Dirichlet series
| L(s) = 1 | + 30·4-s − 30·5-s − 42·7-s − 30·11-s − 38·13-s + 357·16-s + 618·17-s − 318·19-s − 900·20-s + 546·23-s + 450·25-s − 1.26e3·28-s + 948·29-s − 4.08e3·31-s − 426·32-s + 1.26e3·35-s + 4.54e3·37-s − 4.97e3·41-s + 642·43-s − 900·44-s − 516·47-s + 3.31e3·49-s − 1.14e3·52-s + 2.38e3·53-s + 900·55-s − 1.22e4·59-s + 1.16e4·61-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 6/5·5-s − 6/7·7-s − 0.247·11-s − 0.224·13-s + 1.39·16-s + 2.13·17-s − 0.880·19-s − 9/4·20-s + 1.03·23-s + 0.719·25-s − 1.60·28-s + 1.12·29-s − 4.24·31-s − 0.416·32-s + 1.02·35-s + 3.31·37-s − 2.95·41-s + 0.347·43-s − 0.464·44-s − 0.233·47-s + 1.38·49-s − 0.421·52-s + 0.850·53-s + 0.297·55-s − 3.50·59-s + 3.14·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{40} \cdot 13^{20}\right)^{s/2} \, \Gamma_{\C}(s+2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{40} \cdot 13^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.48336\times 10^{21}\) |
| Root analytic conductor: | \(3.47768\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{40} \cdot 13^{20} ,\ ( \ : [2]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.03246962420\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03246962420\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 + 38 T + 71165 T^{2} + 6475934 T^{3} + 97744053 p T^{4} + 1685681236 p^{2} T^{5} - 11580762280 p^{3} T^{6} - 32753614404 p^{4} T^{7} - 3131675520611 p^{5} T^{8} - 90582832692798 p^{6} T^{9} - 28209006056275 p^{8} T^{10} - 90582832692798 p^{10} T^{11} - 3131675520611 p^{13} T^{12} - 32753614404 p^{16} T^{13} - 11580762280 p^{19} T^{14} + 1685681236 p^{22} T^{15} + 97744053 p^{25} T^{16} + 6475934 p^{28} T^{17} + 71165 p^{32} T^{18} + 38 p^{36} T^{19} + p^{40} T^{20} \) | |
| good | 2 | \( 1 - 15 p T^{2} + 543 T^{4} + 213 p T^{5} - 3645 p T^{6} - 6873 p T^{7} + 55117 T^{8} + 147039 p T^{9} - 157095 p T^{10} - 772827 p^{2} T^{11} + 940087 p^{4} T^{12} - 1280919 p^{3} T^{13} - 45744825 p^{3} T^{14} + 24381627 p^{4} T^{15} + 12154427 p^{9} T^{16} - 318150885 p^{5} T^{17} - 2479175307 p^{5} T^{18} + 1579456467 p^{6} T^{19} + 13338220921 p^{6} T^{20} + 1579456467 p^{10} T^{21} - 2479175307 p^{13} T^{22} - 318150885 p^{17} T^{23} + 12154427 p^{25} T^{24} + 24381627 p^{24} T^{25} - 45744825 p^{27} T^{26} - 1280919 p^{31} T^{27} + 940087 p^{36} T^{28} - 772827 p^{38} T^{29} - 157095 p^{41} T^{30} + 147039 p^{45} T^{31} + 55117 p^{48} T^{32} - 6873 p^{53} T^{33} - 3645 p^{57} T^{34} + 213 p^{61} T^{35} + 543 p^{64} T^{36} - 15 p^{73} T^{38} + p^{80} T^{40} \) |
| 5 | \( 1 + 6 p T + 18 p^{2} T^{2} + 22608 T^{3} - 23859 p^{2} T^{4} - 14464116 T^{5} + 90051102 T^{6} - 5618104746 T^{7} + 170293224524 p T^{8} + 14563286115942 T^{9} + 17008654619574 p T^{10} + 5216249014934892 T^{11} - 546501017810171741 T^{12} - 5073989163828041916 T^{13} - 9178044760822729698 T^{14} - \)\(30\!\cdots\!06\)\( p T^{15} + \)\(29\!\cdots\!43\)\( T^{16} + \)\(12\!\cdots\!68\)\( T^{17} - \)\(97\!\cdots\!12\)\( T^{18} - \)\(44\!\cdots\!12\)\( T^{19} - \)\(13\!\cdots\!56\)\( T^{20} - \)\(44\!\cdots\!12\)\( p^{4} T^{21} - \)\(97\!\cdots\!12\)\( p^{8} T^{22} + \)\(12\!\cdots\!68\)\( p^{12} T^{23} + \)\(29\!\cdots\!43\)\( p^{16} T^{24} - \)\(30\!\cdots\!06\)\( p^{21} T^{25} - 9178044760822729698 p^{24} T^{26} - 5073989163828041916 p^{28} T^{27} - 546501017810171741 p^{32} T^{28} + 5216249014934892 p^{36} T^{29} + 17008654619574 p^{41} T^{30} + 14563286115942 p^{44} T^{31} + 170293224524 p^{49} T^{32} - 5618104746 p^{52} T^{33} + 90051102 p^{56} T^{34} - 14464116 p^{60} T^{35} - 23859 p^{66} T^{36} + 22608 p^{68} T^{37} + 18 p^{74} T^{38} + 6 p^{77} T^{39} + p^{80} T^{40} \) | |
