Dirichlet series
| L(s) = 1 | + 2·2-s − 14·4-s − 196·7-s + 52·8-s + 1.62e3·9-s − 392·14-s + 1.03e3·16-s + 3.24e3·18-s − 4.67e3·23-s − 6.25e3·25-s + 2.74e3·28-s + 7.16e3·31-s + 1.97e3·32-s − 2.26e4·36-s + 1.16e4·41-s − 9.35e3·46-s + 4.41e4·47-s − 1.39e5·49-s − 1.25e4·50-s − 1.01e4·56-s + 1.43e4·62-s − 3.17e5·63-s − 9.49e3·64-s − 2.00e5·71-s + 8.42e4·72-s − 1.05e5·73-s + 2.82e5·79-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 0.437·4-s − 1.51·7-s + 0.287·8-s + 20/3·9-s − 0.534·14-s + 1.00·16-s + 2.35·18-s − 1.84·23-s − 2·25-s + 0.661·28-s + 1.33·31-s + 0.341·32-s − 2.91·36-s + 1.07·41-s − 0.651·46-s + 2.91·47-s − 8.29·49-s − 0.707·50-s − 0.434·56-s + 0.473·62-s − 10.0·63-s − 0.289·64-s − 4.71·71-s + 1.91·72-s − 2.30·73-s + 5.08·79-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 5^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{60} \cdot 5^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.39447\times 10^{16}\) |
| Root analytic conductor: | \(2.53285\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{60} \cdot 5^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(20.91245152\) |
| \(L(\frac12)\) | \(\approx\) | \(20.91245152\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T + 9 p T^{2} - 29 p^{2} T^{3} - 111 p^{2} T^{4} - 99 p^{4} T^{5} - 67 p^{5} T^{6} + 1457 p^{7} T^{7} - 1943 p^{8} T^{8} + 277 p^{11} T^{9} - 3119 p^{13} T^{10} + 277 p^{16} T^{11} - 1943 p^{18} T^{12} + 1457 p^{22} T^{13} - 67 p^{25} T^{14} - 99 p^{29} T^{15} - 111 p^{32} T^{16} - 29 p^{37} T^{17} + 9 p^{41} T^{18} - p^{46} T^{19} + p^{50} T^{20} \) |
| 5 | \( ( 1 + p^{4} T^{2} )^{10} \) | |
| good | 3 | \( 1 - 20 p^{4} T^{2} + 464774 p T^{4} - 834927892 T^{6} + 43647016469 p^{2} T^{8} - 51765303184624 p T^{10} + 53814329151823576 T^{12} - 1863599575911640208 p^{2} T^{14} + 19706806085941150310 p^{5} T^{16} - \)\(17\!\cdots\!00\)\( p^{6} T^{18} + \)\(48\!\cdots\!00\)\( p^{8} T^{20} - \)\(17\!\cdots\!00\)\( p^{16} T^{22} + 19706806085941150310 p^{25} T^{24} - 1863599575911640208 p^{32} T^{26} + 53814329151823576 p^{40} T^{28} - 51765303184624 p^{51} T^{30} + 43647016469 p^{62} T^{32} - 834927892 p^{70} T^{34} + 464774 p^{81} T^{36} - 20 p^{94} T^{38} + p^{100} T^{40} \) |
| 7 | \( ( 1 + 2 p^{2} T + 84148 T^{2} + 8017214 T^{3} + 3556570901 T^{4} + 345044190776 T^{5} + 103920201897616 T^{6} + 10159390994080936 T^{7} + 2363994840482709802 T^{8} + \)\(22\!\cdots\!76\)\( T^{9} + \)\(43\!\cdots\!72\)\( T^{10} + \)\(22\!\cdots\!76\)\( p^{5} T^{11} + 2363994840482709802 p^{10} T^{12} + 10159390994080936 p^{15} T^{13} + 103920201897616 p^{20} T^{14} + 345044190776 p^{25} T^{15} + 3556570901 p^{30} T^{16} + 8017214 p^{35} T^{17} + 84148 p^{40} T^{18} + 2 p^{47} T^{19} + p^{50} T^{20} )^{2} \) | |
