Dirichlet series
| L(s) = 1 | − 30·3-s + 160·4-s + 112·5-s − 208·7-s − 2.09e3·9-s − 284·11-s − 4.80e3·12-s + 1.05e4·13-s − 3.36e3·15-s + 1.02e4·16-s − 1.16e4·17-s + 1.28e4·19-s + 1.79e4·20-s + 6.24e3·21-s + 8.48e3·23-s + 5.24e4·25-s + 7.18e4·27-s − 3.32e4·28-s + 1.37e5·29-s + 8.52e3·33-s − 2.32e4·35-s − 3.35e5·36-s − 3.15e5·39-s + 1.09e5·41-s + 3.55e4·43-s − 4.54e4·44-s − 2.34e5·45-s + ⋯ |
| L(s) = 1 | − 1.11·3-s + 5/2·4-s + 0.895·5-s − 0.606·7-s − 2.87·9-s − 0.213·11-s − 2.77·12-s + 4.77·13-s − 0.995·15-s + 5/2·16-s − 2.37·17-s + 1.87·19-s + 2.23·20-s + 0.673·21-s + 0.697·23-s + 3.35·25-s + 3.65·27-s − 1.51·28-s + 5.64·29-s + 0.237·33-s − 0.543·35-s − 7.18·36-s − 5.31·39-s + 1.58·41-s + 0.447·43-s − 0.533·44-s − 2.57·45-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+3)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.79627\times 10^{18}\) |
| Root analytic conductor: | \(2.95669\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 19^{20} ,\ ( \ : [3]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(2.869517081\) |
| \(L(\frac12)\) | \(\approx\) | \(2.869517081\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{5} \) |
| 19 | \( 1 - 12862 T + 2368039 p T^{2} + 1516542790 p^{2} T^{3} - 1172927940100 p^{3} T^{4} + 10440171838462 p^{5} T^{5} + 65171394685854928 p^{5} T^{6} - 46500540537958539264 p^{6} T^{7} + \)\(45\!\cdots\!28\)\( p^{8} T^{8} + \)\(64\!\cdots\!88\)\( p^{10} T^{9} - \)\(11\!\cdots\!12\)\( p^{13} T^{10} + \)\(64\!\cdots\!88\)\( p^{16} T^{11} + \)\(45\!\cdots\!28\)\( p^{20} T^{12} - 46500540537958539264 p^{24} T^{13} + 65171394685854928 p^{29} T^{14} + 10440171838462 p^{35} T^{15} - 1172927940100 p^{39} T^{16} + 1516542790 p^{44} T^{17} + 2368039 p^{49} T^{18} - 12862 p^{54} T^{19} + p^{60} T^{20} \) | |
| good | 3 | \( 1 + 10 p T + 2995 T^{2} + 26950 p T^{3} + 3440770 T^{4} + 2399318 p^{3} T^{5} + 182576317 p^{2} T^{6} - 64941274 p^{3} T^{7} + 23415042734 p^{2} T^{8} - 180793498810 p^{4} T^{9} + 4730485399117 p^{4} T^{10} + 43769663795990 p^{5} T^{11} + 848469692794036 p^{6} T^{12} + 5615551351315762 p^{6} T^{13} + 12977183793543479 p^{7} T^{14} - 8221111962494620054 p^{7} T^{15} - \)\(57\!\cdots\!03\)\( p^{6} T^{16} - \)\(30\!\cdots\!72\)\( p^{10} T^{17} - \)\(34\!\cdots\!42\)\( p^{10} T^{18} - \)\(37\!\cdots\!00\)\( p^{9} T^{19} - \)\(17\!\cdots\!36\)\( p^{8} T^{20} - \)\(37\!\cdots\!00\)\( p^{15} T^{21} - \)\(34\!\cdots\!42\)\( p^{22} T^{22} - \)\(30\!\cdots\!72\)\( p^{28} T^{23} - \)\(57\!\cdots\!03\)\( p^{30} T^{24} - 8221111962494620054 p^{37} T^{25} + 12977183793543479 p^{43} T^{26} + 5615551351315762 p^{48} T^{27} + 848469692794036 p^{54} T^{28} + 43769663795990 p^{59} T^{29} + 4730485399117 p^{64} T^{30} - 180793498810 p^{70} T^{31} + 23415042734 p^{74} T^{32} - 64941274 p^{81} T^{33} + 182576317 p^{86} T^{34} + 2399318 p^{93} T^{35} + 3440770 p^{96} T^{36} + 26950 p^{103} T^{37} + 2995 p^{108} T^{38} + 10 p^{115} T^{39} + p^{120} T^{40} \) |
| 5 | \( 1 - 112 T - 39932 T^{2} + 12139568 T^{3} - 261095854 T^{4} - 66114885868 p T^{5} + 2355113783588 p^{2} T^{6} - 3078811891576 p^{3} T^{7} - 16675060920921 p^{7} T^{8} + 64276193367293648 p^{5} T^{9} - 154949112253089388 p^{6} T^{10} - 43058790093325199812 p^{7} T^{11} + \)\(26\!\cdots\!84\)\( p^{9} T^{12} - \)\(69\!\cdots\!44\)\( p^{9} T^{13} - \)\(65\!\cdots\!36\)\( p^{10} T^{14} + \)\(20\!\cdots\!16\)\( p^{11} T^{15} - \)\(10\!\cdots\!23\)\( p^{12} T^{16} - \)\(88\!\cdots\!44\)\( p^{13} T^{17} + \)\(21\!\cdots\!44\)\( p^{14} T^{18} + \)\(46\!\cdots\!88\)\( p^{17} T^{19} - \)\(20\!\cdots\!18\)\( p^{20} T^{20} + \)\(46\!\cdots\!88\)\( p^{23} T^{21} + \)\(21\!\cdots\!44\)\( p^{26} T^{22} - \)\(88\!\cdots\!44\)\( p^{31} T^{23} - \)\(10\!\cdots\!23\)\( p^{36} T^{24} + \)\(20\!\cdots\!16\)\( p^{41} T^{25} - \)\(65\!\cdots\!36\)\( p^{46} T^{26} - \)\(69\!\cdots\!44\)\( p^{51} T^{27} + \)\(26\!\cdots\!84\)\( p^{57} T^{28} - 43058790093325199812 p^{61} T^{29} - 154949112253089388 p^{66} T^{30} + 64276193367293648 p^{71} T^{31} - 16675060920921 p^{79} T^{32} - 3078811891576 p^{81} T^{33} + 2355113783588 p^{86} T^{34} - 66114885868 p^{91} T^{35} - 261095854 p^{96} T^{36} + 12139568 p^{102} T^{37} - 39932 p^{108} T^{38} - 112 p^{114} T^{39} + p^{120} T^{40} \) | |