| 7 | \( 1 + 6 p T - 222 p T^{2} - 206368 T^{3} + 1189568 T^{4} + 598816902 T^{5} + 13660561772 T^{6} - 27810769226 p T^{7} + 8875332582111 T^{8} + 2864274201410226 T^{9} - 12396929423857844 T^{10} - 4365526724224420338 T^{11} - 24587780303999151146 T^{12} + \)\(19\!\cdots\!04\)\( T^{13} + \)\(10\!\cdots\!86\)\( p T^{14} + \)\(10\!\cdots\!62\)\( T^{15} + \)\(90\!\cdots\!97\)\( T^{16} + \)\(42\!\cdots\!12\)\( T^{17} + \)\(18\!\cdots\!04\)\( T^{18} + \)\(31\!\cdots\!40\)\( T^{19} + \)\(71\!\cdots\!74\)\( T^{20} + \)\(31\!\cdots\!40\)\( p^{4} T^{21} + \)\(18\!\cdots\!04\)\( p^{8} T^{22} + \)\(42\!\cdots\!12\)\( p^{12} T^{23} + \)\(90\!\cdots\!97\)\( p^{16} T^{24} + \)\(10\!\cdots\!62\)\( p^{20} T^{25} + \)\(10\!\cdots\!86\)\( p^{25} T^{26} + \)\(19\!\cdots\!04\)\( p^{28} T^{27} - 24587780303999151146 p^{32} T^{28} - 4365526724224420338 p^{36} T^{29} - 12396929423857844 p^{40} T^{30} + 2864274201410226 p^{44} T^{31} + 8875332582111 p^{48} T^{32} - 27810769226 p^{53} T^{33} + 13660561772 p^{56} T^{34} + 598816902 p^{60} T^{35} + 1189568 p^{64} T^{36} - 206368 p^{68} T^{37} - 222 p^{73} T^{38} + 6 p^{77} T^{39} + p^{80} T^{40} \) | |
| 11 | \( 1 + 30 T - 59964 T^{2} + 1057896 T^{3} + 1939433373 T^{4} - 74920698396 T^{5} - 39121678222056 T^{6} + 178067587429338 p T^{7} + 599293155146823481 T^{8} - 23940917947778518248 T^{9} - \)\(86\!\cdots\!16\)\( T^{10} + \)\(75\!\cdots\!00\)\( T^{11} + \)\(15\!\cdots\!72\)\( T^{12} + \)\(13\!\cdots\!24\)\( T^{13} - \)\(31\!\cdots\!08\)\( T^{14} + \)\(20\!\cdots\!08\)\( T^{15} + \)\(55\!\cdots\!73\)\( T^{16} - \)\(97\!\cdots\!58\)\( T^{17} - \)\(80\!\cdots\!68\)\( T^{18} + \)\(74\!\cdots\!76\)\( p T^{19} + \)\(94\!\cdots\!65\)\( p^{2} T^{20} + \)\(74\!\cdots\!76\)\( p^{5} T^{21} - \)\(80\!\cdots\!68\)\( p^{8} T^{22} - \)\(97\!\cdots\!58\)\( p^{12} T^{23} + \)\(55\!\cdots\!73\)\( p^{16} T^{24} + \)\(20\!\cdots\!08\)\( p^{20} T^{25} - \)\(31\!\cdots\!08\)\( p^{24} T^{26} + \)\(13\!\cdots\!24\)\( p^{28} T^{27} + \)\(15\!\cdots\!72\)\( p^{32} T^{28} + \)\(75\!\cdots\!00\)\( p^{36} T^{29} - \)\(86\!\cdots\!16\)\( p^{40} T^{30} - 23940917947778518248 p^{44} T^{31} + 599293155146823481 p^{48} T^{32} + 178067587429338 p^{53} T^{33} - 39121678222056 p^{56} T^{34} - 74920698396 p^{60} T^{35} + 1939433373 p^{64} T^{36} + 1057896 p^{68} T^{37} - 59964 p^{72} T^{38} + 30 p^{76} T^{39} + p^{80} T^{40} \) | |
| 17 | \( 1 - 618 T + 538693 T^{2} - 254235930 T^{3} + 126549599815 T^{4} - 47076161683248 T^{5} + 16821376302854820 T^{6} - 4881037143786195708 T^{7} + \)\(12\!\cdots\!54\)\( T^{8} - \)\(26\!\cdots\!92\)\( T^{9} + \)\(43\!\cdots\!76\)\( T^{10} - \)\(10\!\cdots\!84\)\( T^{11} - \)\(17\!\cdots\!95\)\( T^{12} + \)\(77\!\cdots\!54\)\( T^{13} - \)\(21\!\cdots\!05\)\( T^{14} + \)\(22\!\cdots\!02\)\( T^{15} + \)\(65\!\cdots\!99\)\( T^{16} - \)\(45\!\cdots\!32\)\( T^{17} + \)\(15\!\cdots\!40\)\( T^{18} - \)\(67\!\cdots\!92\)\( T^{19} + \)\(17\!\cdots\!24\)\( T^{20} - \)\(67\!\cdots\!92\)\( p^{4} T^{21} + \)\(15\!\cdots\!40\)\( p^{8} T^{22} - \)\(45\!\cdots\!32\)\( p^{12} T^{23} + \)\(65\!\cdots\!99\)\( p^{16} T^{24} + \)\(22\!\cdots\!02\)\( p^{20} T^{25} - \)\(21\!\cdots\!05\)\( p^{24} T^{26} + \)\(77\!\cdots\!54\)\( p^{28} T^{27} - \)\(17\!\cdots\!95\)\( p^{32} T^{28} - \)\(10\!\cdots\!84\)\( p^{36} T^{29} + \)\(43\!\cdots\!76\)\( p^{40} T^{30} - \)\(26\!\cdots\!92\)\( p^{44} T^{31} + \)\(12\!\cdots\!54\)\( p^{48} T^{32} - 4881037143786195708 p^{52} T^{33} + 16821376302854820 p^{56} T^{34} - 47076161683248 p^{60} T^{35} + 126549599815 p^{64} T^{36} - 254235930 p^{68} T^{37} + 538693 p^{72} T^{38} - 618 p^{76} T^{39} + p^{80} T^{40} \) | |
