Dirichlet series
| L(s) = 1 | − 4.37e3·3-s − 4.09e4·4-s + 3.35e3·5-s + 1.45e6·7-s − 5.57e6·9-s + 8.40e6·11-s + 1.79e8·12-s − 1.46e7·15-s + 6.71e8·16-s + 2.18e9·17-s − 3.91e9·19-s − 1.37e8·20-s − 6.36e9·21-s − 6.90e9·23-s − 2.34e10·25-s + 5.22e10·27-s − 5.96e10·28-s + 2.78e10·29-s + 4.56e10·31-s − 3.67e10·33-s + 4.88e9·35-s + 2.28e11·36-s − 2.70e10·37-s + 7.26e11·43-s − 3.44e11·44-s − 1.86e10·45-s − 2.04e12·47-s + ⋯ |
| L(s) = 1 | − 2·3-s − 5/2·4-s + 0.0429·5-s + 1.76·7-s − 1.16·9-s + 0.431·11-s + 5·12-s − 0.0858·15-s + 5/2·16-s + 5.31·17-s − 4.38·19-s − 0.107·20-s − 3.53·21-s − 2.02·23-s − 3.83·25-s + 4.99·27-s − 4.41·28-s + 1.61·29-s + 1.65·31-s − 0.862·33-s + 0.0758·35-s + 2.91·36-s − 0.284·37-s + 2.67·43-s − 1.07·44-s − 0.0500·45-s − 4.03·47-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(15-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{20}\right)^{s/2} \, \Gamma_{\C}(s+7)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 7^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.51639\times 10^{24}\) |
| Root analytic conductor: | \(4.17205\) |
| Motivic weight: | \(14\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 7^{20} ,\ ( \ : [7]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{15}{2})\) | \(\approx\) | \(6.438571268\times10^{-5}\) |
| \(L(\frac12)\) | \(\approx\) | \(6.438571268\times10^{-5}\) |
| \(L(8)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{13} T^{2} + p^{26} T^{4} )^{5} \) |
| 7 | \( 1 - 1455616 T - 58654005482 p T^{2} + 12777563469669216 p^{2} T^{3} + \)\(77\!\cdots\!04\)\( p^{4} T^{4} - \)\(23\!\cdots\!36\)\( p^{8} T^{5} - \)\(47\!\cdots\!96\)\( p^{10} T^{6} + \)\(51\!\cdots\!72\)\( p^{14} T^{7} + \)\(14\!\cdots\!52\)\( p^{19} T^{8} - \)\(13\!\cdots\!68\)\( p^{24} T^{9} - \)\(68\!\cdots\!08\)\( p^{31} T^{10} - \)\(13\!\cdots\!68\)\( p^{38} T^{11} + \)\(14\!\cdots\!52\)\( p^{47} T^{12} + \)\(51\!\cdots\!72\)\( p^{56} T^{13} - \)\(47\!\cdots\!96\)\( p^{66} T^{14} - \)\(23\!\cdots\!36\)\( p^{78} T^{15} + \)\(77\!\cdots\!04\)\( p^{88} T^{16} + 12777563469669216 p^{100} T^{17} - 58654005482 p^{113} T^{18} - 1455616 p^{126} T^{19} + p^{140} T^{20} \) | |
| good | 3 | \( 1 + 2 p^{7} T + 8235689 p T^{2} + 12219850 p^{8} T^{3} + 31170005581970 p^{2} T^{4} + 26069291705795138 p^{3} T^{5} + 63952179063316051175 p^{3} T^{6} + \)\(40\!\cdots\!42\)\( p^{4} T^{7} + \)\(58\!\cdots\!30\)\( p^{4} T^{8} + \)\(39\!\cdots\!42\)\( p^{6} T^{9} - \)\(27\!\cdots\!95\)\( p^{8} T^{10} - \)\(38\!\cdots\!34\)\( p^{9} T^{11} - \)\(44\!\cdots\!24\)\( p^{10} T^{12} - \)\(12\!\cdots\!98\)\( p^{14} T^{13} - \)\(94\!\cdots\!41\)\( p^{15} T^{14} - \)\(18\!\cdots\!38\)\( p^{19} T^{15} - \)\(60\!\cdots\!91\)\( p^{20} T^{16} + \)\(59\!\cdots\!24\)\( p^{24} T^{17} + \)\(83\!\cdots\!98\)\( p^{28} T^{18} + \)\(93\!\cdots\!96\)\( p^{31} T^{19} + \)\(94\!\cdots\!32\)\( p^{34} T^{20} + \)\(93\!\cdots\!96\)\( p^{45} T^{21} + \)\(83\!\cdots\!98\)\( p^{56} T^{22} + \)\(59\!\cdots\!24\)\( p^{66} T^{23} - \)\(60\!\cdots\!91\)\( p^{76} T^{24} - \)\(18\!\cdots\!38\)\( p^{89} T^{25} - \)\(94\!\cdots\!41\)\( p^{99} T^{26} - \)\(12\!\cdots\!98\)\( p^{112} T^{27} - \)\(44\!\cdots\!24\)\( p^{122} T^{28} - \)\(38\!\cdots\!34\)\( p^{135} T^{29} - \)\(27\!\cdots\!95\)\( p^{148} T^{30} + \)\(39\!\cdots\!42\)\( p^{160} T^{31} + \)\(58\!\cdots\!30\)\( p^{172} T^{32} + \)\(40\!\cdots\!42\)\( p^{186} T^{33} + 63952179063316051175 p^{199} T^{34} + 26069291705795138 p^{213} T^{35} + 31170005581970 p^{226} T^{36} + 12219850 p^{246} T^{37} + 8235689 p^{253} T^{38} + 2 p^{273} T^{39} + p^{280} T^{40} \) |
