cdots\!60\)\( p^{18} T^{23} + \)\(51\!\cdots\!69\)\( p^{24} T^{24} - \)\(92\!\cdots\!32\)\( p^{30} T^{25} + \)\(16\!\cdots\!98\)\( p^{36} T^{26} - \)\(27\!\cdots\!48\)\( p^{42} T^{27} + \)\(44\!\cdots\!06\)\( p^{48} T^{28} - \)\(69\!\cdots\!84\)\( p^{54} T^{29} + \)\(10\!\cdots\!08\)\( p^{60} T^{30} - \)\(15\!\cdots\!04\)\( p^{66} T^{31} + \)\(21\!\cdots\!05\)\( p^{72} T^{32} - \)\(27\!\cdots\!52\)\( p^{78} T^{33} + \)\(34\!\cdots\!76\)\( p^{84} T^{34} - \)\(38\!\cdots\!96\)\( p^{90} T^{35} + \)\(41\!\cdots\!32\)\( p^{96} T^{36} - 3644549889985182336 p^{102} T^{37} + 3105179824546 p^{108} T^{38} - 1718592 p^{114} T^{39} + p^{120} T^{40} \) | |
| 83 | \( ( 1 - 219806 T + 1316515143093 T^{2} - 732528330922529354 T^{3} + \)\(95\!\cdots\!96\)\( T^{4} - \)\(74\!\cdots\!10\)\( T^{5} + \)\(57\!\cdots\!07\)\( T^{6} - \)\(42\!\cdots\!38\)\( T^{7} + \)\(28\!\cdots\!95\)\( T^{8} - \)\(17\!\cdots\!84\)\( T^{9} + \)\(11\!\cdots\!76\)\( T^{10} - \)\(17\!\cdots\!84\)\( p^{6} T^{11} + \)\(28\!\cdots\!95\)\( p^{12} T^{12} - \)\(42\!\cdots\!38\)\( p^{18} T^{13} + \)\(57\!\cdots\!07\)\( p^{24} T^{14} - \)\(74\!\cdots\!10\)\( p^{30} T^{15} + \)\(95\!\cdots\!96\)\( p^{36} T^{16} - 732528330922529354 p^{42} T^{17} + 1316515143093 p^{48} T^{18} - 219806 p^{54} T^{19} + p^{60} T^{20} )^{2} \) | |
| 89 | \( 1 - 478032 T + 3097324667500 T^{2} - 1444207876143543744 T^{3} + \)\(48\!\cdots\!46\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{5} + \)\(53\!\cdots\!52\)\( T^{6} - \)\(30\!\cdots\!64\)\( T^{7} + \)\(47\!\cdots\!99\)\( T^{8} - \)\(31\!\cdots\!16\)\( T^{9} + \)\(36\!\cdots\!12\)\( T^{10} - \)\(27\!\cdots\!08\)\( T^{11} + \)\(24\!\cdots\!12\)\( T^{12} - \)\(20\!\cdots\!48\)\( T^{13} + \)\(15\!\cdots\!20\)\( T^{14} - \)\(13\!\cdots\!72\)\( T^{15} + \)\(94\!\cdots\!49\)\( T^{16} - \)\(78\!\cdots\!76\)\( T^{17} + \)\(52\!\cdots\!04\)\( T^{18} - \)\(41\!\cdots\!96\)\( T^{19} + \)\(27\!\cdots\!62\)\( T^{20} - \)\(41\!\cdots\!96\)\( p^{6} T^{21} + \)\(52\!\cdots\!04\)\( p^{12} T^{22} - \)\(78\!\cdots\!76\)\( p^{18} T^{23} + \)\(94\!\cdots\!49\)\( p^{24} T^{24} - \)\(13\!\cdots\!72\)\( p^{30} T^{25} + \)\(15\!\cdots\!20\)\( p^{36} T^{26} - \)\(20\!\cdots\!48\)\( p^{42} T^{27} + \)\(24\!\cdots\!12\)\( p^{48} T^{28} - \)\(27\!\cdots\!08\)\( p^{54} T^{29} + \)\(36\!\cdots\!12\)\( p^{60} T^{30} - \)\(31\!\cdots\!16\)\( p^{66} T^{31} + \)\(47\!\cdots\!99\)\( p^{72} T^{32} - \)\(30\!\cdots\!64\)\( p^{78} T^{33} + \)\(53\!\cdots\!52\)\( p^{84} T^{34} - \)\(24\!\cdots\!00\)\( p^{90} T^{35} + \)\(48\!\cdots\!46\)\( p^{96} T^{36} - 1444207876143543744 p^{102} T^{37} + 3097324667500 p^{108} T^{38} - 478032 p^{114} T^{39} + p^{120} T^{40} \) | |