| 19 | \( 1 + 318 T + 14928 T^{2} + 84445976 T^{3} + 23771215889 T^{4} - 1426561334556 T^{5} + 3029138287132676 T^{6} + 180278943768261862 T^{7} - \)\(55\!\cdots\!47\)\( T^{8} - \)\(71\!\cdots\!32\)\( T^{9} - \)\(52\!\cdots\!56\)\( T^{10} - \)\(30\!\cdots\!04\)\( T^{11} - \)\(94\!\cdots\!20\)\( T^{12} - \)\(28\!\cdots\!48\)\( T^{13} - \)\(63\!\cdots\!20\)\( T^{14} - \)\(19\!\cdots\!56\)\( T^{15} - \)\(40\!\cdots\!15\)\( T^{16} + \)\(98\!\cdots\!46\)\( T^{17} + \)\(10\!\cdots\!08\)\( T^{18} + \)\(52\!\cdots\!52\)\( T^{19} + \)\(68\!\cdots\!17\)\( T^{20} + \)\(52\!\cdots\!52\)\( p^{4} T^{21} + \)\(10\!\cdots\!08\)\( p^{8} T^{22} + \)\(98\!\cdots\!46\)\( p^{12} T^{23} - \)\(40\!\cdots\!15\)\( p^{16} T^{24} - \)\(19\!\cdots\!56\)\( p^{20} T^{25} - \)\(63\!\cdots\!20\)\( p^{24} T^{26} - \)\(28\!\cdots\!48\)\( p^{28} T^{27} - \)\(94\!\cdots\!20\)\( p^{32} T^{28} - \)\(30\!\cdots\!04\)\( p^{36} T^{29} - \)\(52\!\cdots\!56\)\( p^{40} T^{30} - \)\(71\!\cdots\!32\)\( p^{44} T^{31} - \)\(55\!\cdots\!47\)\( p^{48} T^{32} + 180278943768261862 p^{52} T^{33} + 3029138287132676 p^{56} T^{34} - 1426561334556 p^{60} T^{35} + 23771215889 p^{64} T^{36} + 84445976 p^{68} T^{37} + 14928 p^{72} T^{38} + 318 p^{76} T^{39} + p^{80} T^{40} \) | |
| 23 | \( 1 - 546 T + 1332190 T^{2} - 673118628 T^{3} + 818323630309 T^{4} - 452622795939504 T^{5} + 390484656560518146 T^{6} - \)\(25\!\cdots\!54\)\( T^{7} + \)\(18\!\cdots\!33\)\( T^{8} - \)\(11\!\cdots\!36\)\( T^{9} + \)\(80\!\cdots\!20\)\( T^{10} - \)\(47\!\cdots\!40\)\( T^{11} + \)\(31\!\cdots\!96\)\( T^{12} - \)\(17\!\cdots\!24\)\( T^{13} + \)\(11\!\cdots\!36\)\( T^{14} - \)\(61\!\cdots\!28\)\( T^{15} + \)\(37\!\cdots\!69\)\( T^{16} - \)\(19\!\cdots\!58\)\( T^{17} + \)\(11\!\cdots\!86\)\( T^{18} - \)\(60\!\cdots\!60\)\( T^{19} + \)\(33\!\cdots\!93\)\( T^{20} - \)\(60\!\cdots\!60\)\( p^{4} T^{21} + \)\(11\!\cdots\!86\)\( p^{8} T^{22} - \)\(19\!\cdots\!58\)\( p^{12} T^{23} + \)\(37\!\cdots\!69\)\( p^{16} T^{24} - \)\(61\!\cdots\!28\)\( p^{20} T^{25} + \)\(11\!\cdots\!36\)\( p^{24} T^{26} - \)\(17\!\cdots\!24\)\( p^{28} T^{27} + \)\(31\!\cdots\!96\)\( p^{32} T^{28} - \)\(47\!\cdots\!40\)\( p^{36} T^{29} + \)\(80\!\cdots\!20\)\( p^{40} T^{30} - \)\(11\!\cdots\!36\)\( p^{44} T^{31} + \)\(18\!\cdots\!33\)\( p^{48} T^{32} - \)\(25\!\cdots\!54\)\( p^{52} T^{33} + 390484656560518146 p^{56} T^{34} - 452622795939504 p^{60} T^{35} + 818323630309 p^{64} T^{36} - 673118628 p^{68} T^{37} + 1332190 p^{72} T^{38} - 546 p^{76} T^{39} + p^{80} T^{40} \) | |
| 29 | \( 1 - 948 T - 2142909 T^{2} + 2626215684 T^{3} + 1253761459769 T^{4} - 2524101680387640 T^{5} + 656317639076839262 T^{6} + \)\(24\!\cdots\!84\)\( T^{7} - \)\(88\!\cdots\!94\)\( T^{8} + \)\(14\!\cdots\!60\)\( T^{9} - \)\(42\!\cdots\!10\)\( p T^{10} - \)\(11\!\cdots\!84\)\( T^{11} + \)\(71\!\cdots\!27\)\( T^{12} + \)\(15\!\cdots\!72\)\( T^{13} - \)\(69\!\cdots\!95\)\( T^{14} + \)\(54\!\cdots\!44\)\( T^{15} + \)\(37\!\cdots\!03\)\( T^{16} - \)\(65\!\cdots\!40\)\( T^{17} + \)\(47\!\cdots\!68\)\( T^{18} + \)\(24\!\cdots\!68\)\( T^{19} - \)\(17\!\cdots\!00\)\( T^{20} + \)\(24\!\cdots\!68\)\( p^{4} T^{21} + \)\(47\!\cdots\!68\)\( p^{8} T^{22} - \)\(65\!\cdots\!40\)\( p^{12} T^{23} + \)\(37\!\cdots\!03\)\( p^{16} T^{24} + \)\(54\!\cdots\!44\)\( p^{20} T^{25} - \)\(69\!\cdots\!95\)\( p^{24} T^{26} + \)\(15\!\cdots\!72\)\( p^{28} T^{27} + \)\(71\!\cdots\!27\)\( p^{32} T^{28} - \)\(11\!\cdots\!84\)\( p^{36} T^{29} - \)\(42\!\cdots\!10\)\( p^{41} T^{30} + \)\(14\!\cdots\!60\)\( p^{44} T^{31} - \)\(88\!\cdots\!94\)\( p^{48} T^{32} + \)\(24\!\cdots\!84\)\( p^{52} T^{33} + 656317639076839262 p^{56} T^{34} - 2524101680387640 p^{60} T^{35} + 1253761459769 p^{64} T^{36} + 2626215684 p^{68} T^{37} - 2142909 p^{72} T^{38} - 948 p^{76} T^{39} + p^{80} T^{40} \) | |
| 31 | \( 1 + 4084 T + 8339528 T^{2} + 12824626816 T^{3} + 19303013086216 T^{4} + 28259150892750228 T^{5} + 36667880613960334032 T^{6} + \)\(42\!\cdots\!32\)\( T^{7} + \)\(47\!\cdots\!62\)\( T^{8} + \)\(50\!\cdots\!04\)\( T^{9} + \)\(50\!\cdots\!92\)\( T^{10} + \)\(45\!\cdots\!76\)\( T^{11} + \)\(37\!\cdots\!98\)\( T^{12} + \)\(25\!\cdots\!48\)\( T^{13} + \)\(13\!\cdots\!72\)\( T^{14} + \)\(63\!\cdots\!04\)\( T^{15} - \)\(13\!\cdots\!39\)\( T^{16} - \)\(24\!\cdots\!80\)\( T^{17} - \)\(32\!\cdots\!72\)\( T^{18} - \)\(36\!\cdots\!04\)\( T^{19} - \)\(37\!\cdots\!60\)\( T^{20} - \)\(36\!