| 11 | \( 1 - 1510004 T^{2} + 1155249727726 T^{4} - 592766540112799924 T^{6} + \)\(22\!\cdots\!17\)\( T^{8} - \)\(70\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!36\)\( T^{12} - \)\(40\!\cdots\!64\)\( T^{14} + \)\(79\!\cdots\!82\)\( T^{16} - \)\(14\!\cdots\!04\)\( T^{18} + \)\(23\!\cdots\!76\)\( T^{20} - \)\(14\!\cdots\!04\)\( p^{10} T^{22} + \)\(79\!\cdots\!82\)\( p^{20} T^{24} - \)\(40\!\cdots\!64\)\( p^{30} T^{26} + \)\(18\!\cdots\!36\)\( p^{40} T^{28} - \)\(70\!\cdots\!04\)\( p^{50} T^{30} + \)\(22\!\cdots\!17\)\( p^{60} T^{32} - 592766540112799924 p^{70} T^{34} + 1155249727726 p^{80} T^{36} - 1510004 p^{90} T^{38} + p^{100} T^{40} \) | |
| 13 | \( 1 - 3520332 T^{2} + 6227853839054 T^{4} - 7441447313525468748 T^{6} + \)\(67\!\cdots\!57\)\( T^{8} - \)\(50\!\cdots\!12\)\( T^{10} + \)\(32\!\cdots\!64\)\( T^{12} - \)\(17\!\cdots\!28\)\( T^{14} + \)\(87\!\cdots\!42\)\( T^{16} - \)\(38\!\cdots\!32\)\( T^{18} + \)\(14\!\cdots\!64\)\( T^{20} - \)\(38\!\cdots\!32\)\( p^{10} T^{22} + \)\(87\!\cdots\!42\)\( p^{20} T^{24} - \)\(17\!\cdots\!28\)\( p^{30} T^{26} + \)\(32\!\cdots\!64\)\( p^{40} T^{28} - \)\(50\!\cdots\!12\)\( p^{50} T^{30} + \)\(67\!\cdots\!57\)\( p^{60} T^{32} - 7441447313525468748 p^{70} T^{34} + 6227853839054 p^{80} T^{36} - 3520332 p^{90} T^{38} + p^{100} T^{40} \) | |
| 17 | \( ( 1 + 5447662 T^{2} + 1859072000 T^{3} + 15317572824845 T^{4} + 7413542230528000 T^{5} + 32518332783929091752 T^{6} + \)\(14\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!10\)\( T^{8} + \)\(21\!\cdots\!00\)\( T^{9} + \)\(84\!\cdots\!72\)\( T^{10} + \)\(21\!\cdots\!00\)\( p^{5} T^{11} + \)\(56\!\cdots\!10\)\( p^{10} T^{12} + \)\(14\!\cdots\!00\)\( p^{15} T^{13} + 32518332783929091752 p^{20} T^{14} + 7413542230528000 p^{25} T^{15} + 15317572824845 p^{30} T^{16} + 1859072000 p^{35} T^{17} + 5447662 p^{40} T^{18} + p^{50} T^{20} )^{2} \) | |
| 19 | \( 1 - 23638692 T^{2} + 296336418074734 T^{4} - \)\(25\!\cdots\!32\)\( T^{6} + \)\(17\!\cdots\!97\)\( T^{8} - \)\(49\!\cdots\!20\)\( p T^{10} + \)\(42\!\cdots\!52\)\( T^{12} - \)\(16\!\cdots\!48\)\( T^{14} + \)\(56\!\cdots\!66\)\( T^{16} - \)\(16\!\cdots\!12\)\( T^{18} + \)\(44\!\cdots\!28\)\( T^{20} - \)\(16\!\cdots\!12\)\( p^{10} T^{22} + \)\(56\!\cdots\!66\)\( p^{20} T^{24} - \)\(16\!\cdots\!48\)\( p^{30} T^{26} + \)\(42\!\cdots\!52\)\( p^{40} T^{28} - \)\(49\!\cdots\!20\)\( p^{51} T^{30} + \)\(17\!\cdots\!97\)\( p^{60} T^{32} - \)\(25\!\cdots\!32\)\( p^{70} T^{34} + 296336418074734 p^{80} T^{36} - 23638692 p^{90} T^{38} + p^{100} T^{40} \) | |