| 7 | \( ( 1 + 104 T + 528718 T^{2} + 63824524 T^{3} + 145021134305 T^{4} + 3606702717916 p T^{5} + 3889487555710460 p T^{6} + 6494707722522763516 T^{7} + \)\(39\!\cdots\!02\)\( T^{8} + \)\(11\!\cdots\!52\)\( T^{9} + \)\(49\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!52\)\( p^{6} T^{11} + \)\(39\!\cdots\!02\)\( p^{12} T^{12} + 6494707722522763516 p^{18} T^{13} + 3889487555710460 p^{25} T^{14} + 3606702717916 p^{31} T^{15} + 145021134305 p^{36} T^{16} + 63824524 p^{42} T^{17} + 528718 p^{48} T^{18} + 104 p^{54} T^{19} + p^{60} T^{20} )^{2} \) | |
| 11 | \( ( 1 + 142 T + 6672981 T^{2} - 2081820698 T^{3} + 26313096523932 T^{4} - 9724295551779526 T^{5} + 81184131041911076199 T^{6} - \)\(28\!\cdots\!02\)\( T^{7} + \)\(19\!\cdots\!87\)\( T^{8} - \)\(66\!\cdots\!76\)\( T^{9} + \)\(37\!\cdots\!00\)\( T^{10} - \)\(66\!\cdots\!76\)\( p^{6} T^{11} + \)\(19\!\cdots\!87\)\( p^{12} T^{12} - \)\(28\!\cdots\!02\)\( p^{18} T^{13} + 81184131041911076199 p^{24} T^{14} - 9724295551779526 p^{30} T^{15} + 26313096523932 p^{36} T^{16} - 2081820698 p^{42} T^{17} + 6672981 p^{48} T^{18} + 142 p^{54} T^{19} + p^{60} T^{20} )^{2} \) | |
| 13 | \( 1 - 10500 T + 68527912 T^{2} - 333668076000 T^{3} + 576755510554 p^{3} T^{4} - 3930882697800669636 T^{5} + \)\(10\!\cdots\!40\)\( T^{6} - \)\(22\!\cdots\!96\)\( T^{7} + \)\(46\!\cdots\!91\)\( T^{8} - \)\(10\!\cdots\!36\)\( T^{9} + \)\(29\!\cdots\!24\)\( T^{10} - \)\(87\!\cdots\!44\)\( T^{11} + \)\(24\!\cdots\!84\)\( T^{12} - \)\(57\!\cdots\!48\)\( T^{13} + \)\(11\!\cdots\!08\)\( T^{14} - \)\(18\!\cdots\!48\)\( T^{15} + \)\(30\!\cdots\!25\)\( T^{16} - \)\(67\!\cdots\!68\)\( T^{17} + \)\(20\!\cdots\!64\)\( T^{18} - \)\(61\!\cdots\!40\)\( T^{19} + \)\(15\!\cdots\!18\)\( T^{20} - \)\(61\!\cdots\!40\)\( p^{6} T^{21} + \)\(20\!\cdots\!64\)\( p^{12} T^{22} - \)\(67\!\cdots\!68\)\( p^{18} T^{23} + \)\(30\!\cdots\!25\)\( p^{24} T^{24} - \)\(18\!\cdots\!48\)\( p^{30} T^{25} + \)\(11\!\cdots\!08\)\( p^{36} T^{26} - \)\(57\!\cdots\!48\)\( p^{42} T^{27} + \)\(24\!\cdots\!84\)\( p^{48} T^{28} - \)\(87\!\cdots\!44\)\( p^{54} T^{29} + \)\(29\!\cdots\!24\)\( p^{60} T^{30} - \)\(10\!\cdots\!36\)\( p^{66} T^{31} + \)\(46\!\cdots\!91\)\( p^{72} T^{32} - \)\(22\!\cdots\!96\)\( p^{78} T^{33} + \)\(10\!\cdots\!40\)\( p^{84} T^{34} - 3930882697800669636 p^{90} T^{35} + 576755510554 p^{99} T^{36} - 333668076000 p^{102} T^{37} + 68527912 p^{108} T^{38} - 10500 p^{114} T^{39} + p^{120} T^{40} \) | |
| 17 | \( 1 + 11684 T - 71315420 T^{2} - 870064891288 T^{3} + 7075766595654050 T^{4} + 47867735906434693588 T^{5} - \)\(44\!\cdots\!52\)\( T^{6} - \)\(11\!\cdots\!56\)\( T^{7} + \)\(23\!\cdots\!67\)\( T^{8} - \)\(37\!\cdots\!12\)\( T^{9} - \)\(83\!\cdots\!16\)\( T^{10} + \)\(24\!\cdots\!40\)\( T^{11} + \)\(21\!\cdots\!40\)\( T^{12} - \)\(76\!\cdots\!80\)\( p T^{13} - \)\(24\!\cdots\!08\)\( T^{14} + \)\(44\!\cdots\!36\)\( T^{15} - \)\(58\!\cdots\!23\)\( T^{16} - \)\(94\!\cdots\!00\)\( T^{17} + \)\(45\!\cdots\!68\)\( T^{18} + \)\(96\!\cdots\!08\)\( T^{19} - \)\(13\!\cdots\!70\)\( T^{20} + \)\(96\!\cdots\!08\)\( p^{6} T^{21} + \)\(45\!\cdots\!68\)\( p^{12} T^{22} - \)\(94\!\cdots\!00\)\( p^{18} T^{23} - \)\(58\!\cdots\!23\)\( p^{24} T^{24} + \)\(44\!\cdots\!36\)\( p^{30} T^{25} - \)\(24\!\cdots\!08\)\( p^{36} T^{26} - \)\(76\!\cdots\!80\)\( p^{43} T^{27} + \)\(21\!\cdots\!40\)\( p^{48} T^{28} + \)\(24\!\cdots\!40\)\( p^{54} T^{29} - \)\(83\!\cdots\!16\)\( p^{60} T^{30} - \)\(37\!\cdots\!12\)\( p^{66} T^{31} + \)\(23\!\cdots\!67\)\( p^{72} T^{32} - \)\(11\!\cdots\!56\)\( p^{78} T^{33} - \)\(44\!\cdots\!52\)\( p^{84} T^{34} + 47867735906434693588 p^{90} T^{35} + 7075766595654050 p^{96} T^{36} - 870064891288 p^{102} T^{37} - 71315420 p^{108} T^{38} + 11684 p^{114} T^{39} + p^{120} T^{40} \) | |