\cdots\!04\)\( p^{4} T^{21} - \)\(32\!\cdots\!72\)\( p^{8} T^{22} - \)\(24\!\cdots\!80\)\( p^{12} T^{23} - \)\(13\!\cdots\!39\)\( p^{16} T^{24} + \)\(63\!\cdots\!04\)\( p^{20} T^{25} + \)\(13\!\cdots\!72\)\( p^{24} T^{26} + \)\(25\!\cdots\!48\)\( p^{28} T^{27} + \)\(37\!\cdots\!98\)\( p^{32} T^{28} + \)\(45\!\cdots\!76\)\( p^{36} T^{29} + \)\(50\!\cdots\!92\)\( p^{40} T^{30} + \)\(50\!\cdots\!04\)\( p^{44} T^{31} + \)\(47\!\cdots\!62\)\( p^{48} T^{32} + \)\(42\!\cdots\!32\)\( p^{52} T^{33} + 36667880613960334032 p^{56} T^{34} + 28259150892750228 p^{60} T^{35} + 19303013086216 p^{64} T^{36} + 12824626816 p^{68} T^{37} + 8339528 p^{72} T^{38} + 4084 p^{76} T^{39} + p^{80} T^{40} \) | |
| 37 | \( 1 - 4544 T + 18271013 T^{2} - 45249478640 T^{3} + 102271966551355 T^{4} - 160143548453544780 T^{5} + \)\(22\!\cdots\!60\)\( T^{6} - \)\(13\!\cdots\!56\)\( T^{7} - \)\(64\!\cdots\!50\)\( T^{8} + \)\(72\!\cdots\!32\)\( T^{9} - \)\(14\!\cdots\!28\)\( T^{10} + \)\(26\!\cdots\!20\)\( T^{11} - \)\(27\!\cdots\!47\)\( T^{12} + \)\(28\!\cdots\!88\)\( T^{13} + \)\(23\!\cdots\!75\)\( T^{14} - \)\(22\!\cdots\!12\)\( T^{15} + \)\(85\!\cdots\!03\)\( T^{16} - \)\(21\!\cdots\!68\)\( p T^{17} + \)\(13\!\cdots\!24\)\( T^{18} - \)\(47\!\cdots\!00\)\( T^{19} + \)\(14\!\cdots\!68\)\( T^{20} - \)\(47\!\cdots\!00\)\( p^{4} T^{21} + \)\(13\!\cdots\!24\)\( p^{8} T^{22} - \)\(21\!\cdots\!68\)\( p^{13} T^{23} + \)\(85\!\cdots\!03\)\( p^{16} T^{24} - \)\(22\!\cdots\!12\)\( p^{20} T^{25} + \)\(23\!\cdots\!75\)\( p^{24} T^{26} + \)\(28\!\cdots\!88\)\( p^{28} T^{27} - \)\(27\!\cdots\!47\)\( p^{32} T^{28} + \)\(26\!\cdots\!20\)\( p^{36} T^{29} - \)\(14\!\cdots\!28\)\( p^{40} T^{30} + \)\(72\!\cdots\!32\)\( p^{44} T^{31} - \)\(64\!\cdots\!50\)\( p^{48} T^{32} - \)\(13\!\cdots\!56\)\( p^{52} T^{33} + \)\(22\!\cdots\!60\)\( p^{56} T^{34} - 160143548453544780 p^{60} T^{35} + 102271966551355 p^{64} T^{36} - 45249478640 p^{68} T^{37} + 18271013 p^{72} T^{38} - 4544 p^{76} T^{39} + p^{80} T^{40} \) | |
| 41 | \( 1 + 4974 T + 9119061 T^{2} - 7511891418 T^{3} - 70229191532133 T^{4} - 120459790048495152 T^{5} + 58635740045516288508 T^{6} + \)\(64\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!62\)\( T^{8} - \)\(94\!\cdots\!44\)\( T^{9} - \)\(39\!\cdots\!40\)\( T^{10} - \)\(73\!\cdots\!72\)\( T^{11} - \)\(18\!\cdots\!99\)\( T^{12} + \)\(20\!\cdots\!74\)\( T^{13} + \)\(11\!\cdots\!31\)\( p T^{14} + \)\(30\!\cdots\!78\)\( T^{15} - \)\(88\!\cdots\!21\)\( T^{16} - \)\(26\!\cdots\!64\)\( T^{17} - \)\(20\!\cdots\!12\)\( T^{18} + \)\(41\!\cdots\!48\)\( T^{19} + \)\(13\!\cdots\!28\)\( T^{20} + \)\(41\!\cdots\!48\)\( p^{4} T^{21} - \)\(20\!\cdots\!12\)\( p^{8} T^{22} - \)\(26\!\cdots\!64\)\( p^{12} T^{23} - \)\(88\!\cdots\!21\)\( p^{16} T^{24} + \)\(30\!\cdots\!78\)\( p^{20} T^{25} + \)\(11\!\cdots\!31\)\( p^{25} T^{26} + \)\(20\!\cdots\!74\)\( p^{28} T^{27} - \)\(18\!\cdots\!99\)\( p^{32} T^{28} - \)\(73\!\cdots\!72\)\( p^{36} T^{29} - \)\(39\!\cdots\!40\)\( p^{40} T^{30} - \)\(94\!\cdots\!44\)\( p^{44} T^{31} + \)\(10\!\cdots\!62\)\( p^{48} T^{32} + \)\(64\!\cdots\!36\)\( p^{52} T^{33} + 58635740045516288508 p^{56} T^{34} - 120459790048495152 p^{60} T^{35} - 70229191532133 p^{64} T^{36} - 7511891418 p^{68} T^{37} + 9119061 p^{72} T^{38} + 4974 p^{76} T^{39} + p^{80} T^{40} \) | |
| 43 | \( 1 - 642 T + 23983744 T^{2} - 15309360552 T^{3} + 297707167583584 T^{4} - 144179089972852194 T^{5} + \)\(25\!\cdots\!76\)\( T^{6} - \)\(63\!\cdots\!70\)\( T^{7} + \)\(16\!\cdots\!03\)\( T^{8} + \)\(32\!\cdots\!18\)\( T^{9} + \)\(84\!\cdots\!68\)\( T^{10} + \)\(24\!\cdots\!70\)\( T^{11} + \)\(39\!\cdots\!14\)\( T^{12} + \)\(19\!\cdots\!36\)\( T^{13} + \)\(17\!\cdots\!40\)\( T^{14} + \)\(93\!\cdots\!30\)\( T^{15} + \)\(71\!\cdots\!85\)\( T^{16} + \)\(35\!\cdots\!84\)\( T^{17} + \)\(27\!\cdots\!84\)\( T^{18} + \)\(12\!\cdots\!00\)\( T^{19} + \)\(10\!\cdots\!38\)\( T^{20} + \)\(12\!