| 23 | \( ( 1 + 2338 T + 35997660 T^{2} + 45007042654 T^{3} + 520972471777845 T^{4} + 50426407997609208 T^{5} + \)\(39\!\cdots\!60\)\( T^{6} - \)\(66\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!10\)\( T^{8} - \)\(91\!\cdots\!52\)\( T^{9} + \)\(76\!\cdots\!60\)\( T^{10} - \)\(91\!\cdots\!52\)\( p^{5} T^{11} + \)\(17\!\cdots\!10\)\( p^{10} T^{12} - \)\(66\!\cdots\!96\)\( p^{15} T^{13} + \)\(39\!\cdots\!60\)\( p^{20} T^{14} + 50426407997609208 p^{25} T^{15} + 520972471777845 p^{30} T^{16} + 45007042654 p^{35} T^{17} + 35997660 p^{40} T^{18} + 2338 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 29 | \( 1 - 215092900 T^{2} + 23243889276296494 T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(90\!\cdots\!13\)\( T^{8} - \)\(38\!\cdots\!00\)\( T^{10} + \)\(13\!\cdots\!92\)\( T^{12} - \)\(42\!\cdots\!00\)\( T^{14} + \)\(11\!\cdots\!86\)\( T^{16} - \)\(27\!\cdots\!00\)\( T^{18} + \)\(58\!\cdots\!28\)\( T^{20} - \)\(27\!\cdots\!00\)\( p^{10} T^{22} + \)\(11\!\cdots\!86\)\( p^{20} T^{24} - \)\(42\!\cdots\!00\)\( p^{30} T^{26} + \)\(13\!\cdots\!92\)\( p^{40} T^{28} - \)\(38\!\cdots\!00\)\( p^{50} T^{30} + \)\(90\!\cdots\!13\)\( p^{60} T^{32} - \)\(16\!\cdots\!00\)\( p^{70} T^{34} + 23243889276296494 p^{80} T^{36} - 215092900 p^{90} T^{38} + p^{100} T^{40} \) | |
| 31 | \( ( 1 - 3580 T + 132421614 T^{2} - 484936307876 T^{3} + 8983138546835629 T^{4} - 33755686429634218608 T^{5} + \)\(43\!\cdots\!88\)\( T^{6} - \)\(16\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!38\)\( T^{8} - \)\(60\!\cdots\!32\)\( T^{9} + \)\(54\!\cdots\!44\)\( T^{10} - \)\(60\!\cdots\!32\)\( p^{5} T^{11} + \)\(17\!\cdots\!38\)\( p^{10} T^{12} - \)\(16\!\cdots\!96\)\( p^{15} T^{13} + \)\(43\!\cdots\!88\)\( p^{20} T^{14} - 33755686429634218608 p^{25} T^{15} + 8983138546835629 p^{30} T^{16} - 484936307876 p^{35} T^{17} + 132421614 p^{40} T^{18} - 3580 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 37 | \( 1 - 565372668 T^{2} + 171401952644913934 T^{4} - \)\(36\!\cdots\!00\)\( T^{6} + \)\(59\!\cdots\!09\)\( T^{8} - \)\(80\!\cdots\!44\)\( T^{10} + \)\(93\!\cdots\!16\)\( T^{12} - \)\(93\!\cdots\!40\)\( T^{14} + \)\(83\!\cdots\!86\)\( T^{16} - \)\(67\!\cdots\!48\)\( T^{18} + \)\(48\!\cdots\!08\)\( T^{20} - \)\(67\!\cdots\!48\)\( p^{10} T^{22} + \)\(83\!\cdots\!86\)\( p^{20} T^{24} - \)\(93\!\cdots\!40\)\( p^{30} T^{26} + \)\(93\!\cdots\!16\)\( p^{40} T^{28} - \)\(80\!\cdots\!44\)\( p^{50} T^{30} + \)\(59\!\cdots\!09\)\( p^{60} T^{32} - \)\(36\!\cdots\!00\)\( p^{70} T^{34} + 171401952644913934 p^{80} T^{36} - 565372668 p^{90} T^{38} + p^{100} T^{40} \) | |