| 97 | \( 1 + 191286 T + 2619395773405 T^{2} + 498720667048048278 T^{3} + \)\(25\!\cdots\!66\)\( T^{4} + \)\(26\!\cdots\!42\)\( T^{5} + \)\(14\!\cdots\!31\)\( T^{6} + \)\(64\!\cdots\!94\)\( T^{7} + \)\(10\!\cdots\!66\)\( T^{8} + \)\(73\!\cdots\!14\)\( T^{9} + \)\(28\!\cdots\!91\)\( T^{10} + \)\(43\!\cdots\!18\)\( T^{11} + \)\(71\!\cdots\!52\)\( T^{12} + \)\(17\!\cdots\!46\)\( T^{13} + \)\(93\!\cdots\!07\)\( T^{14} + \)\(18\!\cdots\!10\)\( T^{15} + \)\(65\!\cdots\!81\)\( T^{16} + \)\(45\!\cdots\!36\)\( T^{17} + \)\(25\!\cdots\!06\)\( T^{18} + \)\(69\!\cdots\!04\)\( T^{19} + \)\(10\!\cdots\!44\)\( T^{20} + \)\(69\!\cdots\!04\)\( p^{6} T^{21} + \)\(25\!\cdots\!06\)\( p^{12} T^{22} + \)\(45\!\cdots\!36\)\( p^{18} T^{23} + \)\(65\!\cdots\!81\)\( p^{24} T^{24} + \)\(18\!\cdots\!10\)\( p^{30} T^{25} + \)\(93\!\cdots\!07\)\( p^{36} T^{26} + \)\(17\!\cdots\!46\)\( p^{42} T^{27} + \)\(71\!\cdots\!52\)\( p^{48} T^{28} + \)\(43\!\cdots\!18\)\( p^{54} T^{29} + \)\(28\!\cdots\!91\)\( p^{60} T^{30} + \)\(73\!\cdots\!14\)\( p^{66} T^{31} + \)\(10\!\cdots\!66\)\( p^{72} T^{32} + \)\(64\!\cdots\!94\)\( p^{78} T^{33} + \)\(14\!\cdots\!31\)\( p^{84} T^{34} + \)\(26\!\cdots\!42\)\( p^{90} T^{35} + \)\(25\!\cdots\!66\)\( p^{96} T^{36} + 498720667048048278 p^{102} T^{37} + 2619395773405 p^{108} T^{38} + 191286 p^{114} T^{39} + p^{120} T^{40} \) | |
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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
−2.59829953324573887032163298306, −2.42280089764526351065859906134, −2.38912845659969279232769124854, −2.30931977752403383909887841011, −2.28940875075613432738622615698, −2.28213532720738458652423437280, −1.78164105760461402834019404856, −1.75790941902490299817907776420, −1.71372195301612806596437219452, −1.65716064563745683917984983411, −1.63025465822378329672620634505, −1.47346970664464864385399208950, −1.32106620960967366059075378720, −1.31913763456951994581828109545, −1.09090425494615279811938828701, −1.02903538685125471156043023579, −0.68318998453920798954089737268, −0.67038410425217523208772420823, −0.57185899386372993101540557460, −0.44290755579948686345639275123, −0.40063367897167111406575995821, −0.38529353738924129363794753347, −0.29728475631547845227186954640, −0.26893030853151177977659685327, −0.25244328278950112742684180105, 0.25244328278950112742684180105, 0.26893030853151177977659685327, 0.29728475631547845227186954640, 0.38529353738924129363794753347, 0.40063367897167111406575995821, 0.44290755579948686345639275123, 0.57185899386372993101540557460, 0.67038410425217523208772420823, 0.68318998453920798954089737268, 1.02903538685125471156043023579, 1.09090425494615279811938828701, 1.31913763456951994581828109545, 1.32106620960967366059075378720, 1.47346970664464864385399208950, 1.63025465822378329672620634505, 1.65716064563745683917984983411, 1.71372195301612806596437219452, 1.75790941902490299817907776420, 1.78164105760461402834019404856, 2.28213532720738458652423437280, 2.28940875075613432738622615698, 2.30931977752403383909887841011, 2.38912845659969279232769124854, 2.42280089764526351065859906134, 2.59829953324573887032163298306
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.