| 5 | \( 1 - 3354 T + 23440661461 T^{2} - 78607401804906 T^{3} + \)\(28\!\cdots\!86\)\( T^{4} + \)\(83\!\cdots\!78\)\( p T^{5} + \)\(44\!\cdots\!99\)\( p T^{6} + \)\(38\!\cdots\!86\)\( p^{2} T^{7} + \)\(44\!\cdots\!62\)\( p^{2} T^{8} + \)\(16\!\cdots\!58\)\( p^{4} T^{9} + \)\(30\!\cdots\!31\)\( p^{3} T^{10} + \)\(22\!\cdots\!18\)\( p^{5} T^{11} + \)\(17\!\cdots\!96\)\( p^{4} T^{12} + \)\(36\!\cdots\!82\)\( p^{7} T^{13} + \)\(15\!\cdots\!91\)\( p^{8} T^{14} + \)\(14\!\cdots\!22\)\( p^{11} T^{15} + \)\(75\!\cdots\!21\)\( p^{14} T^{16} + \)\(33\!\cdots\!52\)\( p^{15} T^{17} + \)\(18\!\cdots\!06\)\( p^{16} T^{18} + \)\(12\!\cdots\!92\)\( p^{19} T^{19} + \)\(18\!\cdots\!76\)\( p^{20} T^{20} + \)\(12\!\cdots\!92\)\( p^{33} T^{21} + \)\(18\!\cdots\!06\)\( p^{44} T^{22} + \)\(33\!\cdots\!52\)\( p^{57} T^{23} + \)\(75\!\cdots\!21\)\( p^{70} T^{24} + \)\(14\!\cdots\!22\)\( p^{81} T^{25} + \)\(15\!\cdots\!91\)\( p^{92} T^{26} + \)\(36\!\cdots\!82\)\( p^{105} T^{27} + \)\(17\!\cdots\!96\)\( p^{116} T^{28} + \)\(22\!\cdots\!18\)\( p^{131} T^{29} + \)\(30\!\cdots\!31\)\( p^{143} T^{30} + \)\(16\!\cdots\!58\)\( p^{158} T^{31} + \)\(44\!\cdots\!62\)\( p^{170} T^{32} + \)\(38\!\cdots\!86\)\( p^{184} T^{33} + \)\(44\!\cdots\!99\)\( p^{197} T^{34} + \)\(83\!\cdots\!78\)\( p^{211} T^{35} + \)\(28\!\cdots\!86\)\( p^{224} T^{36} - 78607401804906 p^{238} T^{37} + 23440661461 p^{252} T^{38} - 3354 p^{266} T^{39} + p^{280} T^{40} \) | |
| 11 | \( 1 - 8400426 T - 168532024130471 p T^{2} - \)\(96\!\cdots\!90\)\( T^{3} + \)\(17\!\cdots\!30\)\( T^{4} + \)\(13\!\cdots\!58\)\( p T^{5} - \)\(73\!\cdots\!13\)\( p^{3} T^{6} - \)\(12\!\cdots\!58\)\( p^{3} T^{7} + \)\(23\!\cdots\!38\)\( p^{4} T^{8} + \)\(61\!\cdots\!94\)\( p^{5} T^{9} - \)\(40\!\cdots\!23\)\( p^{6} T^{10} - \)\(16\!\cdots\!46\)\( p^{7} T^{11} + \)\(35\!\cdots\!88\)\( p^{8} T^{12} + \)\(16\!\cdots\!18\)\( p^{9} T^{13} - \)\(25\!\cdots\!95\)\( p^{10} T^{14} + \)\(77\!\cdots\!10\)\( p^{11} T^{15} + \)\(30\!\cdots\!93\)\( p^{12} T^{16} - \)\(41\!\cdots\!76\)\( p^{13} T^{17} - \)\(18\!\cdots\!42\)\( p^{14} T^{18} + \)\(56\!\cdots\!44\)\( p^{16} T^{19} + \)\(72\!\cdots\!08\)\( p^{16} T^{20} + \)\(56\!\cdots\!44\)\( p^{30} T^{21} - \)\(18\!\cdots\!42\)\( p^{42} T^{22} - \)\(41\!\cdots\!76\)\( p^{55} T^{23} + \)\(30\!\cdots\!93\)\( p^{68} T^{24} + \)\(77\!\cdots\!10\)\( p^{81} T^{25} - \)\(25\!\cdots\!95\)\( p^{94} T^{26} + \)\(16\!\cdots\!18\)\( p^{107} T^{27} + \)\(35\!\cdots\!88\)\( p^{120} T^{28} - \)\(16\!\cdots\!46\)\( p^{133} T^{29} - \)\(40\!\cdots\!23\)\( p^{146} T^{30} + \)\(61\!\cdots\!94\)\( p^{159} T^{31} + \)\(23\!\cdots\!38\)\( p^{172} T^{32} - \)\(12\!\cdots\!58\)\( p^{185} T^{33} - \)\(73\!\cdots\!13\)\( p^{199} T^{34} + \)\(13\!\cdots\!58\)\( p^{211} T^{35} + \)\(17\!\cdots\!30\)\( p^{224} T^{36} - \)\(96\!\cdots\!90\)\( p^{238} T^{37} - 168532024130471 p^{253} T^{38} - 8400426 p^{266} T^{39} + p^{280} T^{40} \) | |
| 13 | \( 1 - 41995265002650332 T^{2} + \)\(88\!\cdots\!54\)\( T^{4} - \)\(57\!\cdots\!52\)\( p^{3} T^{6} + \)\(47\!\cdots\!61\)\( p^{4} T^{8} - \)\(23\!\cdots\!96\)\( p^{6} T^{10} + \)\(10\!\cdots\!60\)\( p^{8} T^{12} - \)\(36\!\cdots\!00\)\( p^{10} T^{14} + \)\(11\!\cdots\!18\)\( p^{12} T^{16} - \)\(32\!\cdots\!68\)\( p^{14} T^{18} + \)\(79\!\cdots\!44\)\( p^{16} T^{20} - \)\(32\!\cdots\!68\)\( p^{42} T^{22} + \)\(11\!\cdots\!18\)\( p^{68} T^{24} - \)\(36\!\cdots\!00\)\( p^{94} T^{26} + \)\(10\!\cdots\!60\)\( p^{120} T^{28} - \)\(23\!\cdots\!96\)\( p^{146} T^{30} + \)\(47\!\cdots\!61\)\( p^{172} T^{32} - \)\(57\!\cdots\!52\)\( p^{199} T^{34} + \)\(88\!\cdots\!54\)\( p^{224} T^{36} - 41995265002650332 p^{252} T^{38} + p^{280} T^{40} \) | |
| 17 | \( 1 - 2180481042 T + 3123278663210181661 T^{2} - \)\(33\!\cdots\!66\)\( T^{3} + \)\(29\!\cdots\!18\)\( T^{4} - \)\(22\!\cdots\!54\)\( T^{5} + \)\(15\!\cdots\!03\)\( T^{6} - \)\(96\!\cdots\!70\)\( T^{7} + \)\(55\!\cdots\!66\)\( T^{8} - \)\(29\!\cdots\!06\)\( T^{9} + \)\(14\!\cdots\!23\)\( T^{10} - \)\(69\!\cdots\!74\)\( T^{11} + \)\(31\!\cdots\!48\)\( T^{12} - \)\(13\!\cdots\!22\)\( T^{13} + \)\(54\!\cdots\!07\)\( T^{14} - \)\(21\!\cdots\!42\)\( T^{15} + \)\(80\!\cdots\!17\)\( T^{16} - \)\(29\!\cdots\!88\)\( T^{17} + \)\(10\!\cdots\!70\)\( T^{18} - \)\(40\!\cdots\!40\)\( T^{19} + \)\(16\!\cdots\!92\)\( T^{20} - \)\(40\!