| 23 | \( 1 - 8488 T - 597873998 T^{2} + 5829963927344 T^{3} + 140125585271312960 T^{4} - 68442974964261524656 p T^{5} - \)\(16\!\cdots\!88\)\( T^{6} + \)\(21\!\cdots\!52\)\( T^{7} + \)\(19\!\cdots\!33\)\( T^{8} - \)\(15\!\cdots\!40\)\( T^{9} - \)\(40\!\cdots\!36\)\( T^{10} + \)\(57\!\cdots\!96\)\( T^{11} + \)\(86\!\cdots\!02\)\( T^{12} + \)\(39\!\cdots\!84\)\( T^{13} - \)\(16\!\cdots\!26\)\( T^{14} + \)\(21\!\cdots\!72\)\( T^{15} + \)\(29\!\cdots\!33\)\( T^{16} - \)\(11\!\cdots\!96\)\( T^{17} - \)\(38\!\cdots\!52\)\( T^{18} + \)\(49\!\cdots\!36\)\( p T^{19} + \)\(88\!\cdots\!62\)\( p^{2} T^{20} + \)\(49\!\cdots\!36\)\( p^{7} T^{21} - \)\(38\!\cdots\!52\)\( p^{12} T^{22} - \)\(11\!\cdots\!96\)\( p^{18} T^{23} + \)\(29\!\cdots\!33\)\( p^{24} T^{24} + \)\(21\!\cdots\!72\)\( p^{30} T^{25} - \)\(16\!\cdots\!26\)\( p^{36} T^{26} + \)\(39\!\cdots\!84\)\( p^{42} T^{27} + \)\(86\!\cdots\!02\)\( p^{48} T^{28} + \)\(57\!\cdots\!96\)\( p^{54} T^{29} - \)\(40\!\cdots\!36\)\( p^{60} T^{30} - \)\(15\!\cdots\!40\)\( p^{66} T^{31} + \)\(19\!\cdots\!33\)\( p^{72} T^{32} + \)\(21\!\cdots\!52\)\( p^{78} T^{33} - \)\(16\!\cdots\!88\)\( p^{84} T^{34} - 68442974964261524656 p^{91} T^{35} + 140125585271312960 p^{96} T^{36} + 5829963927344 p^{102} T^{37} - 597873998 p^{108} T^{38} - 8488 p^{114} T^{39} + p^{120} T^{40} \) | |
| 29 | \( 1 - 137760 T + 12416969908 T^{2} - 839100390334080 T^{3} + 46994721297201913834 T^{4} - \)\(22\!\cdots\!96\)\( T^{5} + \)\(96\!\cdots\!92\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(13\!\cdots\!91\)\( T^{8} - \)\(44\!\cdots\!44\)\( T^{9} + \)\(14\!\cdots\!24\)\( T^{10} - \)\(14\!\cdots\!20\)\( p T^{11} + \)\(12\!\cdots\!04\)\( T^{12} - \)\(36\!\cdots\!00\)\( T^{13} + \)\(10\!\cdots\!44\)\( T^{14} - \)\(28\!\cdots\!84\)\( T^{15} + \)\(75\!\cdots\!81\)\( T^{16} - \)\(67\!\cdots\!44\)\( p T^{17} + \)\(50\!\cdots\!48\)\( T^{18} - \)\(12\!\cdots\!88\)\( T^{19} + \)\(30\!\cdots\!18\)\( T^{20} - \)\(12\!\cdots\!88\)\( p^{6} T^{21} + \)\(50\!\cdots\!48\)\( p^{12} T^{22} - \)\(67\!\cdots\!44\)\( p^{19} T^{23} + \)\(75\!\cdots\!81\)\( p^{24} T^{24} - \)\(28\!\cdots\!84\)\( p^{30} T^{25} + \)\(10\!\cdots\!44\)\( p^{36} T^{26} - \)\(36\!\cdots\!00\)\( p^{42} T^{27} + \)\(12\!\cdots\!04\)\( p^{48} T^{28} - \)\(14\!\cdots\!20\)\( p^{55} T^{29} + \)\(14\!\cdots\!24\)\( p^{60} T^{30} - \)\(44\!\cdots\!44\)\( p^{66} T^{31} + \)\(13\!\cdots\!91\)\( p^{72} T^{32} - \)\(37\!\cdots\!28\)\( p^{78} T^{33} + \)\(96\!\cdots\!92\)\( p^{84} T^{34} - \)\(22\!\cdots\!96\)\( p^{90} T^{35} + 46994721297201913834 p^{96} T^{36} - 839100390334080 p^{102} T^{37} + 12416969908 p^{108} T^{38} - 137760 p^{114} T^{39} + p^{120} T^{40} \) | |
| 31 | \( 1 - 11009634428 T^{2} + 59381383689340146838 T^{4} - \)\(20\!\cdots\!08\)\( T^{6} + \)\(54\!\cdots\!33\)\( T^{8} - \)\(11\!\cdots\!64\)\( T^{10} + \)\(18\!\cdots\!64\)\( T^{12} - \)\(26\!\cdots\!92\)\( T^{14} + \)\(32\!\cdots\!46\)\( T^{16} - \)\(34\!\cdots\!60\)\( T^{18} + \)\(33\!\cdots\!56\)\( T^{20} - \)\(34\!\cdots\!60\)\( p^{12} T^{22} + \)\(32\!\cdots\!46\)\( p^{24} T^{24} - \)\(26\!\cdots\!92\)\( p^{36} T^{26} + \)\(18\!\cdots\!64\)\( p^{48} T^{28} - \)\(11\!\cdots\!64\)\( p^{60} T^{30} + \)\(54\!\cdots\!33\)\( p^{72} T^{32} - \)\(20\!\cdots\!08\)\( p^{84} T^{34} + 59381383689340146838 p^{96} T^{36} - 11009634428 p^{108} T^{38} + p^{120} T^{40} \) | |