\cdots\!00\)\( p^{4} T^{21} + \)\(27\!\cdots\!84\)\( p^{8} T^{22} + \)\(35\!\cdots\!84\)\( p^{12} T^{23} + \)\(71\!\cdots\!85\)\( p^{16} T^{24} + \)\(93\!\cdots\!30\)\( p^{20} T^{25} + \)\(17\!\cdots\!40\)\( p^{24} T^{26} + \)\(19\!\cdots\!36\)\( p^{28} T^{27} + \)\(39\!\cdots\!14\)\( p^{32} T^{28} + \)\(24\!\cdots\!70\)\( p^{36} T^{29} + \)\(84\!\cdots\!68\)\( p^{40} T^{30} + \)\(32\!\cdots\!18\)\( p^{44} T^{31} + \)\(16\!\cdots\!03\)\( p^{48} T^{32} - \)\(63\!\cdots\!70\)\( p^{52} T^{33} + \)\(25\!\cdots\!76\)\( p^{56} T^{34} - 144179089972852194 p^{60} T^{35} + 297707167583584 p^{64} T^{36} - 15309360552 p^{68} T^{37} + 23983744 p^{72} T^{38} - 642 p^{76} T^{39} + p^{80} T^{40} \) | |
| 47 | \( 1 + 516 T + 133128 T^{2} - 9948036648 T^{3} + 13762932040974 T^{4} + 122411928031291200 T^{5} + \)\(11\!\cdots\!80\)\( T^{6} + \)\(96\!\cdots\!88\)\( p T^{7} + \)\(45\!\cdots\!01\)\( T^{8} + \)\(62\!\cdots\!88\)\( T^{9} + \)\(36\!\cdots\!56\)\( p T^{10} + \)\(58\!\cdots\!20\)\( T^{11} + \)\(79\!\cdots\!96\)\( T^{12} + \)\(96\!\cdots\!64\)\( T^{13} + \)\(85\!\cdots\!76\)\( T^{14} + \)\(20\!\cdots\!96\)\( T^{15} + \)\(10\!\cdots\!26\)\( T^{16} + \)\(26\!\cdots\!52\)\( T^{17} + \)\(29\!\cdots\!52\)\( T^{18} + \)\(11\!\cdots\!68\)\( T^{19} + \)\(41\!\cdots\!68\)\( T^{20} + \)\(11\!\cdots\!68\)\( p^{4} T^{21} + \)\(29\!\cdots\!52\)\( p^{8} T^{22} + \)\(26\!\cdots\!52\)\( p^{12} T^{23} + \)\(10\!\cdots\!26\)\( p^{16} T^{24} + \)\(20\!\cdots\!96\)\( p^{20} T^{25} + \)\(85\!\cdots\!76\)\( p^{24} T^{26} + \)\(96\!\cdots\!64\)\( p^{28} T^{27} + \)\(79\!\cdots\!96\)\( p^{32} T^{28} + \)\(58\!\cdots\!20\)\( p^{36} T^{29} + \)\(36\!\cdots\!56\)\( p^{41} T^{30} + \)\(62\!\cdots\!88\)\( p^{44} T^{31} + \)\(45\!\cdots\!01\)\( p^{48} T^{32} + \)\(96\!\cdots\!88\)\( p^{53} T^{33} + \)\(11\!\cdots\!80\)\( p^{56} T^{34} + 122411928031291200 p^{60} T^{35} + 13762932040974 p^{64} T^{36} - 9948036648 p^{68} T^{37} + 133128 p^{72} T^{38} + 516 p^{76} T^{39} + p^{80} T^{40} \) | |
| 53 | \( ( 1 - 1194 T + 54638937 T^{2} - 47610302814 T^{3} + 1423572865069108 T^{4} - 913745011043542062 T^{5} + \)\(23\!\cdots\!67\)\( T^{6} - \)\(11\!\cdots\!58\)\( T^{7} + \)\(28\!\cdots\!35\)\( T^{8} - \)\(11\!\cdots\!76\)\( T^{9} + \)\(26\!\cdots\!24\)\( T^{10} - \)\(11\!\cdots\!76\)\( p^{4} T^{11} + \)\(28\!\cdots\!35\)\( p^{8} T^{12} - \)\(11\!\cdots\!58\)\( p^{12} T^{13} + \)\(23\!\cdots\!67\)\( p^{16} T^{14} - 913745011043542062 p^{20} T^{15} + 1423572865069108 p^{24} T^{16} - 47610302814 p^{28} T^{17} + 54638937 p^{32} T^{18} - 1194 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 59 | \( 1 + 12204 T + 23727816 T^{2} - 161945990340 T^{3} + 481248778020651 T^{4} + 8875411778671037364 T^{5} + \)\(62\!\cdots\!48\)\( T^{6} - \)\(92\!\cdots\!68\)\( T^{7} + \)\(25\!\cdots\!35\)\( T^{8} + \)\(23\!\cdots\!60\)\( T^{9} - \)\(19\!\cdots\!56\)\( T^{10} - \)\(19\!\cdots\!80\)\( T^{11} + \)\(83\!\cdots\!46\)\( T^{12} + \)\(29\!\cdots\!52\)\( T^{13} - \)\(19\!\cdots\!88\)\( p T^{14} - \)\(25\!\cdots\!60\)\( T^{15} + \)\(17\!\cdots\!61\)\( T^{16} + \)\(22\!\cdots\!80\)\( p T^{17} - \)\(28\!\cdots\!40\)\( T^{18} - \)\(17\!\cdots\!60\)\( T^{19} + \)\(29\!\cdots\!65\)\( T^{20} - \)\(17\!\cdots\!60\)\( p^{4} T^{21} - \)\(28\!\cdots\!40\)\( p^{8} T^{22} + \)\(22\!\cdots\!80\)\( p^{13} T^{23} + \)\(17\!\cdots\!61\)\( p^{16} T^{24} - \)\(25\!\cdots\!60\)\( p^{20} T^{25} - \)\(19\!\cdots\!88\)\( p^{25} T^{26} + \)\(29\!\cdots\!52\)\( p^{28} T^{27} + \)\(83\!\cdots\!46\)\( p^{32} T^{28} - \)\(19\!\cdots\!80\)\( p^{36} T^{29} - \)\(19\!\cdots\!56\)\( p^{40} T^{30} + \)\(23\!\cdots\!60\)\( p^{44} T^{31} + \)\(25\!\cdots\!35\)\( p^{48} T^{32} - \)\(92\!\cdots\!68\)\( p^{52} T^{33} + \)\(62\!\cdots\!48\)\( p^{56} T^{34} + 8875411778671037364 p^{60} T^{35} + 481248778020651 p^{64} T^{36} - 161945990340 p^{68} T^{37} + 23727816 p^{72} T^{38} + 12204 p^{76} T^{39} + p^{80} T^{40} \) | |