\cdots\!40\)\( p^{14} T^{21} + \)\(10\!\cdots\!70\)\( p^{28} T^{22} - \)\(29\!\cdots\!88\)\( p^{42} T^{23} + \)\(80\!\cdots\!17\)\( p^{56} T^{24} - \)\(21\!\cdots\!42\)\( p^{70} T^{25} + \)\(54\!\cdots\!07\)\( p^{84} T^{26} - \)\(13\!\cdots\!22\)\( p^{98} T^{27} + \)\(31\!\cdots\!48\)\( p^{112} T^{28} - \)\(69\!\cdots\!74\)\( p^{126} T^{29} + \)\(14\!\cdots\!23\)\( p^{140} T^{30} - \)\(29\!\cdots\!06\)\( p^{154} T^{31} + \)\(55\!\cdots\!66\)\( p^{168} T^{32} - \)\(96\!\cdots\!70\)\( p^{182} T^{33} + \)\(15\!\cdots\!03\)\( p^{196} T^{34} - \)\(22\!\cdots\!54\)\( p^{210} T^{35} + \)\(29\!\cdots\!18\)\( p^{224} T^{36} - \)\(33\!\cdots\!66\)\( p^{238} T^{37} + 3123278663210181661 p^{252} T^{38} - 2180481042 p^{266} T^{39} + p^{280} T^{40} \) | |
| 19 | \( 1 + 3919727442 T + 10076647080033984715 T^{2} + \)\(19\!\cdots\!34\)\( T^{3} + \)\(30\!\cdots\!30\)\( T^{4} + \)\(40\!\cdots\!86\)\( T^{5} + \)\(50\!\cdots\!13\)\( T^{6} + \)\(58\!\cdots\!30\)\( T^{7} + \)\(34\!\cdots\!06\)\( p T^{8} + \)\(71\!\cdots\!94\)\( T^{9} + \)\(74\!\cdots\!77\)\( T^{10} + \)\(74\!\cdots\!38\)\( T^{11} + \)\(71\!\cdots\!64\)\( T^{12} + \)\(68\!\cdots\!94\)\( T^{13} + \)\(64\!\cdots\!93\)\( T^{14} + \)\(61\!\cdots\!30\)\( T^{15} + \)\(57\!\cdots\!73\)\( T^{16} + \)\(53\!\cdots\!96\)\( T^{17} + \)\(48\!\cdots\!54\)\( T^{18} + \)\(43\!\cdots\!76\)\( T^{19} + \)\(38\!\cdots\!88\)\( T^{20} + \)\(43\!\cdots\!76\)\( p^{14} T^{21} + \)\(48\!\cdots\!54\)\( p^{28} T^{22} + \)\(53\!\cdots\!96\)\( p^{42} T^{23} + \)\(57\!\cdots\!73\)\( p^{56} T^{24} + \)\(61\!\cdots\!30\)\( p^{70} T^{25} + \)\(64\!\cdots\!93\)\( p^{84} T^{26} + \)\(68\!\cdots\!94\)\( p^{98} T^{27} + \)\(71\!\cdots\!64\)\( p^{112} T^{28} + \)\(74\!\cdots\!38\)\( p^{126} T^{29} + \)\(74\!\cdots\!77\)\( p^{140} T^{30} + \)\(71\!\cdots\!94\)\( p^{154} T^{31} + \)\(34\!\cdots\!06\)\( p^{169} T^{32} + \)\(58\!\cdots\!30\)\( p^{182} T^{33} + \)\(50\!\cdots\!13\)\( p^{196} T^{34} + \)\(40\!\cdots\!86\)\( p^{210} T^{35} + \)\(30\!\cdots\!30\)\( p^{224} T^{36} + \)\(19\!\cdots\!34\)\( p^{238} T^{37} + 10076647080033984715 p^{252} T^{38} + 3919727442 p^{266} T^{39} + p^{280} T^{40} \) | |
| 23 | \( 1 + 6905098386 T - 26744280514725789709 T^{2} - \)\(11\!\cdots\!42\)\( T^{3} + \)\(11\!\cdots\!02\)\( T^{4} + \)\(26\!\cdots\!58\)\( T^{5} - \)\(24\!\cdots\!39\)\( T^{6} + \)\(41\!\cdots\!58\)\( T^{7} + \)\(24\!\cdots\!50\)\( T^{8} - \)\(98\!\cdots\!18\)\( T^{9} - \)\(94\!\cdots\!59\)\( T^{10} + \)\(56\!\cdots\!58\)\( T^{11} + \)\(69\!\cdots\!84\)\( T^{12} + \)\(43\!\cdots\!26\)\( T^{13} - \)\(50\!\cdots\!19\)\( T^{14} - \)\(11\!\cdots\!66\)\( p T^{15} + \)\(89\!\cdots\!21\)\( T^{16} - \)\(70\!\cdots\!28\)\( p T^{17} - \)\(50\!\cdots\!50\)\( T^{18} + \)\(13\!\cdots\!72\)\( T^{19} + \)\(24\!\cdots\!96\)\( T^{20} + \)\(13\!\cdots\!72\)\( p^{14} T^{21} - \)\(50\!\cdots\!50\)\( p^{28} T^{22} - \)\(70\!\cdots\!28\)\( p^{43} T^{23} + \)\(89\!\cdots\!21\)\( p^{56} T^{24} - \)\(11\!\cdots\!66\)\( p^{71} T^{25} - \)\(50\!\cdots\!19\)\( p^{84} T^{26} + \)\(43\!\cdots\!26\)\( p^{98} T^{27} + \)\(69\!\cdots\!84\)\( p^{112} T^{28} + \)\(56\!\cdots\!58\)\( p^{126} T^{29} - \)\(94\!\cdots\!59\)\( p^{140} T^{30} - \)\(98\!\cdots\!18\)\( p^{154} T^{31} + \)\(24\!\cdots\!50\)\( p^{168} T^{32} + \)\(41\!\cdots\!58\)\( p^{182} T^{33} - \)\(24\!\cdots\!39\)\( p^{196} T^{34} + \)\(26\!\cdots\!58\)\( p^{210} T^{35} + \)\(11\!\cdots\!02\)\( p^{224} T^{36} - \)\(11\!\cdots\!42\)\( p^{238} T^{37} - 26744280514725789709 p^{252} T^{38} + 6905098386 p^{266} T^{39} + p^{280} T^{40} \) | |
| 29 | \( ( 1 - 480774288 p T + \)\(11\!\cdots\!78\)\( T^{2} - \)\(14\!\cdots\!48\)\( T^{3} + \)\(67\!\cdots\!77\)\( T^{4} - \)\(97\!\cdots\!56\)\( T^{5} + \)\(29\!\cdots\!08\)\( T^{6} - \)\(49\!\cdots\!88\)\( T^{7} + \)\(10\!\cdots\!86\)\( T^{8} - \)\(19\!\cdots\!44\)\( T^{9} + \)\(33\!\cdots\!48\)\( T^{10} - \)\(19\!\cdots\!44\)\( p^{14} T^{11} + \)\(10\!\cdots\!86\)\( p^{28} T^{12} - \)\(49\!\cdots\!88\)\( p^{42} T^{13} + \)\(29\!\cdots\!08\)\( p^{56} T^{14} - \)\(97\!\cdots\!56\)\( p^{70} T^{15} + \)\(67\!\cdots\!77\)\( p^{84} T^{16} - \)\(14\!\cdots\!48\)\( p^{98} T^{17} + \)\(11\!\cdots\!78\)\( p^{112} T^{18} - 480774288 p^{127} T^{19} + p^{140} T^{20} )^{2} \) | |