| 37 | \( 1 - 28489653836 T^{2} + \)\(40\!\cdots\!66\)\( T^{4} - \)\(38\!\cdots\!64\)\( T^{6} + \)\(27\!\cdots\!33\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{10} + \)\(72\!\cdots\!40\)\( T^{12} - \)\(28\!\cdots\!52\)\( T^{14} + \)\(99\!\cdots\!62\)\( T^{16} - \)\(30\!\cdots\!24\)\( T^{18} + \)\(82\!\cdots\!40\)\( T^{20} - \)\(30\!\cdots\!24\)\( p^{12} T^{22} + \)\(99\!\cdots\!62\)\( p^{24} T^{24} - \)\(28\!\cdots\!52\)\( p^{36} T^{26} + \)\(72\!\cdots\!40\)\( p^{48} T^{28} - \)\(15\!\cdots\!80\)\( p^{60} T^{30} + \)\(27\!\cdots\!33\)\( p^{72} T^{32} - \)\(38\!\cdots\!64\)\( p^{84} T^{34} + \)\(40\!\cdots\!66\)\( p^{96} T^{36} - 28489653836 p^{108} T^{38} + p^{120} T^{40} \) | |
| 41 | \( 1 - 109206 T + 29669841493 T^{2} - 2805996262313286 T^{3} + \)\(43\!\cdots\!78\)\( T^{4} - \)\(38\!\cdots\!86\)\( T^{5} + \)\(42\!\cdots\!15\)\( T^{6} - \)\(89\!\cdots\!98\)\( p T^{7} + \)\(18\!\cdots\!50\)\( p^{2} T^{8} - \)\(26\!\cdots\!74\)\( T^{9} + \)\(18\!\cdots\!15\)\( T^{10} - \)\(14\!\cdots\!62\)\( T^{11} + \)\(87\!\cdots\!72\)\( T^{12} - \)\(63\!\cdots\!22\)\( T^{13} + \)\(33\!\cdots\!59\)\( T^{14} - \)\(20\!\cdots\!66\)\( T^{15} + \)\(98\!\cdots\!69\)\( T^{16} - \)\(38\!\cdots\!88\)\( T^{17} + \)\(23\!\cdots\!22\)\( T^{18} + \)\(73\!\cdots\!52\)\( T^{19} + \)\(69\!\cdots\!40\)\( T^{20} + \)\(73\!\cdots\!52\)\( p^{6} T^{21} + \)\(23\!\cdots\!22\)\( p^{12} T^{22} - \)\(38\!\cdots\!88\)\( p^{18} T^{23} + \)\(98\!\cdots\!69\)\( p^{24} T^{24} - \)\(20\!\cdots\!66\)\( p^{30} T^{25} + \)\(33\!\cdots\!59\)\( p^{36} T^{26} - \)\(63\!\cdots\!22\)\( p^{42} T^{27} + \)\(87\!\cdots\!72\)\( p^{48} T^{28} - \)\(14\!\cdots\!62\)\( p^{54} T^{29} + \)\(18\!\cdots\!15\)\( p^{60} T^{30} - \)\(26\!\cdots\!74\)\( p^{66} T^{31} + \)\(18\!\cdots\!50\)\( p^{74} T^{32} - \)\(89\!\cdots\!98\)\( p^{79} T^{33} + \)\(42\!\cdots\!15\)\( p^{84} T^{34} - \)\(38\!\cdots\!86\)\( p^{90} T^{35} + \)\(43\!\cdots\!78\)\( p^{96} T^{36} - 2805996262313286 p^{102} T^{37} + 29669841493 p^{108} T^{38} - 109206 p^{114} T^{39} + p^{120} T^{40} \) | |
| 43 | \( 1 - 35572 T - 44740981446 T^{2} + 2696909230114472 T^{3} + \)\(10\!\cdots\!92\)\( T^{4} - \)\(84\!\cdots\!84\)\( T^{5} - \)\(16\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!68\)\( T^{7} + \)\(17\!\cdots\!85\)\( T^{8} - \)\(23\!\cdots\!68\)\( T^{9} - \)\(13\!\cdots\!24\)\( T^{10} + \)\(26\!\cdots\!84\)\( T^{11} + \)\(57\!\cdots\!22\)\( T^{12} - \)\(22\!\cdots\!96\)\( T^{13} + \)\(78\!\cdots\!14\)\( T^{14} + \)\(14\!\cdots\!24\)\( T^{15} - \)\(43\!\cdots\!91\)\( T^{16} - \)\(71\!\cdots\!96\)\( T^{17} + \)\(48\!\cdots\!64\)\( T^{18} + \)\(17\!\cdots\!68\)\( T^{19} - \)\(35\!\cdots\!30\)\( T^{20} + \)\(17\!\cdots\!68\)\( p^{6} T^{21} + \)\(48\!\cdots\!64\)\( p^{12} T^{22} - \)\(71\!\cdots\!96\)\( p^{18} T^{23} - \)\(43\!\cdots\!91\)\( p^{24} T^{24} + \)\(14\!\cdots\!24\)\( p^{30} T^{25} + \)\(78\!\cdots\!14\)\( p^{36} T^{26} - \)\(22\!\cdots\!96\)\( p^{42} T^{27} + \)\(57\!\cdots\!22\)\( p^{48} T^{28} + \)\(26\!\cdots\!84\)\( p^{54} T^{29} - \)\(13\!\cdots\!24\)\( p^{60} T^{30} - \)\(23\!\cdots\!68\)\( p^{66} T^{31} + \)\(17\!\cdots\!85\)\( p^{72} T^{32} + \)\(16\!\cdots\!68\)\( p^{78} T^{33} - \)\(16\!\cdots\!40\)\( p^{84} T^{34} - \)\(84\!\cdots\!84\)\( p^{90} T^{35} + \)\(10\!\cdots\!92\)\( p^{96} T^{36} + 2696909230114472 p^{102} T^{37} - 44740981446 p^{108} T^{38} - 35572 p^{114} T^{39} + p^{120} T^{40} \) | |
| 47 | \( 1 + 361184 T - 3537913970 T^{2} - 14170984074593752 T^{3} - \)\(36\!\cdots\!72\)\( T^{4} + \)\(35\!\cdots\!36\)\( T^{5} + \)\(64\!\cdots\!52\)\( T^{6} - \)\(75\!\cdots\!92\)\( T^{7} + \)\(31\!\cdots\!53\)\( T^{8} + \)\(29\!\cdots\!48\)\( p T^{9} - \)\(39\!\cdots\!36\)\( T^{10} - \)\(21\!\cdots\!52\)\( T^{11} + \)\(11\!\cdots\!22\)\( T^{12} + \)\(28\!\cdots\!64\)\( T^{13} - \)\(24\!\cdots\!14\)\( T^{14} - \)\(29\!\cdots\!72\)\( T^{15} + \)\(40\!\cdots\!41\)\( T^{16} + \)\(23\!\cdots\!32\)\( T^{17} - \)\(55\!\cdots\!52\)\( T^{18} - \)\(42\!\cdots\!20\)\( p^{2} T^{19} + \)\(13\!\cdots\!34\)\( p^{4} T^{20} - \)\(42\!\cdots\!20\)\( p^{8} T^{21} - \)\(55\!