| 61 | \( 1 - 11684 T + 6941039 T^{2} + 376010127876 T^{3} - 357728663070840 T^{4} - 12027820700964601976 T^{5} + \)\(29\!\cdots\!55\)\( T^{6} + \)\(17\!\cdots\!48\)\( T^{7} - \)\(56\!\cdots\!64\)\( T^{8} - \)\(20\!\cdots\!08\)\( T^{9} + \)\(10\!\cdots\!45\)\( T^{10} - \)\(90\!\cdots\!68\)\( T^{11} - \)\(75\!\cdots\!28\)\( T^{12} + \)\(49\!\cdots\!56\)\( T^{13} + \)\(16\!\cdots\!93\)\( T^{14} - \)\(12\!\cdots\!44\)\( T^{15} + \)\(23\!\cdots\!43\)\( T^{16} + \)\(13\!\cdots\!68\)\( T^{17} - \)\(42\!\cdots\!24\)\( T^{18} - \)\(92\!\cdots\!88\)\( T^{19} + \)\(79\!\cdots\!56\)\( T^{20} - \)\(92\!\cdots\!88\)\( p^{4} T^{21} - \)\(42\!\cdots\!24\)\( p^{8} T^{22} + \)\(13\!\cdots\!68\)\( p^{12} T^{23} + \)\(23\!\cdots\!43\)\( p^{16} T^{24} - \)\(12\!\cdots\!44\)\( p^{20} T^{25} + \)\(16\!\cdots\!93\)\( p^{24} T^{26} + \)\(49\!\cdots\!56\)\( p^{28} T^{27} - \)\(75\!\cdots\!28\)\( p^{32} T^{28} - \)\(90\!\cdots\!68\)\( p^{36} T^{29} + \)\(10\!\cdots\!45\)\( p^{40} T^{30} - \)\(20\!\cdots\!08\)\( p^{44} T^{31} - \)\(56\!\cdots\!64\)\( p^{48} T^{32} + \)\(17\!\cdots\!48\)\( p^{52} T^{33} + \)\(29\!\cdots\!55\)\( p^{56} T^{34} - 12027820700964601976 p^{60} T^{35} - 357728663070840 p^{64} T^{36} + 376010127876 p^{68} T^{37} + 6941039 p^{72} T^{38} - 11684 p^{76} T^{39} + p^{80} T^{40} \) | |
| 67 | \( 1 - 27162 T + 362208918 T^{2} - 3218727738424 T^{3} + 21480390537417392 T^{4} - \)\(11\!\cdots\!78\)\( T^{5} + \)\(53\!\cdots\!92\)\( T^{6} - \)\(24\!\cdots\!42\)\( T^{7} + \)\(10\!\cdots\!63\)\( T^{8} - \)\(48\!\cdots\!18\)\( T^{9} + \)\(19\!\cdots\!56\)\( T^{10} - \)\(68\!\cdots\!78\)\( T^{11} + \)\(19\!\cdots\!70\)\( T^{12} - \)\(42\!\cdots\!72\)\( T^{13} + \)\(10\!\cdots\!10\)\( T^{14} - \)\(64\!\cdots\!50\)\( T^{15} + \)\(76\!\cdots\!55\)\( p T^{16} - \)\(34\!\cdots\!88\)\( T^{17} + \)\(18\!\cdots\!72\)\( T^{18} - \)\(89\!\cdots\!88\)\( T^{19} + \)\(40\!\cdots\!58\)\( T^{20} - \)\(89\!\cdots\!88\)\( p^{4} T^{21} + \)\(18\!\cdots\!72\)\( p^{8} T^{22} - \)\(34\!\cdots\!88\)\( p^{12} T^{23} + \)\(76\!\cdots\!55\)\( p^{17} T^{24} - \)\(64\!\cdots\!50\)\( p^{20} T^{25} + \)\(10\!\cdots\!10\)\( p^{24} T^{26} - \)\(42\!\cdots\!72\)\( p^{28} T^{27} + \)\(19\!\cdots\!70\)\( p^{32} T^{28} - \)\(68\!\cdots\!78\)\( p^{36} T^{29} + \)\(19\!\cdots\!56\)\( p^{40} T^{30} - \)\(48\!\cdots\!18\)\( p^{44} T^{31} + \)\(10\!\cdots\!63\)\( p^{48} T^{32} - \)\(24\!\cdots\!42\)\( p^{52} T^{33} + \)\(53\!\cdots\!92\)\( p^{56} T^{34} - \)\(11\!\cdots\!78\)\( p^{60} T^{35} + 21480390537417392 p^{64} T^{36} - 3218727738424 p^{68} T^{37} + 362208918 p^{72} T^{38} - 27162 p^{76} T^{39} + p^{80} T^{40} \) | |
| 71 | \( 1 - 41910 T + 899393100 T^{2} - 12957739460388 T^{3} + 139281968370982389 T^{4} - \)\(11\!\cdots\!88\)\( T^{5} + \)\(82\!\cdots\!52\)\( T^{6} - \)\(49\!\cdots\!18\)\( T^{7} + \)\(26\!\cdots\!57\)\( T^{8} - \)\(13\!\cdots\!36\)\( T^{9} + \)\(65\!\cdots\!04\)\( T^{10} - \)\(29\!\cdots\!68\)\( T^{11} + \)\(10\!\cdots\!76\)\( T^{12} - \)\(19\!\cdots\!56\)\( T^{13} - \)\(91\!\cdots\!04\)\( T^{14} + \)\(13\!\cdots\!28\)\( T^{15} - \)\(10\!\cdots\!35\)\( T^{16} + \)\(69\!\cdots\!90\)\( T^{17} - \)\(45\!\cdots\!24\)\( T^{18} + \)\(27\!\cdots\!36\)\( T^{19} - \)\(14\!\cdots\!63\)\( T^{20} + \)\(27\!\cdots\!36\)\( p^{4} T^{21} - \)\(45\!\cdots\!24\)\( p^{8} T^{22} + \)\(69\!\cdots\!90\)\( p^{12} T^{23} - \)\(10\!\cdots\!35\)\( p^{16} T^{24} + \)\(13\!\cdots\!28\)\( p^{20} T^{25} - \)\(91\!\cdots\!04\)\( p^{24} T^{26} - \)\(19\!\cdots\!56\)\( p^{28} T^{27} + \)\(10\!\cdots\!76\)\( p^{32} T^{28} - \)\(29\!\cdots\!68\)\( p^{36} T^{29} + \)\(65\!\cdots\!04\)\( p^{40} T^{30} - \)\(13\!\cdots\!36\)\( p^{44} T^{31} + \)\(26\!\cdots\!57\)\( p^{48} T^{32} - \)\(49\!\cdots\!18\)\( p^{52} T^{33} + \)\(82\!\cdots\!52\)\( p^{56} T^{34} - \)\(11\!\cdots\!88\)\( p^{60} T^{35} + 139281968370982389 p^{64} T^{36} - 12957739460388 p^{68} T^{37} + 899393100 p^{72} T^{38} - 41910 p^{76} T^{39} + p^{80} T^{40} \) | |