| 31 | \( 1 - 45638710782 T + \)\(58\!\cdots\!35\)\( T^{2} - \)\(23\!\cdots\!14\)\( T^{3} + \)\(17\!\cdots\!58\)\( T^{4} - \)\(60\!\cdots\!22\)\( T^{5} + \)\(32\!\cdots\!05\)\( T^{6} - \)\(99\!\cdots\!70\)\( T^{7} + \)\(45\!\cdots\!22\)\( T^{8} - \)\(11\!\cdots\!90\)\( T^{9} + \)\(46\!\cdots\!77\)\( T^{10} - \)\(94\!\cdots\!82\)\( T^{11} + \)\(34\!\cdots\!76\)\( T^{12} - \)\(39\!\cdots\!86\)\( T^{13} + \)\(14\!\cdots\!53\)\( T^{14} + \)\(21\!\cdots\!66\)\( T^{15} - \)\(37\!\cdots\!31\)\( T^{16} + \)\(63\!\cdots\!32\)\( T^{17} - \)\(13\!\cdots\!06\)\( T^{18} + \)\(71\!\cdots\!24\)\( T^{19} - \)\(13\!\cdots\!88\)\( T^{20} + \)\(71\!\cdots\!24\)\( p^{14} T^{21} - \)\(13\!\cdots\!06\)\( p^{28} T^{22} + \)\(63\!\cdots\!32\)\( p^{42} T^{23} - \)\(37\!\cdots\!31\)\( p^{56} T^{24} + \)\(21\!\cdots\!66\)\( p^{70} T^{25} + \)\(14\!\cdots\!53\)\( p^{84} T^{26} - \)\(39\!\cdots\!86\)\( p^{98} T^{27} + \)\(34\!\cdots\!76\)\( p^{112} T^{28} - \)\(94\!\cdots\!82\)\( p^{126} T^{29} + \)\(46\!\cdots\!77\)\( p^{140} T^{30} - \)\(11\!\cdots\!90\)\( p^{154} T^{31} + \)\(45\!\cdots\!22\)\( p^{168} T^{32} - \)\(99\!\cdots\!70\)\( p^{182} T^{33} + \)\(32\!\cdots\!05\)\( p^{196} T^{34} - \)\(60\!\cdots\!22\)\( p^{210} T^{35} + \)\(17\!\cdots\!58\)\( p^{224} T^{36} - \)\(23\!\cdots\!14\)\( p^{238} T^{37} + \)\(58\!\cdots\!35\)\( p^{252} T^{38} - 45638710782 p^{266} T^{39} + p^{280} T^{40} \) | |
| 37 | \( 1 + 27026027926 T - \)\(52\!\cdots\!91\)\( T^{2} - \)\(73\!\cdots\!62\)\( T^{3} + \)\(16\!\cdots\!02\)\( T^{4} + \)\(54\!\cdots\!90\)\( T^{5} - \)\(35\!\cdots\!29\)\( T^{6} + \)\(33\!\cdots\!46\)\( T^{7} + \)\(59\!\cdots\!86\)\( T^{8} - \)\(13\!\cdots\!02\)\( T^{9} - \)\(75\!\cdots\!17\)\( T^{10} + \)\(29\!\cdots\!62\)\( T^{11} + \)\(74\!\cdots\!36\)\( T^{12} - \)\(43\!\cdots\!34\)\( T^{13} - \)\(52\!\cdots\!09\)\( T^{14} + \)\(48\!\cdots\!58\)\( T^{15} + \)\(19\!\cdots\!97\)\( T^{16} - \)\(36\!\cdots\!92\)\( T^{17} + \)\(80\!\cdots\!74\)\( T^{18} + \)\(13\!\cdots\!88\)\( T^{19} - \)\(16\!\cdots\!08\)\( T^{20} + \)\(13\!\cdots\!88\)\( p^{14} T^{21} + \)\(80\!\cdots\!74\)\( p^{28} T^{22} - \)\(36\!\cdots\!92\)\( p^{42} T^{23} + \)\(19\!\cdots\!97\)\( p^{56} T^{24} + \)\(48\!\cdots\!58\)\( p^{70} T^{25} - \)\(52\!\cdots\!09\)\( p^{84} T^{26} - \)\(43\!\cdots\!34\)\( p^{98} T^{27} + \)\(74\!\cdots\!36\)\( p^{112} T^{28} + \)\(29\!\cdots\!62\)\( p^{126} T^{29} - \)\(75\!\cdots\!17\)\( p^{140} T^{30} - \)\(13\!\cdots\!02\)\( p^{154} T^{31} + \)\(59\!\cdots\!86\)\( p^{168} T^{32} + \)\(33\!\cdots\!46\)\( p^{182} T^{33} - \)\(35\!\cdots\!29\)\( p^{196} T^{34} + \)\(54\!\cdots\!90\)\( p^{210} T^{35} + \)\(16\!\cdots\!02\)\( p^{224} T^{36} - \)\(73\!\cdots\!62\)\( p^{238} T^{37} - \)\(52\!\cdots\!91\)\( p^{252} T^{38} + 27026027926 p^{266} T^{39} + p^{280} T^{40} \) | |
| 41 | \( 1 - \)\(47\!\cdots\!76\)\( T^{2} + \)\(11\!\cdots\!10\)\( T^{4} - \)\(17\!\cdots\!72\)\( T^{6} + \)\(21\!\cdots\!29\)\( T^{8} - \)\(19\!\cdots\!72\)\( T^{10} + \)\(15\!\cdots\!04\)\( T^{12} - \)\(98\!\cdots\!12\)\( T^{14} + \)\(53\!\cdots\!70\)\( T^{16} - \)\(25\!\cdots\!36\)\( T^{18} + \)\(10\!\cdots\!32\)\( T^{20} - \)\(25\!\cdots\!36\)\( p^{28} T^{22} + \)\(53\!\cdots\!70\)\( p^{56} T^{24} - \)\(98\!\cdots\!12\)\( p^{84} T^{26} + \)\(15\!\cdots\!04\)\( p^{112} T^{28} - \)\(19\!\cdots\!72\)\( p^{140} T^{30} + \)\(21\!\cdots\!29\)\( p^{168} T^{32} - \)\(17\!\cdots\!72\)\( p^{196} T^{34} + \)\(11\!\cdots\!10\)\( p^{224} T^{36} - \)\(47\!\cdots\!76\)\( p^{252} T^{38} + p^{280} T^{40} \) | |