\cdots\!52\)\( p^{12} T^{22} + \)\(23\!\cdots\!32\)\( p^{18} T^{23} + \)\(40\!\cdots\!41\)\( p^{24} T^{24} - \)\(29\!\cdots\!72\)\( p^{30} T^{25} - \)\(24\!\cdots\!14\)\( p^{36} T^{26} + \)\(28\!\cdots\!64\)\( p^{42} T^{27} + \)\(11\!\cdots\!22\)\( p^{48} T^{28} - \)\(21\!\cdots\!52\)\( p^{54} T^{29} - \)\(39\!\cdots\!36\)\( p^{60} T^{30} + \)\(29\!\cdots\!48\)\( p^{67} T^{31} + \)\(31\!\cdots\!53\)\( p^{72} T^{32} - \)\(75\!\cdots\!92\)\( p^{78} T^{33} + \)\(64\!\cdots\!52\)\( p^{84} T^{34} + \)\(35\!\cdots\!36\)\( p^{90} T^{35} - \)\(36\!\cdots\!72\)\( p^{96} T^{36} - 14170984074593752 p^{102} T^{37} - 3537913970 p^{108} T^{38} + 361184 p^{114} T^{39} + p^{120} T^{40} \) | |
| 53 | \( 1 + 236172 T + 169761171292 T^{2} + 35701829984187408 T^{3} + \)\(14\!\cdots\!78\)\( T^{4} + \)\(28\!\cdots\!52\)\( T^{5} + \)\(83\!\cdots\!96\)\( T^{6} + \)\(15\!\cdots\!48\)\( T^{7} + \)\(36\!\cdots\!11\)\( T^{8} + \)\(62\!\cdots\!80\)\( T^{9} + \)\(13\!\cdots\!44\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{11} + \)\(38\!\cdots\!44\)\( T^{12} + \)\(53\!\cdots\!00\)\( T^{13} + \)\(91\!\cdots\!84\)\( T^{14} + \)\(11\!\cdots\!96\)\( T^{15} + \)\(18\!\cdots\!73\)\( T^{16} + \)\(22\!\cdots\!00\)\( T^{17} + \)\(36\!\cdots\!12\)\( T^{18} + \)\(15\!\cdots\!32\)\( p^{2} T^{19} + \)\(95\!\cdots\!54\)\( p^{4} T^{20} + \)\(15\!\cdots\!32\)\( p^{8} T^{21} + \)\(36\!\cdots\!12\)\( p^{12} T^{22} + \)\(22\!\cdots\!00\)\( p^{18} T^{23} + \)\(18\!\cdots\!73\)\( p^{24} T^{24} + \)\(11\!\cdots\!96\)\( p^{30} T^{25} + \)\(91\!\cdots\!84\)\( p^{36} T^{26} + \)\(53\!\cdots\!00\)\( p^{42} T^{27} + \)\(38\!\cdots\!44\)\( p^{48} T^{28} + \)\(20\!\cdots\!56\)\( p^{54} T^{29} + \)\(13\!\cdots\!44\)\( p^{60} T^{30} + \)\(62\!\cdots\!80\)\( p^{66} T^{31} + \)\(36\!\cdots\!11\)\( p^{72} T^{32} + \)\(15\!\cdots\!48\)\( p^{78} T^{33} + \)\(83\!\cdots\!96\)\( p^{84} T^{34} + \)\(28\!\cdots\!52\)\( p^{90} T^{35} + \)\(14\!\cdots\!78\)\( p^{96} T^{36} + 35701829984187408 p^{102} T^{37} + 169761171292 p^{108} T^{38} + 236172 p^{114} T^{39} + p^{120} T^{40} \) | |
| 59 | \( 1 - 1310610 T + 1063192429015 T^{2} - 643019654198022150 T^{3} + \)\(31\!\cdots\!98\)\( T^{4} - \)\(13\!\cdots\!62\)\( T^{5} + \)\(50\!\cdots\!37\)\( T^{6} - \)\(17\!\cdots\!46\)\( T^{7} + \)\(54\!\cdots\!94\)\( T^{8} - \)\(16\!\cdots\!66\)\( T^{9} + \)\(44\!\cdots\!09\)\( T^{10} - \)\(11\!\cdots\!74\)\( T^{11} + \)\(29\!\cdots\!72\)\( T^{12} - \)\(71\!\cdots\!14\)\( T^{13} + \)\(16\!\cdots\!25\)\( T^{14} - \)\(37\!\cdots\!22\)\( T^{15} + \)\(82\!\cdots\!97\)\( T^{16} - \)\(17\!\cdots\!04\)\( T^{17} + \)\(37\!\cdots\!54\)\( T^{18} - \)\(78\!\cdots\!16\)\( T^{19} + \)\(16\!\cdots\!24\)\( T^{20} - \)\(78\!\cdots\!16\)\( p^{6} T^{21} + \)\(37\!\cdots\!54\)\( p^{12} T^{22} - \)\(17\!\cdots\!04\)\( p^{18} T^{23} + \)\(82\!\cdots\!97\)\( p^{24} T^{24} - \)\(37\!\cdots\!22\)\( p^{30} T^{25} + \)\(16\!\cdots\!25\)\( p^{36} T^{26} - \)\(71\!\cdots\!14\)\( p^{42} T^{27} + \)\(29\!\cdots\!72\)\( p^{48} T^{28} - \)\(11\!\cdots\!74\)\( p^{54} T^{29} + \)\(44\!\cdots\!09\)\( p^{60} T^{30} - \)\(16\!\cdots\!66\)\( p^{66} T^{31} + \)\(54\!\cdots\!94\)\( p^{72} T^{32} - \)\(17\!\cdots\!46\)\( p^{78} T^{33} + \)\(50\!\cdots\!37\)\( p^{84} T^{34} - \)\(13\!\cdots\!62\)\( p^{90} T^{35} + \)\(31\!\cdots\!98\)\( p^{96} T^{36} - 643019654198022150 p^{102} T^{37} + 1063192429015 p^{108} T^{38} - 1310610 p^{114} T^{39} + p^{120} T^{40} \) | |