| 41 | \( ( 1 - 5804 T + 580737234 T^{2} - 4176614475628 T^{3} + 181136735899530493 T^{4} - \)\(13\!\cdots\!48\)\( T^{5} + \)\(39\!\cdots\!40\)\( T^{6} - \)\(28\!\cdots\!68\)\( T^{7} + \)\(63\!\cdots\!42\)\( T^{8} - \)\(43\!\cdots\!24\)\( T^{9} + \)\(82\!\cdots\!24\)\( T^{10} - \)\(43\!\cdots\!24\)\( p^{5} T^{11} + \)\(63\!\cdots\!42\)\( p^{10} T^{12} - \)\(28\!\cdots\!68\)\( p^{15} T^{13} + \)\(39\!\cdots\!40\)\( p^{20} T^{14} - \)\(13\!\cdots\!48\)\( p^{25} T^{15} + 181136735899530493 p^{30} T^{16} - 4176614475628 p^{35} T^{17} + 580737234 p^{40} T^{18} - 5804 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 43 | \( 1 - 1208126740 T^{2} + 787740581508660018 T^{4} - \)\(36\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!73\)\( T^{8} - \)\(37\!\cdots\!80\)\( T^{10} + \)\(95\!\cdots\!40\)\( T^{12} - \)\(20\!\cdots\!56\)\( T^{14} + \)\(40\!\cdots\!70\)\( T^{16} - \)\(70\!\cdots\!00\)\( T^{18} + \)\(10\!\cdots\!96\)\( T^{20} - \)\(70\!\cdots\!00\)\( p^{10} T^{22} + \)\(40\!\cdots\!70\)\( p^{20} T^{24} - \)\(20\!\cdots\!56\)\( p^{30} T^{26} + \)\(95\!\cdots\!40\)\( p^{40} T^{28} - \)\(37\!\cdots\!80\)\( p^{50} T^{30} + \)\(12\!\cdots\!73\)\( p^{60} T^{32} - \)\(36\!\cdots\!96\)\( p^{70} T^{34} + 787740581508660018 p^{80} T^{36} - 1208126740 p^{90} T^{38} + p^{100} T^{40} \) | |
| 47 | \( ( 1 - 10 p^{2} T + 1548510076 T^{2} - 25779450599270 T^{3} + 1065218725316011845 T^{4} - \)\(14\!\cdots\!20\)\( T^{5} + \)\(45\!\cdots\!96\)\( T^{6} - \)\(51\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!10\)\( T^{8} - \)\(30\!\cdots\!00\)\( p T^{9} + \)\(36\!\cdots\!56\)\( T^{10} - \)\(30\!\cdots\!00\)\( p^{6} T^{11} + \)\(14\!\cdots\!10\)\( p^{10} T^{12} - \)\(51\!\cdots\!60\)\( p^{15} T^{13} + \)\(45\!\cdots\!96\)\( p^{20} T^{14} - \)\(14\!\cdots\!20\)\( p^{25} T^{15} + 1065218725316011845 p^{30} T^{16} - 25779450599270 p^{35} T^{17} + 1548510076 p^{40} T^{18} - 10 p^{47} T^{19} + p^{50} T^{20} )^{2} \) | |
| 53 | \( 1 - 4003669356 T^{2} + 7954725909905649454 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!77\)\( T^{8} - \)\(91\!\cdots\!76\)\( T^{10} + \)\(65\!\cdots\!16\)\( T^{12} - \)\(40\!\cdots\!80\)\( T^{14} + \)\(22\!\cdots\!54\)\( T^{16} - \)\(11\!\cdots\!56\)\( T^{18} + \)\(48\!\cdots\!96\)\( T^{20} - \)\(11\!\cdots\!56\)\( p^{10} T^{22} + \)\(22\!\cdots\!54\)\( p^{20} T^{24} - \)\(40\!\cdots\!80\)\( p^{30} T^{26} + \)\(65\!\cdots\!16\)\( p^{40} T^{28} - \)\(91\!\cdots\!76\)\( p^{50} T^{30} + \)\(10\!\cdots\!77\)\( p^{60} T^{32} - \)\(10\!\cdots\!60\)\( p^{70} T^{34} + 7954725909905649454 p^{80} T^{36} - 4003669356 p^{90} T^{38} + p^{100} T^{40} \) | |