| 73 | \( 1 + 51078 T + 1304481042 T^{2} + 22943924129144 T^{3} + 319062651143318243 T^{4} + \)\(37\!\cdots\!28\)\( T^{5} + \)\(38\!\cdots\!46\)\( T^{6} + \)\(35\!\cdots\!54\)\( T^{7} + \)\(30\!\cdots\!71\)\( T^{8} + \)\(23\!\cdots\!56\)\( T^{9} + \)\(17\!\cdots\!76\)\( T^{10} + \)\(11\!\cdots\!12\)\( T^{11} + \)\(76\!\cdots\!18\)\( T^{12} + \)\(47\!\cdots\!24\)\( T^{13} + \)\(28\!\cdots\!36\)\( T^{14} + \)\(16\!\cdots\!76\)\( T^{15} + \)\(95\!\cdots\!45\)\( T^{16} + \)\(53\!\cdots\!94\)\( T^{17} + \)\(29\!\cdots\!02\)\( T^{18} + \)\(15\!\cdots\!04\)\( T^{19} + \)\(85\!\cdots\!09\)\( T^{20} + \)\(15\!\cdots\!04\)\( p^{4} T^{21} + \)\(29\!\cdots\!02\)\( p^{8} T^{22} + \)\(53\!\cdots\!94\)\( p^{12} T^{23} + \)\(95\!\cdots\!45\)\( p^{16} T^{24} + \)\(16\!\cdots\!76\)\( p^{20} T^{25} + \)\(28\!\cdots\!36\)\( p^{24} T^{26} + \)\(47\!\cdots\!24\)\( p^{28} T^{27} + \)\(76\!\cdots\!18\)\( p^{32} T^{28} + \)\(11\!\cdots\!12\)\( p^{36} T^{29} + \)\(17\!\cdots\!76\)\( p^{40} T^{30} + \)\(23\!\cdots\!56\)\( p^{44} T^{31} + \)\(30\!\cdots\!71\)\( p^{48} T^{32} + \)\(35\!\cdots\!54\)\( p^{52} T^{33} + \)\(38\!\cdots\!46\)\( p^{56} T^{34} + \)\(37\!\cdots\!28\)\( p^{60} T^{35} + 319062651143318243 p^{64} T^{36} + 22943924129144 p^{68} T^{37} + 1304481042 p^{72} T^{38} + 51078 p^{76} T^{39} + p^{80} T^{40} \) | |
| 79 | \( ( 1 - 4168 T + 241897796 T^{2} - 1169510780128 T^{3} + 29243554461332142 T^{4} - \)\(14\!\cdots\!40\)\( T^{5} + \)\(23\!\cdots\!30\)\( T^{6} - \)\(10\!\cdots\!24\)\( T^{7} + \)\(13\!\cdots\!33\)\( T^{8} - \)\(58\!\cdots\!96\)\( T^{9} + \)\(61\!\cdots\!24\)\( T^{10} - \)\(58\!\cdots\!96\)\( p^{4} T^{11} + \)\(13\!\cdots\!33\)\( p^{8} T^{12} - \)\(10\!\cdots\!24\)\( p^{12} T^{13} + \)\(23\!\cdots\!30\)\( p^{16} T^{14} - \)\(14\!\cdots\!40\)\( p^{20} T^{15} + 29243554461332142 p^{24} T^{16} - 1169510780128 p^{28} T^{17} + 241897796 p^{32} T^{18} - 4168 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 83 | \( 1 + 17232 T + 148470912 T^{2} + 1316721217356 T^{3} + 16140108003448686 T^{4} + \)\(15\!\cdots\!16\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} + \)\(97\!\cdots\!84\)\( T^{7} + \)\(89\!\cdots\!57\)\( T^{8} + \)\(67\!\cdots\!40\)\( T^{9} + \)\(45\!\cdots\!00\)\( T^{10} + \)\(34\!\cdots\!88\)\( T^{11} + \)\(26\!\cdots\!20\)\( T^{12} + \)\(17\!\cdots\!60\)\( T^{13} + \)\(10\!\cdots\!40\)\( T^{14} + \)\(74\!\cdots\!92\)\( T^{15} + \)\(50\!\cdots\!38\)\( T^{16} + \)\(29\!\cdots\!00\)\( T^{17} + \)\(18\!\cdots\!48\)\( T^{18} + \)\(13\!\cdots\!40\)\( T^{19} + \)\(91\!\cdots\!60\)\( T^{20} + \)\(13\!\cdots\!40\)\( p^{4} T^{21} + \)\(18\!\cdots\!48\)\( p^{8} T^{22} + \)\(29\!\cdots\!00\)\( p^{12} T^{23} + \)\(50\!\cdots\!38\)\( p^{16} T^{24} + \)\(74\!\cdots\!92\)\( p^{20} T^{25} + \)\(10\!\cdots\!40\)\( p^{24} T^{26} + \)\(17\!\cdots\!60\)\( p^{28} T^{27} + \)\(26\!\cdots\!20\)\( p^{32} T^{28} + \)\(34\!\cdots\!88\)\( p^{36} T^{29} + \)\(45\!\cdots\!00\)\( p^{40} T^{30} + \)\(67\!\cdots\!40\)\( p^{44} T^{31} + \)\(89\!\cdots\!57\)\( p^{48} T^{32} + \)\(97\!\cdots\!84\)\( p^{52} T^{33} + \)\(11\!\cdots\!48\)\( p^{56} T^{34} + \)\(15\!\cdots\!16\)\( p^{60} T^{35} + 16140108003448686 p^{64} T^{36} + 1316721217356 p^{68} T^{37} + 148470912 p^{72} T^{38} + 17232 p^{76} T^{39} + p^{80} T^{40} \) | |