| 61 | \( 1 - 83552 T - 268922049252 T^{2} + 9127153116882912 T^{3} + \)\(35\!\cdots\!34\)\( T^{4} + \)\(60\!\cdots\!24\)\( T^{5} - \)\(31\!\cdots\!16\)\( T^{6} - \)\(19\!\cdots\!60\)\( T^{7} + \)\(21\!\cdots\!83\)\( T^{8} + \)\(21\!\cdots\!44\)\( T^{9} - \)\(12\!\cdots\!52\)\( T^{10} - \)\(17\!\cdots\!72\)\( T^{11} + \)\(67\!\cdots\!52\)\( T^{12} + \)\(12\!\cdots\!76\)\( T^{13} - \)\(29\!\cdots\!56\)\( T^{14} - \)\(64\!\cdots\!44\)\( T^{15} + \)\(10\!\cdots\!85\)\( T^{16} + \)\(25\!\cdots\!52\)\( T^{17} - \)\(40\!\cdots\!64\)\( T^{18} - \)\(47\!\cdots\!36\)\( T^{19} + \)\(18\!\cdots\!22\)\( T^{20} - \)\(47\!\cdots\!36\)\( p^{6} T^{21} - \)\(40\!\cdots\!64\)\( p^{12} T^{22} + \)\(25\!\cdots\!52\)\( p^{18} T^{23} + \)\(10\!\cdots\!85\)\( p^{24} T^{24} - \)\(64\!\cdots\!44\)\( p^{30} T^{25} - \)\(29\!\cdots\!56\)\( p^{36} T^{26} + \)\(12\!\cdots\!76\)\( p^{42} T^{27} + \)\(67\!\cdots\!52\)\( p^{48} T^{28} - \)\(17\!\cdots\!72\)\( p^{54} T^{29} - \)\(12\!\cdots\!52\)\( p^{60} T^{30} + \)\(21\!\cdots\!44\)\( p^{66} T^{31} + \)\(21\!\cdots\!83\)\( p^{72} T^{32} - \)\(19\!\cdots\!60\)\( p^{78} T^{33} - \)\(31\!\cdots\!16\)\( p^{84} T^{34} + \)\(60\!\cdots\!24\)\( p^{90} T^{35} + \)\(35\!\cdots\!34\)\( p^{96} T^{36} + 9127153116882912 p^{102} T^{37} - 268922049252 p^{108} T^{38} - 83552 p^{114} T^{39} + p^{120} T^{40} \) | |
| 67 | \( 1 - 806646 T + 938925242311 T^{2} - 582424751039974194 T^{3} + \)\(41\!\cdots\!86\)\( T^{4} - \)\(21\!\cdots\!58\)\( T^{5} + \)\(11\!\cdots\!77\)\( T^{6} - \)\(53\!\cdots\!66\)\( T^{7} + \)\(24\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!34\)\( T^{9} + \)\(42\!\cdots\!73\)\( T^{10} - \)\(16\!\cdots\!22\)\( T^{11} + \)\(61\!\cdots\!92\)\( T^{12} - \)\(21\!\cdots\!82\)\( T^{13} + \)\(77\!\cdots\!21\)\( T^{14} - \)\(25\!\cdots\!50\)\( T^{15} + \)\(86\!\cdots\!41\)\( T^{16} - \)\(27\!\cdots\!36\)\( T^{17} + \)\(87\!\cdots\!14\)\( T^{18} - \)\(26\!\cdots\!36\)\( T^{19} + \)\(82\!\cdots\!12\)\( T^{20} - \)\(26\!\cdots\!36\)\( p^{6} T^{21} + \)\(87\!\cdots\!14\)\( p^{12} T^{22} - \)\(27\!\cdots\!36\)\( p^{18} T^{23} + \)\(86\!\cdots\!41\)\( p^{24} T^{24} - \)\(25\!\cdots\!50\)\( p^{30} T^{25} + \)\(77\!\cdots\!21\)\( p^{36} T^{26} - \)\(21\!\cdots\!82\)\( p^{42} T^{27} + \)\(61\!\cdots\!92\)\( p^{48} T^{28} - \)\(16\!\cdots\!22\)\( p^{54} T^{29} + \)\(42\!\cdots\!73\)\( p^{60} T^{30} - \)\(10\!\cdots\!34\)\( p^{66} T^{31} + \)\(24\!\cdots\!66\)\( p^{72} T^{32} - \)\(53\!\cdots\!66\)\( p^{78} T^{33} + \)\(11\!\cdots\!77\)\( p^{84} T^{34} - \)\(21\!\cdots\!58\)\( p^{90} T^{35} + \)\(41\!\cdots\!86\)\( p^{96} T^{36} - 582424751039974194 p^{102} T^{37} + 938925242311 p^{108} T^{38} - 806646 p^{114} T^{39} + p^{120} T^{40} \) | |
| 71 | \( 1 + 869220 T + 1196037946510 T^{2} + 820708956715606200 T^{3} + \)\(67\!\cdots\!84\)\( T^{4} + \)\(39\!\cdots\!56\)\( T^{5} + \)\(25\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!60\)\( T^{7} + \)\(68\!\cdots\!33\)\( T^{8} + \)\(32\!\cdots\!44\)\( T^{9} + \)\(15\!\cdots\!28\)\( T^{10} + \)\(68\!\cdots\!28\)\( T^{11} + \)\(30\!\cdots\!86\)\( T^{12} + \)\(12\!\cdots\!60\)\( T^{13} + \)\(53\!\cdots\!38\)\( T^{14} + \)\(21\!\cdots\!08\)\( T^{15} + \)\(84\!\cdots\!89\)\( T^{16} + \)\(33\!\cdots\!80\)\( T^{17} + \)\(12\!\cdots\!48\)\( T^{18} + \)\(46\!\cdots\!60\)\( T^{19} + \)\(16\!\cdots\!98\)\( T^{20} + \)\(46\!\cdots\!60\)\( p^{6} T^{21} + \)\(12\!\cdots\!48\)\( p^{12} T^{22} + \)\(33\!\cdots\!80\)\( p^{18} T^{23} + \)\(84\!\cdots\!89\)\( p^{24} T^{24} + \)\(21\!\cdots\!08\)\( p^{30} T^{25} + \)\(53\!\cdots\!38\)\( p^{36} T^{26} + \)\(12\!\cdots\!60\)\( p^{42} T^{27} + \)\(30\!\cdots\!86\)\( p^{48} T^{28} + \)\(68\!\cdots\!28\)\( p^{54} T^{29} + \)\(15\!\cdots\!28\)\( p^{60} T^{30} + \)\(32\!\cdots\!44\)\( p^{66} T^{31} + \)\(68\!\cdots\!33\)\( p^{72} T^{32} + \)\(12\!\cdots\!60\)\( p^{78} T^{33} + \)\(25\!\cdots\!80\)\( p^{84} T^{34} + \)\(39\!\cdots\!56\)\( p^{90} T^{35} + \)\(67\!\cdots\!84\)\( p^{96} T^{36} + 820708956715606200 p^{102} T^{37} + 1196037946510 p^{108} T^{38} + 869220 p^{114} T^{39} + p^{120} T^{40} \) | |