| 59 | \( 1 - 9996949828 T^{2} + 49650800837889885966 T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(39\!\cdots\!17\)\( T^{8} - \)\(73\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(15\!\cdots\!54\)\( T^{16} - \)\(13\!\cdots\!88\)\( T^{18} + \)\(10\!\cdots\!24\)\( T^{20} - \)\(13\!\cdots\!88\)\( p^{10} T^{22} + \)\(15\!\cdots\!54\)\( p^{20} T^{24} - \)\(14\!\cdots\!80\)\( p^{30} T^{26} + \)\(11\!\cdots\!44\)\( p^{40} T^{28} - \)\(73\!\cdots\!28\)\( p^{50} T^{30} + \)\(39\!\cdots\!17\)\( p^{60} T^{32} - \)\(16\!\cdots\!20\)\( p^{70} T^{34} + 49650800837889885966 p^{80} T^{36} - 9996949828 p^{90} T^{38} + p^{100} T^{40} \) | |
| 61 | \( 1 - 7152852348 T^{2} + 26523669582205677166 T^{4} - \)\(67\!\cdots\!00\)\( T^{6} + \)\(13\!\cdots\!17\)\( T^{8} - \)\(22\!\cdots\!48\)\( T^{10} + \)\(31\!\cdots\!84\)\( T^{12} - \)\(38\!\cdots\!60\)\( T^{14} + \)\(42\!\cdots\!14\)\( T^{16} - \)\(41\!\cdots\!08\)\( T^{18} + \)\(36\!\cdots\!64\)\( T^{20} - \)\(41\!\cdots\!08\)\( p^{10} T^{22} + \)\(42\!\cdots\!14\)\( p^{20} T^{24} - \)\(38\!\cdots\!60\)\( p^{30} T^{26} + \)\(31\!\cdots\!84\)\( p^{40} T^{28} - \)\(22\!\cdots\!48\)\( p^{50} T^{30} + \)\(13\!\cdots\!17\)\( p^{60} T^{32} - \)\(67\!\cdots\!00\)\( p^{70} T^{34} + 26523669582205677166 p^{80} T^{36} - 7152852348 p^{90} T^{38} + p^{100} T^{40} \) | |
| 67 | \( 1 - 11828518964 T^{2} + 68669252417166303634 T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(76\!\cdots\!57\)\( T^{8} - \)\(18\!\cdots\!04\)\( T^{10} + \)\(38\!\cdots\!16\)\( T^{12} - \)\(70\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!54\)\( T^{16} - \)\(18\!\cdots\!64\)\( T^{18} + \)\(25\!\cdots\!76\)\( T^{20} - \)\(18\!\cdots\!64\)\( p^{10} T^{22} + \)\(11\!\cdots\!54\)\( p^{20} T^{24} - \)\(70\!\cdots\!40\)\( p^{30} T^{26} + \)\(38\!\cdots\!16\)\( p^{40} T^{28} - \)\(18\!\cdots\!04\)\( p^{50} T^{30} + \)\(76\!\cdots\!57\)\( p^{60} T^{32} - \)\(26\!\cdots\!60\)\( p^{70} T^{34} + 68669252417166303634 p^{80} T^{36} - 11828518964 p^{90} T^{38} + p^{100} T^{40} \) | |
| 71 | \( ( 1 + 100156 T + 13448448446 T^{2} + 957445823975748 T^{3} + 76873855601451932317 T^{4} + \)\(43\!\cdots\!20\)\( T^{5} + \)\(26\!\cdots\!28\)\( T^{6} + \)\(12\!\cdots\!92\)\( T^{7} + \)\(67\!\cdots\!74\)\( T^{8} + \)\(28\!\cdots\!84\)\( T^{9} + \)\(13\!\cdots\!68\)\( T^{10} + \)\(28\!\cdots\!84\)\( p^{5} T^{11} + \)\(67\!\cdots\!74\)\( p^{10} T^{12} + \)\(12\!\cdots\!92\)\( p^{15} T^{13} + \)\(26\!\cdots\!28\)\( p^{20} T^{14} + \)\(43\!