| 43 | \( ( 1 - 363341476828 T + \)\(46\!\cdots\!70\)\( T^{2} - \)\(12\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!09\)\( T^{4} - \)\(21\!\cdots\!36\)\( T^{5} + \)\(14\!\cdots\!68\)\( T^{6} - \)\(25\!\cdots\!92\)\( T^{7} + \)\(15\!\cdots\!14\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{9} + \)\(12\!\cdots\!04\)\( T^{10} - \)\(24\!\cdots\!00\)\( p^{14} T^{11} + \)\(15\!\cdots\!14\)\( p^{28} T^{12} - \)\(25\!\cdots\!92\)\( p^{42} T^{13} + \)\(14\!\cdots\!68\)\( p^{56} T^{14} - \)\(21\!\cdots\!36\)\( p^{70} T^{15} + \)\(10\!\cdots\!09\)\( p^{84} T^{16} - \)\(12\!\cdots\!36\)\( p^{98} T^{17} + \)\(46\!\cdots\!70\)\( p^{112} T^{18} - 363341476828 p^{126} T^{19} + p^{140} T^{20} )^{2} \) | |
| 47 | \( 1 + 2044625353338 T + \)\(34\!\cdots\!67\)\( T^{2} + \)\(42\!\cdots\!22\)\( T^{3} + \)\(46\!\cdots\!74\)\( T^{4} + \)\(42\!\cdots\!42\)\( T^{5} + \)\(34\!\cdots\!65\)\( T^{6} + \)\(25\!\cdots\!50\)\( T^{7} + \)\(17\!\cdots\!46\)\( T^{8} + \)\(10\!\cdots\!82\)\( T^{9} + \)\(61\!\cdots\!25\)\( T^{10} + \)\(31\!\cdots\!34\)\( T^{11} + \)\(14\!\cdots\!32\)\( T^{12} + \)\(60\!\cdots\!30\)\( T^{13} + \)\(20\!\cdots\!33\)\( T^{14} + \)\(47\!\cdots\!46\)\( T^{15} - \)\(53\!\cdots\!11\)\( T^{16} - \)\(15\!\cdots\!04\)\( T^{17} - \)\(13\!\cdots\!18\)\( T^{18} - \)\(83\!\cdots\!08\)\( T^{19} - \)\(44\!\cdots\!80\)\( T^{20} - \)\(83\!\cdots\!08\)\( p^{14} T^{21} - \)\(13\!\cdots\!18\)\( p^{28} T^{22} - \)\(15\!\cdots\!04\)\( p^{42} T^{23} - \)\(53\!\cdots\!11\)\( p^{56} T^{24} + \)\(47\!\cdots\!46\)\( p^{70} T^{25} + \)\(20\!\cdots\!33\)\( p^{84} T^{26} + \)\(60\!\cdots\!30\)\( p^{98} T^{27} + \)\(14\!\cdots\!32\)\( p^{112} T^{28} + \)\(31\!\cdots\!34\)\( p^{126} T^{29} + \)\(61\!\cdots\!25\)\( p^{140} T^{30} + \)\(10\!\cdots\!82\)\( p^{154} T^{31} + \)\(17\!\cdots\!46\)\( p^{168} T^{32} + \)\(25\!\cdots\!50\)\( p^{182} T^{33} + \)\(34\!\cdots\!65\)\( p^{196} T^{34} + \)\(42\!\cdots\!42\)\( p^{210} T^{35} + \)\(46\!\cdots\!74\)\( p^{224} T^{36} + \)\(42\!\cdots\!22\)\( p^{238} T^{37} + \)\(34\!\cdots\!67\)\( p^{252} T^{38} + 2044625353338 p^{266} T^{39} + p^{280} T^{40} \) | |
| 53 | \( 1 - 1546271487546 T - \)\(23\!\cdots\!59\)\( T^{2} + \)\(63\!\cdots\!10\)\( T^{3} - \)\(42\!\cdots\!14\)\( T^{4} - \)\(23\!\cdots\!78\)\( T^{5} + \)\(12\!\cdots\!87\)\( T^{6} - \)\(20\!\cdots\!22\)\( T^{7} + \)\(95\!\cdots\!42\)\( T^{8} + \)\(21\!\cdots\!50\)\( T^{9} - \)\(40\!\cdots\!61\)\( T^{10} + \)\(23\!\cdots\!90\)\( T^{11} + \)\(41\!\cdots\!80\)\( T^{12} - \)\(20\!\cdots\!02\)\( T^{13} + \)\(31\!\cdots\!79\)\( T^{14} - \)\(34\!\cdots\!46\)\( T^{15} + \)\(21\!\cdots\!01\)\( T^{16} - \)\(17\!\cdots\!44\)\( T^{17} + \)\(15\!\cdots\!42\)\( T^{18} + \)\(60\!\cdots\!72\)\( T^{19} - \)\(15\!\cdots\!64\)\( T^{20} + \)\(60\!\cdots\!72\)\( p^{14} T^{21} + \)\(15\!\cdots\!42\)\( p^{28} T^{22} - \)\(17\!\cdots\!44\)\( p^{42} T^{23} + \)\(21\!\cdots\!01\)\( p^{56} T^{24} - \)\(34\!\cdots\!46\)\( p^{70} T^{25} + \)\(31\!\cdots\!79\)\( p^{84} T^{26} - \)\(20\!\cdots\!02\)\( p^{98} T^{27} + \)\(41\!\cdots\!80\)\( p^{112} T^{28} + \)\(23\!\cdots\!90\)\( p^{126} T^{29} - \)\(40\!\cdots\!61\)\( p^{140} T^{30} + \)\(21\!\cdots\!50\)\( p^{154} T^{31} + \)\(95\!\cdots\!42\)\( p^{168} T^{32} - \)\(20\!\cdots\!22\)\( p^{182} T^{33} + \)\(12\!\cdots\!87\)\( p^{196} T^{34} - \)\(23\!\cdots\!78\)\( p^{210} T^{35} - \)\(42\!\cdots\!14\)\( p^{224} T^{36} + \)\(63\!\cdots\!10\)\( p^{238} T^{37} - \)\(23\!\cdots\!59\)\( p^{252} T^{38} - 1546271487546 p^{266} T^{39} + p^{280} T^{40} \) | |