| 73 | \( 1 + 207422 T - 710639910291 T^{2} + 30464971710184310 T^{3} + \)\(30\!\cdots\!90\)\( T^{4} - \)\(61\!\cdots\!82\)\( T^{5} - \)\(78\!\cdots\!49\)\( T^{6} + \)\(29\!\cdots\!26\)\( T^{7} + \)\(11\!\cdots\!54\)\( T^{8} - \)\(78\!\cdots\!18\)\( T^{9} - \)\(40\!\cdots\!25\)\( T^{10} + \)\(13\!\cdots\!14\)\( T^{11} - \)\(25\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!78\)\( T^{13} + \)\(69\!\cdots\!99\)\( T^{14} - \)\(23\!\cdots\!50\)\( T^{15} - \)\(93\!\cdots\!67\)\( T^{16} + \)\(20\!\cdots\!56\)\( T^{17} + \)\(73\!\cdots\!82\)\( T^{18} - \)\(16\!\cdots\!48\)\( T^{19} - \)\(51\!\cdots\!32\)\( T^{20} - \)\(16\!\cdots\!48\)\( p^{6} T^{21} + \)\(73\!\cdots\!82\)\( p^{12} T^{22} + \)\(20\!\cdots\!56\)\( p^{18} T^{23} - \)\(93\!\cdots\!67\)\( p^{24} T^{24} - \)\(23\!\cdots\!50\)\( p^{30} T^{25} + \)\(69\!\cdots\!99\)\( p^{36} T^{26} - \)\(11\!\cdots\!78\)\( p^{42} T^{27} - \)\(25\!\cdots\!80\)\( p^{48} T^{28} + \)\(13\!\cdots\!14\)\( p^{54} T^{29} - \)\(40\!\cdots\!25\)\( p^{60} T^{30} - \)\(78\!\cdots\!18\)\( p^{66} T^{31} + \)\(11\!\cdots\!54\)\( p^{72} T^{32} + \)\(29\!\cdots\!26\)\( p^{78} T^{33} - \)\(78\!\cdots\!49\)\( p^{84} T^{34} - \)\(61\!\cdots\!82\)\( p^{90} T^{35} + \)\(30\!\cdots\!90\)\( p^{96} T^{36} + 30464971710184310 p^{102} T^{37} - 710639910291 p^{108} T^{38} + 207422 p^{114} T^{39} + p^{120} T^{40} \) | |
| 79 | \( 1 + 1706808 T + 2556980606986 T^{2} + 2706854270932071984 T^{3} + \)\(24\!\cdots\!52\)\( T^{4} + \)\(19\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!00\)\( T^{6} + \)\(78\!\cdots\!36\)\( T^{7} + \)\(43\!\cdots\!73\)\( T^{8} + \)\(22\!\cdots\!52\)\( T^{9} + \)\(11\!\cdots\!08\)\( T^{10} + \)\(60\!\cdots\!64\)\( T^{11} + \)\(33\!\cdots\!74\)\( T^{12} + \)\(19\!\cdots\!60\)\( T^{13} + \)\(11\!\cdots\!94\)\( T^{14} + \)\(59\!\cdots\!92\)\( T^{15} + \)\(30\!\cdots\!97\)\( T^{16} + \)\(14\!\cdots\!68\)\( T^{17} + \)\(66\!\cdots\!92\)\( T^{18} + \)\(30\!\cdots\!44\)\( T^{19} + \)\(14\!\cdots\!18\)\( T^{20} + \)\(30\!\cdots\!44\)\( p^{6} T^{21} + \)\(66\!\cdots\!92\)\( p^{12} T^{22} + \)\(14\!\cdots\!68\)\( p^{18} T^{23} + \)\(30\!\cdots\!97\)\( p^{24} T^{24} + \)\(59\!\cdots\!92\)\( p^{30} T^{25} + \)\(11\!\cdots\!94\)\( p^{36} T^{26} + \)\(19\!\cdots\!60\)\( p^{42} T^{27} + \)\(33\!\cdots\!74\)\( p^{48} T^{28} + \)\(60\!\cdots\!64\)\( p^{54} T^{29} + \)\(11\!\cdots\!08\)\( p^{60} T^{30} + \)\(22\!\cdots\!52\)\( p^{66} T^{31} + \)\(43\!\cdots\!73\)\( p^{72} T^{32} + \)\(78\!\cdots\!36\)\( p^{78} T^{33} + \)\(12\!\cdots\!00\)\( p^{84} T^{34} + \)\(19\!\cdots\!76\)\( p^{90} T^{35} + \)\(24\!\cdots\!52\)\( p^{96} T^{36} + 2706854270932071984 p^{102} T^{37} + 2556980606986 p^{108} T^{38} + 1706808 p^{114} T^{39} + p^{120} T^{40} \) | |
| 83 | \( ( 1 - 1763774 T + 3161936397093 T^{2} - 3219192037451900522 T^{3} + \)\(32\!\cdots\!92\)\( T^{4} - \)\(23\!\cdots\!22\)\( T^{5} + \)\(17\!\cdots\!15\)\( T^{6} - \)\(96\!\cdots\!66\)\( T^{7} + \)\(61\!\cdots\!67\)\( T^{8} - \)\(30\!\cdots\!16\)\( T^{9} + \)\(19\!\cdots\!44\)\( T^{10} - \)\(30\!\cdots\!16\)\( p^{6} T^{11} + \)\(61\!\cdots\!67\)\( p^{12} T^{12} - \)\(96\!\cdots\!66\)\( p^{18} T^{13} + \)\(17\!\cdots\!15\)\( p^{24} T^{14} - \)\(23\!\cdots\!22\)\( p^{30} T^{15} + \)\(32\!\cdots\!92\)\( p^{36} T^{16} - 3219192037451900522 p^{42} T^{17} + 3161936397093 p^{48} T^{18} - 1763774 p^{54} T^{19} + p^{60} T^{20} )^{2} \) | |
| 89 | \( 1 - 708432 T + 2065246836748 T^{2} - 1344571964846393280 T^{3} + \)\(17\!\cdots\!74\)\( T^{4} - \)\(13\!\cdots\!60\)\( T^{5} + \)\(88\!\cdots\!00\)\( T^{6} - \)\(88\!\cdots\!68\)\( T^{7} + \)\(30\!\cdots\!19\)\( T^{8} - \)\(40\!\cdots\!68\)\( T^{9} + \)\(12\!\cdots\!68\)\( T^{10} - \)\(79\!\cdots\!44\)\( T^{11} + \)\(70\!\cdots\!60\)\( T^{12} + \)\(22\!\cdots\!40\)\( T^{13} + \)\(52\!\cdots\!76\)\( T^{14} + \)\(79\!\cdots\!04\)\( T^{15} + \)\(33\!\cdots\!49\)\( T^{16} - \)\(24\!\cdots\!08\)\( T^{17} + \)\(15\!\cdots\!32\)\( T^{18} - \)\(22\!\cdots\!60\)\( T^{19} + \)\(67\!\cdots\!78\)\( T^{20} - \)\(22\!\cdots\!60\)\( p^{6} T^{21} + \)\(15\!\cdots\!32\)\( p^{12} T^{22} - \)\(24\!