\cdots\!20\)\( p^{25} T^{15} + 76873855601451932317 p^{30} T^{16} + 957445823975748 p^{35} T^{17} + 13448448446 p^{40} T^{18} + 100156 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 73 | \( ( 1 + 52568 T + 9379342894 T^{2} + 374528688123736 T^{3} + 37721970904921608509 T^{4} + \)\(11\!\cdots\!56\)\( T^{5} + \)\(82\!\cdots\!04\)\( T^{6} + \)\(19\!\cdots\!04\)\( T^{7} + \)\(11\!\cdots\!58\)\( T^{8} + \)\(24\!\cdots\!96\)\( T^{9} + \)\(16\!\cdots\!04\)\( T^{10} + \)\(24\!\cdots\!96\)\( p^{5} T^{11} + \)\(11\!\cdots\!58\)\( p^{10} T^{12} + \)\(19\!\cdots\!04\)\( p^{15} T^{13} + \)\(82\!\cdots\!04\)\( p^{20} T^{14} + \)\(11\!\cdots\!56\)\( p^{25} T^{15} + 37721970904921608509 p^{30} T^{16} + 374528688123736 p^{35} T^{17} + 9379342894 p^{40} T^{18} + 52568 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 79 | \( ( 1 - 141040 T + 30508439526 T^{2} - 3027236690742416 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} - \)\(29\!\cdots\!88\)\( T^{5} + \)\(27\!\cdots\!48\)\( T^{6} - \)\(17\!\cdots\!16\)\( T^{7} + \)\(13\!\cdots\!30\)\( T^{8} - \)\(92\!\cdots\!60\)\( p T^{9} + \)\(47\!\cdots\!04\)\( T^{10} - \)\(92\!\cdots\!60\)\( p^{6} T^{11} + \)\(13\!\cdots\!30\)\( p^{10} T^{12} - \)\(17\!\cdots\!16\)\( p^{15} T^{13} + \)\(27\!\cdots\!48\)\( p^{20} T^{14} - \)\(29\!\cdots\!88\)\( p^{25} T^{15} + \)\(38\!\cdots\!93\)\( p^{30} T^{16} - 3027236690742416 p^{35} T^{17} + 30508439526 p^{40} T^{18} - 141040 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 83 | \( 1 - 34768653380 T^{2} + \)\(55\!\cdots\!58\)\( T^{4} - \)\(53\!\cdots\!56\)\( T^{6} + \)\(34\!\cdots\!33\)\( T^{8} - \)\(15\!\cdots\!20\)\( T^{10} + \)\(44\!\cdots\!40\)\( T^{12} - \)\(24\!\cdots\!36\)\( T^{14} - \)\(64\!\cdots\!10\)\( T^{16} + \)\(49\!\cdots\!20\)\( T^{18} - \)\(23\!\cdots\!44\)\( T^{20} + \)\(49\!\cdots\!20\)\( p^{10} T^{22} - \)\(64\!\cdots\!10\)\( p^{20} T^{24} - \)\(24\!\cdots\!36\)\( p^{30} T^{26} + \)\(44\!\cdots\!40\)\( p^{40} T^{28} - \)\(15\!\cdots\!20\)\( p^{50} T^{30} + \)\(34\!\cdots\!33\)\( p^{60} T^{32} - \)\(53\!\cdots\!56\)\( p^{70} T^{34} + \)\(55\!\cdots\!58\)\( p^{80} T^{36} - 34768653380 p^{90} T^{38} + p^{100} T^{40} \) | |
| 89 | \( ( 1 + 1580 T + 27398194046 T^{2} + 460040940498284 T^{3} + \)\(38\!\cdots\!93\)\( T^{4} + \)\(90\!\cdots\!72\)\( T^{5} + \)\(38\!\cdots\!08\)\( T^{6} + \)\(95\!\cdots\!24\)\( T^{7} + \)\(29\!\cdots\!70\)\( T^{8} + \)\(73\!\cdots\!60\)\( T^{9} + \)\(18\!\cdots\!64\)\( T^{10} + \)\(73\!\cdots\!60\)\( p^{5} T^{11} + \)\(29\!\cdots\!70\)\( p^{10} T^{12} + \)\(95\!\cdots\!24\)\( p^{15} T^{13} + \)\(38\!\cdots\!08\)\( p^{20} T^{14} + \)\(90\!\cdots\!72\)\( p^{25} T^{15} + \)\(38\!\cdots\!93\)\( p^{30} T^{16} + 460040940498284 p^{35} T^{17} + 27398194046 p^{40} T^{18} + 1580 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