| 59 | \( 1 + 6798944731566 T + \)\(45\!\cdots\!23\)\( T^{2} + \)\(20\!\cdots\!86\)\( T^{3} + \)\(80\!\cdots\!82\)\( T^{4} + \)\(27\!\cdots\!50\)\( T^{5} + \)\(80\!\cdots\!77\)\( T^{6} + \)\(21\!\cdots\!38\)\( T^{7} + \)\(49\!\cdots\!90\)\( T^{8} + \)\(10\!\cdots\!38\)\( T^{9} + \)\(30\!\cdots\!63\)\( p T^{10} + \)\(22\!\cdots\!90\)\( T^{11} + \)\(74\!\cdots\!64\)\( T^{12} - \)\(10\!\cdots\!02\)\( T^{13} - \)\(49\!\cdots\!11\)\( T^{14} - \)\(17\!\cdots\!02\)\( T^{15} - \)\(56\!\cdots\!35\)\( T^{16} - \)\(16\!\cdots\!40\)\( T^{17} - \)\(46\!\cdots\!06\)\( T^{18} - \)\(12\!\cdots\!24\)\( T^{19} - \)\(31\!\cdots\!60\)\( T^{20} - \)\(12\!\cdots\!24\)\( p^{14} T^{21} - \)\(46\!\cdots\!06\)\( p^{28} T^{22} - \)\(16\!\cdots\!40\)\( p^{42} T^{23} - \)\(56\!\cdots\!35\)\( p^{56} T^{24} - \)\(17\!\cdots\!02\)\( p^{70} T^{25} - \)\(49\!\cdots\!11\)\( p^{84} T^{26} - \)\(10\!\cdots\!02\)\( p^{98} T^{27} + \)\(74\!\cdots\!64\)\( p^{112} T^{28} + \)\(22\!\cdots\!90\)\( p^{126} T^{29} + \)\(30\!\cdots\!63\)\( p^{141} T^{30} + \)\(10\!\cdots\!38\)\( p^{154} T^{31} + \)\(49\!\cdots\!90\)\( p^{168} T^{32} + \)\(21\!\cdots\!38\)\( p^{182} T^{33} + \)\(80\!\cdots\!77\)\( p^{196} T^{34} + \)\(27\!\cdots\!50\)\( p^{210} T^{35} + \)\(80\!\cdots\!82\)\( p^{224} T^{36} + \)\(20\!\cdots\!86\)\( p^{238} T^{37} + \)\(45\!\cdots\!23\)\( p^{252} T^{38} + 6798944731566 p^{266} T^{39} + p^{280} T^{40} \) | |
| 61 | \( 1 + 2214453865554 T + \)\(61\!\cdots\!57\)\( T^{2} + \)\(13\!\cdots\!90\)\( T^{3} + \)\(18\!\cdots\!22\)\( T^{4} + \)\(31\!\cdots\!30\)\( T^{5} + \)\(35\!\cdots\!35\)\( T^{6} + \)\(42\!\cdots\!50\)\( T^{7} + \)\(54\!\cdots\!94\)\( T^{8} + \)\(38\!\cdots\!70\)\( T^{9} + \)\(75\!\cdots\!71\)\( T^{10} + \)\(37\!\cdots\!98\)\( T^{11} + \)\(10\!\cdots\!72\)\( T^{12} + \)\(46\!\cdots\!30\)\( T^{13} + \)\(12\!\cdots\!15\)\( T^{14} + \)\(55\!\cdots\!74\)\( T^{15} + \)\(14\!\cdots\!61\)\( T^{16} + \)\(56\!\cdots\!72\)\( T^{17} + \)\(15\!\cdots\!70\)\( T^{18} + \)\(52\!\cdots\!84\)\( T^{19} + \)\(15\!\cdots\!44\)\( T^{20} + \)\(52\!\cdots\!84\)\( p^{14} T^{21} + \)\(15\!\cdots\!70\)\( p^{28} T^{22} + \)\(56\!\cdots\!72\)\( p^{42} T^{23} + \)\(14\!\cdots\!61\)\( p^{56} T^{24} + \)\(55\!\cdots\!74\)\( p^{70} T^{25} + \)\(12\!\cdots\!15\)\( p^{84} T^{26} + \)\(46\!\cdots\!30\)\( p^{98} T^{27} + \)\(10\!\cdots\!72\)\( p^{112} T^{28} + \)\(37\!\cdots\!98\)\( p^{126} T^{29} + \)\(75\!\cdots\!71\)\( p^{140} T^{30} + \)\(38\!\cdots\!70\)\( p^{154} T^{31} + \)\(54\!\cdots\!94\)\( p^{168} T^{32} + \)\(42\!\cdots\!50\)\( p^{182} T^{33} + \)\(35\!\cdots\!35\)\( p^{196} T^{34} + \)\(31\!\cdots\!30\)\( p^{210} T^{35} + \)\(18\!\cdots\!22\)\( p^{224} T^{36} + \)\(13\!\cdots\!90\)\( p^{238} T^{37} + \)\(61\!\cdots\!57\)\( p^{252} T^{38} + 2214453865554 p^{266} T^{39} + p^{280} T^{40} \) | |
| 67 | \( 1 + 4655820763226 T - \)\(20\!\cdots\!17\)\( T^{2} - \)\(19\!\cdots\!58\)\( T^{3} + \)\(26\!\cdots\!02\)\( T^{4} - \)\(43\!\cdots\!46\)\( T^{5} - \)\(31\!\cdots\!97\)\( p T^{6} + \)\(82\!\cdots\!66\)\( T^{7} + \)\(11\!\cdots\!30\)\( T^{8} - \)\(72\!\cdots\!10\)\( T^{9} - \)\(38\!\cdots\!11\)\( T^{10} + \)\(40\!\cdots\!30\)\( T^{11} + \)\(93\!\cdots\!12\)\( p T^{12} - \)\(16\!\cdots\!78\)\( T^{13} + \)\(15\!\cdots\!21\)\( T^{14} + \)\(46\!\cdots\!86\)\( T^{15} - \)\(15\!\cdots\!99\)\( T^{16} - \)\(97\!\cdots\!64\)\( T^{17} + \)\(69\!\cdots\!86\)\( T^{18} + \)\(10\!\cdots\!84\)\( T^{19} - \)\(26\!\cdots\!00\)\( T^{20} + \)\(10\!\cdots\!84\)\( p^{14} T^{21} + \)\(69\!\cdots\!86\)\( p^{28} T^{22} - \)\(97\!\cdots\!64\)\( p^{42} T^{23} - \)\(15\!\cdots\!99\)\( p^{56} T^{24} + \)\(46\!\cdots\!86\)\( p^{70} T^{25} + \)\(15\!\cdots\!21\)\( p^{84} T^{26} - \)\(16\!\cdots\!78\)\( p^{98} T^{27} + \)\(93\!\cdots\!12\)\( p^{113} T^{28} + \)\(40\!\cdots\!30\)\( p^{126} T^{29} - \)\(38\!\cdots\!11\)\( p^{140} T^{30} - \)\(72\!\cdots\!10\)\( p^{154} T^{31} + \)\(11\!\cdots\!30\)\( p^{168} T^{32} + \)\(82\!\cdots\!66\)\( p^{182} T^{33} - \)\(31\!\cdots\!97\)\( p^{197} T^{34} - \)\(43\!\cdots\!46\)\( p^{210} T^{35} + \)\(26\!\cdots\!02\)\( p^{224} T^{36} - \)\(19\!\cdots\!58\)\( p^{238} T^{37} - \)\(20\!\cdots\!17\)\( p^{252} T^{38} + 4655820763226 p^{266} T^{39} + p^{280} T^{40} \) | |