\cdots\!08\)\( p^{18} T^{23} + \)\(33\!\cdots\!49\)\( p^{24} T^{24} + \)\(79\!\cdots\!04\)\( p^{30} T^{25} + \)\(52\!\cdots\!76\)\( p^{36} T^{26} + \)\(22\!\cdots\!40\)\( p^{42} T^{27} + \)\(70\!\cdots\!60\)\( p^{48} T^{28} - \)\(79\!\cdots\!44\)\( p^{54} T^{29} + \)\(12\!\cdots\!68\)\( p^{60} T^{30} - \)\(40\!\cdots\!68\)\( p^{66} T^{31} + \)\(30\!\cdots\!19\)\( p^{72} T^{32} - \)\(88\!\cdots\!68\)\( p^{78} T^{33} + \)\(88\!\cdots\!00\)\( p^{84} T^{34} - \)\(13\!\cdots\!60\)\( p^{90} T^{35} + \)\(17\!\cdots\!74\)\( p^{96} T^{36} - 1344571964846393280 p^{102} T^{37} + 2065246836748 p^{108} T^{38} - 708432 p^{114} T^{39} + p^{120} T^{40} \) | |
| 97 | \( 1 - 5113242 T + 16536828207181 T^{2} - 39994485053868801306 T^{3} + \)\(79\!\cdots\!42\)\( T^{4} - \)\(13\!\cdots\!18\)\( T^{5} + \)\(20\!\cdots\!39\)\( T^{6} - \)\(26\!\cdots\!82\)\( T^{7} + \)\(31\!\cdots\!58\)\( T^{8} - \)\(33\!\cdots\!86\)\( T^{9} + \)\(30\!\cdots\!51\)\( T^{10} - \)\(24\!\cdots\!94\)\( T^{11} + \)\(17\!\cdots\!16\)\( T^{12} - \)\(96\!\cdots\!50\)\( T^{13} + \)\(41\!\cdots\!31\)\( T^{14} - \)\(18\!\cdots\!54\)\( T^{15} + \)\(26\!\cdots\!25\)\( T^{16} - \)\(55\!\cdots\!88\)\( T^{17} + \)\(86\!\cdots\!10\)\( T^{18} - \)\(10\!\cdots\!60\)\( T^{19} + \)\(10\!\cdots\!28\)\( T^{20} - \)\(10\!\cdots\!60\)\( p^{6} T^{21} + \)\(86\!\cdots\!10\)\( p^{12} T^{22} - \)\(55\!\cdots\!88\)\( p^{18} T^{23} + \)\(26\!\cdots\!25\)\( p^{24} T^{24} - \)\(18\!\cdots\!54\)\( p^{30} T^{25} + \)\(41\!\cdots\!31\)\( p^{36} T^{26} - \)\(96\!\cdots\!50\)\( p^{42} T^{27} + \)\(17\!\cdots\!16\)\( p^{48} T^{28} - \)\(24\!\cdots\!94\)\( p^{54} T^{29} + \)\(30\!\cdots\!51\)\( p^{60} T^{30} - \)\(33\!\cdots\!86\)\( p^{66} T^{31} + \)\(31\!\cdots\!58\)\( p^{72} T^{32} - \)\(26\!\cdots\!82\)\( p^{78} T^{33} + \)\(20\!\cdots\!39\)\( p^{84} T^{34} - \)\(13\!\cdots\!18\)\( p^{90} T^{35} + \)\(79\!\cdots\!42\)\( p^{96} T^{36} - 39994485053868801306 p^{102} T^{37} + 16536828207181 p^{108} T^{38} - 5113242 p^{114} T^{39} + p^{120} T^{40} \) | |
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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
−3.01457383786774659377733268149, −3.01303481610322638773998526851, −2.99156869507346526541003749994, −2.91762585712602011115613685210, −2.80796658453286458879065856963, −2.53929413407139905289595316375, −2.52292398991067705345825654411, −2.43438143973389611894490039219, −2.25454233562225832459568722139, −2.21532040376035524749229859578, −1.86488723470750523902349089654, −1.85511204593162294668419589405, −1.78381567047040756663599643534, −1.61539360929056145258913848708, −1.39421478371780753909725431132, −1.14401602645131437978175627077, −1.14054043821287817513721788011, −1.10286054525131288642926118236, −1.09244457141995681434074952480, −0.924552797410353258145755070293, −0.72831394409718254240022842354, −0.51927127556220499212215393926, −0.36136860725363034160675944416, −0.30121329566324430660212376901, −0.04912507524734915601592143299, 0.04912507524734915601592143299, 0.30121329566324430660212376901, 0.36136860725363034160675944416, 0.51927127556220499212215393926, 0.72831394409718254240022842354, 0.924552797410353258145755070293, 1.09244457141995681434074952480, 1.10286054525131288642926118236, 1.14054043821287817513721788011, 1.14401602645131437978175627077, 1.39421478371780753909725431132, 1.61539360929056145258913848708, 1.78381567047040756663599643534, 1.85511204593162294668419589405, 1.86488723470750523902349089654, 2.21532040376035524749229859578, 2.25454233562225832459568722139, 2.43438143973389611894490039219, 2.52292398991067705345825654411, 2.53929413407139905289595316375, 2.80796658453286458879065856963, 2.91762585712602011115613685210, 2.99156869507346526541003749994, 3.01303481610322638773998526851, 3.01457383786774659377733268149
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.