| 89 | \( 1 - 366 T + 77077326 T^{2} + 31501580772 T^{3} + 8851317550046271 T^{4} + 17232568753117946004 T^{5} + \)\(52\!\cdots\!82\)\( T^{6} + \)\(39\!\cdots\!02\)\( T^{7} + \)\(28\!\cdots\!07\)\( T^{8} + \)\(38\!\cdots\!40\)\( T^{9} + \)\(12\!\cdots\!68\)\( p T^{10} + \)\(33\!\cdots\!68\)\( T^{11} + \)\(64\!\cdots\!42\)\( T^{12} + \)\(23\!\cdots\!00\)\( T^{13} + \)\(94\!\cdots\!92\)\( T^{14} + \)\(12\!\cdots\!44\)\( T^{15} + \)\(79\!\cdots\!81\)\( T^{16} + \)\(59\!\cdots\!34\)\( T^{17} + \)\(80\!\cdots\!30\)\( T^{18} + \)\(28\!\cdots\!80\)\( T^{19} + \)\(54\!\cdots\!17\)\( T^{20} + \)\(28\!\cdots\!80\)\( p^{4} T^{21} + \)\(80\!\cdots\!30\)\( p^{8} T^{22} + \)\(59\!\cdots\!34\)\( p^{12} T^{23} + \)\(79\!\cdots\!81\)\( p^{16} T^{24} + \)\(12\!\cdots\!44\)\( p^{20} T^{25} + \)\(94\!\cdots\!92\)\( p^{24} T^{26} + \)\(23\!\cdots\!00\)\( p^{28} T^{27} + \)\(64\!\cdots\!42\)\( p^{32} T^{28} + \)\(33\!\cdots\!68\)\( p^{36} T^{29} + \)\(12\!\cdots\!68\)\( p^{41} T^{30} + \)\(38\!\cdots\!40\)\( p^{44} T^{31} + \)\(28\!\cdots\!07\)\( p^{48} T^{32} + \)\(39\!\cdots\!02\)\( p^{52} T^{33} + \)\(52\!\cdots\!82\)\( p^{56} T^{34} + 17232568753117946004 p^{60} T^{35} + 8851317550046271 p^{64} T^{36} + 31501580772 p^{68} T^{37} + 77077326 p^{72} T^{38} - 366 p^{76} T^{39} + p^{80} T^{40} \) | |
| 97 | \( 1 + 47902 T + 985831472 T^{2} + 9709218537808 T^{3} + 4691166750537250 T^{4} - \)\(10\!\cdots\!10\)\( T^{5} - \)\(11\!\cdots\!48\)\( T^{6} + \)\(28\!\cdots\!10\)\( T^{7} + \)\(11\!\cdots\!15\)\( T^{8} + \)\(11\!\cdots\!14\)\( T^{9} + \)\(41\!\cdots\!92\)\( p T^{10} - \)\(47\!\cdots\!50\)\( T^{11} + \)\(82\!\cdots\!56\)\( T^{12} + \)\(17\!\cdots\!56\)\( T^{13} - \)\(11\!\cdots\!36\)\( T^{14} - \)\(30\!\cdots\!26\)\( T^{15} + \)\(16\!\cdots\!85\)\( T^{16} + \)\(49\!\cdots\!52\)\( T^{17} + \)\(47\!\cdots\!04\)\( T^{18} + \)\(11\!\cdots\!16\)\( T^{19} - \)\(25\!\cdots\!26\)\( T^{20} + \)\(11\!\cdots\!16\)\( p^{4} T^{21} + \)\(47\!\cdots\!04\)\( p^{8} T^{22} + \)\(49\!\cdots\!52\)\( p^{12} T^{23} + \)\(16\!\cdots\!85\)\( p^{16} T^{24} - \)\(30\!\cdots\!26\)\( p^{20} T^{25} - \)\(11\!\cdots\!36\)\( p^{24} T^{26} + \)\(17\!\cdots\!56\)\( p^{28} T^{27} + \)\(82\!\cdots\!56\)\( p^{32} T^{28} - \)\(47\!\cdots\!50\)\( p^{36} T^{29} + \)\(41\!\cdots\!92\)\( p^{41} T^{30} + \)\(11\!\cdots\!14\)\( p^{44} T^{31} + \)\(11\!\cdots\!15\)\( p^{48} T^{32} + \)\(28\!\cdots\!10\)\( p^{52} T^{33} - \)\(11\!\cdots\!48\)\( p^{56} T^{34} - \)\(10\!\cdots\!10\)\( p^{60} T^{35} + 4691166750537250 p^{64} T^{36} + 9709218537808 p^{68} T^{37} + 985831472 p^{72} T^{38} + 47902 p^{76} T^{39} + p^{80} T^{40} \) | |
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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
−2.71884337733503151973748872377, −2.49795462218532418409454043181, −2.37653968017731229222761290470, −2.37241033384310754339355845163, −2.28966477775927236710446034027, −2.28312280482914968727330871068, −2.11999658146306152122045655695, −1.98894551669582543284864952732, −1.95702827761849320905175965804, −1.88597387577751088524241494871, −1.83322081469585816865622875457, −1.67096600691181396308827866630, −1.35513441472492868261791020554, −1.31270921263362413897496407871, −1.15401569267968174246174237424, −1.12903257870175688246099558150, −1.02607776705769352406165350865, −0.928073916052450503920336261510, −0.904291107586533491029149260871, −0.892672374101653012543402092367, −0.55206047836432890471532646706, −0.41760799155531203143171468216, −0.16522254444449679023852847945, −0.10135279339017681441894626205, −0.02288856130833384472044178327, 0.02288856130833384472044178327, 0.10135279339017681441894626205, 0.16522254444449679023852847945, 0.41760799155531203143171468216, 0.55206047836432890471532646706, 0.892672374101653012543402092367, 0.904291107586533491029149260871, 0.928073916052450503920336261510, 1.02607776705769352406165350865, 1.12903257870175688246099558150, 1.15401569267968174246174237424, 1.31270921263362413897496407871, 1.35513441472492868261791020554, 1.67096600691181396308827866630, 1.83322081469585816865622875457, 1.88597387577751088524241494871, 1.95702827761849320905175965804, 1.98894551669582543284864952732, 2.11999658146306152122045655695, 2.28312280482914968727330871068, 2.28966477775927236710446034027, 2.37241033384310754339355845163, 2.37653968017731229222761290470, 2.49795462218532418409454043181, 2.71884337733503151973748872377
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.