| 71 | \( ( 1 - 48303068747076 T + \)\(15\!\cdots\!86\)\( T^{2} - \)\(35\!\cdots\!88\)\( T^{3} + \)\(67\!\cdots\!97\)\( T^{4} - \)\(10\!\cdots\!52\)\( T^{5} + \)\(15\!\cdots\!76\)\( T^{6} - \)\(19\!\cdots\!12\)\( T^{7} + \)\(21\!\cdots\!78\)\( T^{8} - \)\(22\!\cdots\!24\)\( T^{9} + \)\(21\!\cdots\!44\)\( T^{10} - \)\(22\!\cdots\!24\)\( p^{14} T^{11} + \)\(21\!\cdots\!78\)\( p^{28} T^{12} - \)\(19\!\cdots\!12\)\( p^{42} T^{13} + \)\(15\!\cdots\!76\)\( p^{56} T^{14} - \)\(10\!\cdots\!52\)\( p^{70} T^{15} + \)\(67\!\cdots\!97\)\( p^{84} T^{16} - \)\(35\!\cdots\!88\)\( p^{98} T^{17} + \)\(15\!\cdots\!86\)\( p^{112} T^{18} - 48303068747076 p^{126} T^{19} + p^{140} T^{20} )^{2} \) | |
| 73 | \( 1 + 65348368908666 T + \)\(29\!\cdots\!25\)\( T^{2} + \)\(99\!\cdots\!18\)\( T^{3} + \)\(28\!\cdots\!30\)\( T^{4} + \)\(69\!\cdots\!06\)\( T^{5} + \)\(15\!\cdots\!23\)\( T^{6} + \)\(30\!\cdots\!06\)\( T^{7} + \)\(55\!\cdots\!38\)\( T^{8} + \)\(96\!\cdots\!58\)\( T^{9} + \)\(15\!\cdots\!79\)\( T^{10} + \)\(24\!\cdots\!54\)\( T^{11} + \)\(35\!\cdots\!32\)\( T^{12} + \)\(49\!\cdots\!78\)\( T^{13} + \)\(67\!\cdots\!27\)\( T^{14} + \)\(88\!\cdots\!26\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} + \)\(13\!\cdots\!92\)\( T^{17} + \)\(16\!\cdots\!42\)\( T^{18} + \)\(18\!\cdots\!48\)\( T^{19} + \)\(21\!\cdots\!16\)\( T^{20} + \)\(18\!\cdots\!48\)\( p^{14} T^{21} + \)\(16\!\cdots\!42\)\( p^{28} T^{22} + \)\(13\!\cdots\!92\)\( p^{42} T^{23} + \)\(11\!\cdots\!01\)\( p^{56} T^{24} + \)\(88\!\cdots\!26\)\( p^{70} T^{25} + \)\(67\!\cdots\!27\)\( p^{84} T^{26} + \)\(49\!\cdots\!78\)\( p^{98} T^{27} + \)\(35\!\cdots\!32\)\( p^{112} T^{28} + \)\(24\!\cdots\!54\)\( p^{126} T^{29} + \)\(15\!\cdots\!79\)\( p^{140} T^{30} + \)\(96\!\cdots\!58\)\( p^{154} T^{31} + \)\(55\!\cdots\!38\)\( p^{168} T^{32} + \)\(30\!\cdots\!06\)\( p^{182} T^{33} + \)\(15\!\cdots\!23\)\( p^{196} T^{34} + \)\(69\!\cdots\!06\)\( p^{210} T^{35} + \)\(28\!\cdots\!30\)\( p^{224} T^{36} + \)\(99\!\cdots\!18\)\( p^{238} T^{37} + \)\(29\!\cdots\!25\)\( p^{252} T^{38} + 65348368908666 p^{266} T^{39} + p^{280} T^{40} \) | |
| 79 | \( 1 + 60517474082978 T - \)\(54\!\cdots\!21\)\( T^{2} - \)\(89\!\cdots\!46\)\( T^{3} + \)\(26\!\cdots\!62\)\( T^{4} + \)\(96\!\cdots\!86\)\( T^{5} + \)\(13\!\cdots\!77\)\( T^{6} - \)\(73\!\cdots\!98\)\( T^{7} - \)\(25\!\cdots\!46\)\( T^{8} + \)\(44\!\cdots\!50\)\( T^{9} + \)\(25\!\cdots\!09\)\( T^{10} - \)\(22\!\cdots\!18\)\( T^{11} - \)\(18\!\cdots\!00\)\( T^{12} + \)\(91\!\cdots\!18\)\( T^{13} + \)\(11\!\cdots\!05\)\( T^{14} - \)\(31\!\cdots\!30\)\( T^{15} - \)\(64\!\cdots\!15\)\( T^{16} + \)\(80\!\cdots\!92\)\( T^{17} + \)\(29\!\cdots\!70\)\( T^{18} - \)\(10\!\cdots\!52\)\( T^{19} - \)\(11\!\cdots\!68\)\( T^{20} - \)\(10\!\cdots\!52\)\( p^{14} T^{21} + \)\(29\!\cdots\!70\)\( p^{28} T^{22} + \)\(80\!\cdots\!92\)\( p^{42} T^{23} - \)\(64\!\cdots\!15\)\( p^{56} T^{24} - \)\(31\!\cdots\!30\)\( p^{70} T^{25} + \)\(11\!\cdots\!05\)\( p^{84} T^{26} + \)\(91\!\cdots\!18\)\( p^{98} T^{27} - \)\(18\!\cdots\!00\)\( p^{112} T^{28} - \)\(22\!\cdots\!18\)\( p^{126} T^{29} + \)\(25\!\cdots\!09\)\( p^{140} T^{30} + \)\(44\!\cdots\!50\)\( p^{154} T^{31} - \)\(25\!\cdots\!46\)\( p^{168} T^{32} - \)\(73\!\cdots\!98\)\( p^{182} T^{33} + \)\(13\!\cdots\!77\)\( p^{196} T^{34} + \)\(96\!\cdots\!86\)\( p^{210} T^{35} + \)\(26\!\cdots\!62\)\( p^{224} T^{36} - \)\(89\!\cdots\!46\)\( p^{238} T^{37} - \)\(54\!\cdots\!21\)\( p^{252} T^{38} + 60517474082978 p^{266} T^{39} + p^{280} T^{40} \) | |
| 83 | \( 1 - \)\(80\!\cdots\!20\)\( T^{2} + \)\(29\!\cdots\!46\)\( T^{4} - \)\(65\!\cdots\!40\)\( T^{6} + \)\(94\!\cdots\!25\)\( T^{8} - \)\(94\!\cdots\!40\)\( T^{10} + \)\(68\!\cdots\!68\)\( T^{12} - \)\(46\!\cdots\!60\)\( T^{14} + \)\(41\!\cdots\!18\)\( T^{16} - \)\(44\!\cdots\!80\)\( T^{18} + \)\(38\!\cdots\!00\)\( T^{20} - \)\(44\!\cdots\!80\)\( p^{28} T^{22} + \)\(41\!\cdots\!18\)\( p^{56} T^{24} - \)\(46\!\cdots\!60\)\( p^{84} T^{26} + \)\(68\!\cdots\!68\)\( p^{112} T^{28} - \)\(94\!\cdots\!40\)\( p^{140} T^{30} + \)\(94\!\cdots\!25\)\( p^{168} T^{32} - \)\(65\!\cdots\!40\)\( p^{196} T^{34} + \)\(29\!\cdots\!46\)\( p^{224} T^{36} - \)\(80\!\cdots\!20\)\( p^{252} T^{38} + p^{280} T^{40} \) | |