| 97 | \( ( 1 - 73688 T + 43672916862 T^{2} - 2817316775448856 T^{3} + \)\(98\!\cdots\!25\)\( T^{4} - \)\(59\!\cdots\!28\)\( T^{5} + \)\(15\!\cdots\!52\)\( T^{6} - \)\(87\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!70\)\( T^{8} - \)\(96\!\cdots\!88\)\( T^{9} + \)\(17\!\cdots\!72\)\( T^{10} - \)\(96\!\cdots\!88\)\( p^{5} T^{11} + \)\(18\!\cdots\!70\)\( p^{10} T^{12} - \)\(87\!\cdots\!16\)\( p^{15} T^{13} + \)\(15\!\cdots\!52\)\( p^{20} T^{14} - \)\(59\!\cdots\!28\)\( p^{25} T^{15} + \)\(98\!\cdots\!25\)\( p^{30} T^{16} - 2817316775448856 p^{35} T^{17} + 43672916862 p^{40} T^{18} - 73688 p^{45} T^{19} + p^{50} T^{20} )^{2} \) | |
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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
−3.40461411700401652480889856081, −3.33361433855163399261073277473, −3.12592839107919773823061693697, −3.05127455088470852280197709193, −3.01074785981801008334792860425, −2.94121891379844951877156945477, −2.93664359031754054838196958755, −2.64642627274454206792165093060, −2.32198114007091358806659784116, −2.05377301906553763848797910005, −2.05107904257099029181520697953, −2.00629340232989276197903019190, −1.85198993067242620407059917037, −1.75341189498998976147199956875, −1.73691780736693295545467658513, −1.57389231704485260251250727618, −1.38196599346536565210767609909, −1.23457771820016463361882648431, −1.18163940505072871177166965518, −1.00633471131885433822313354909, −0.74602661335797141341683292836, −0.69280420922858683866645964031, −0.38345842016112602331330427621, −0.30165867765224509857655236375, −0.18641905283586113437691629914, 0.18641905283586113437691629914, 0.30165867765224509857655236375, 0.38345842016112602331330427621, 0.69280420922858683866645964031, 0.74602661335797141341683292836, 1.00633471131885433822313354909, 1.18163940505072871177166965518, 1.23457771820016463361882648431, 1.38196599346536565210767609909, 1.57389231704485260251250727618, 1.73691780736693295545467658513, 1.75341189498998976147199956875, 1.85198993067242620407059917037, 2.00629340232989276197903019190, 2.05107904257099029181520697953, 2.05377301906553763848797910005, 2.32198114007091358806659784116, 2.64642627274454206792165093060, 2.93664359031754054838196958755, 2.94121891379844951877156945477, 3.01074785981801008334792860425, 3.05127455088470852280197709193, 3.12592839107919773823061693697, 3.33361433855163399261073277473, 3.40461411700401652480889856081
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.