| 89 | \( 1 + 237147002561826 T + \)\(37\!\cdots\!65\)\( T^{2} + \)\(44\!\cdots\!98\)\( T^{3} + \)\(43\!\cdots\!26\)\( T^{4} + \)\(37\!\cdots\!62\)\( T^{5} + \)\(27\!\cdots\!71\)\( T^{6} + \)\(18\!\cdots\!14\)\( T^{7} + \)\(11\!\cdots\!18\)\( T^{8} + \)\(64\!\cdots\!06\)\( T^{9} + \)\(32\!\cdots\!27\)\( T^{10} + \)\(14\!\cdots\!50\)\( T^{11} + \)\(57\!\cdots\!28\)\( T^{12} + \)\(18\!\cdots\!66\)\( T^{13} + \)\(31\!\cdots\!91\)\( T^{14} - \)\(12\!\cdots\!46\)\( T^{15} - \)\(17\!\cdots\!11\)\( T^{16} - \)\(13\!\cdots\!24\)\( T^{17} - \)\(76\!\cdots\!10\)\( T^{18} - \)\(38\!\cdots\!60\)\( T^{19} - \)\(18\!\cdots\!32\)\( T^{20} - \)\(38\!\cdots\!60\)\( p^{14} T^{21} - \)\(76\!\cdots\!10\)\( p^{28} T^{22} - \)\(13\!\cdots\!24\)\( p^{42} T^{23} - \)\(17\!\cdots\!11\)\( p^{56} T^{24} - \)\(12\!\cdots\!46\)\( p^{70} T^{25} + \)\(31\!\cdots\!91\)\( p^{84} T^{26} + \)\(18\!\cdots\!66\)\( p^{98} T^{27} + \)\(57\!\cdots\!28\)\( p^{112} T^{28} + \)\(14\!\cdots\!50\)\( p^{126} T^{29} + \)\(32\!\cdots\!27\)\( p^{140} T^{30} + \)\(64\!\cdots\!06\)\( p^{154} T^{31} + \)\(11\!\cdots\!18\)\( p^{168} T^{32} + \)\(18\!\cdots\!14\)\( p^{182} T^{33} + \)\(27\!\cdots\!71\)\( p^{196} T^{34} + \)\(37\!\cdots\!62\)\( p^{210} T^{35} + \)\(43\!\cdots\!26\)\( p^{224} T^{36} + \)\(44\!\cdots\!98\)\( p^{238} T^{37} + \)\(37\!\cdots\!65\)\( p^{252} T^{38} + 237147002561826 p^{266} T^{39} + p^{280} T^{40} \) | |
| 97 | \( 1 - \)\(66\!\cdots\!20\)\( T^{2} + \)\(23\!\cdots\!10\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{6} + \)\(98\!\cdots\!85\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!80\)\( T^{12} - \)\(18\!\cdots\!60\)\( T^{14} + \)\(16\!\cdots\!10\)\( T^{16} - \)\(13\!\cdots\!00\)\( T^{18} + \)\(98\!\cdots\!36\)\( p^{2} T^{20} - \)\(13\!\cdots\!00\)\( p^{28} T^{22} + \)\(16\!\cdots\!10\)\( p^{56} T^{24} - \)\(18\!\cdots\!60\)\( p^{84} T^{26} + \)\(17\!\cdots\!80\)\( p^{112} T^{28} - \)\(14\!\cdots\!00\)\( p^{140} T^{30} + \)\(98\!\cdots\!85\)\( p^{168} T^{32} - \)\(54\!\cdots\!00\)\( p^{196} T^{34} + \)\(23\!\cdots\!10\)\( p^{224} T^{36} - \)\(66\!\cdots\!20\)\( p^{252} T^{38} + p^{280} T^{40} \) | |
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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
−2.63911518138859897580802014608, −2.53543342547992782440943768853, −2.37492052508269786173851714352, −2.36513306573165323237428106939, −2.24042454102078924469013354647, −2.09800297584096273207199381292, −1.92325815960845896350303976343, −1.78966370199260471053543189647, −1.73340714952662175673632930977, −1.68270049384606095610651839271, −1.47401212529188262838662974002, −1.43209136641335523416168528939, −1.41410052147240746022303207320, −1.28147778712006646595331550507, −1.04165492530931547247547726871, −0.899263361815036445163496445790, −0.78574526028332738067867616311, −0.77147476059681361058640481250, −0.66757527708247795920912827642, −0.49202539503614101735372906570, −0.46882384487146108395447628795, −0.34456148547312468466968508363, −0.19022937715260855038714366863, −0.13145027009017257288098280500, −0.00150590488972926791058820847, 0.00150590488972926791058820847, 0.13145027009017257288098280500, 0.19022937715260855038714366863, 0.34456148547312468466968508363, 0.46882384487146108395447628795, 0.49202539503614101735372906570, 0.66757527708247795920912827642, 0.77147476059681361058640481250, 0.78574526028332738067867616311, 0.899263361815036445163496445790, 1.04165492530931547247547726871, 1.28147778712006646595331550507, 1.41410052147240746022303207320, 1.43209136641335523416168528939, 1.47401212529188262838662974002, 1.68270049384606095610651839271, 1.73340714952662175673632930977, 1.78966370199260471053543189647, 1.92325815960845896350303976343, 2.09800297584096273207199381292, 2.24042454102078924469013354647, 2.36513306573165323237428106939, 2.37492052508269786173851714352, 2.53543342547992782440943768853, 2.63911518138859897580802014608
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.