Dirichlet series
| L(s) = 1 | − 40·4-s − 2·5-s − 90·9-s − 114·11-s + 880·16-s + 370·19-s + 80·20-s − 43·25-s + 368·29-s − 620·31-s + 3.60e3·36-s − 872·41-s + 4.56e3·44-s + 180·45-s + 4.09e3·49-s + 228·55-s + 3.22e3·59-s − 2.60e3·61-s − 1.40e4·64-s − 2.29e3·71-s − 1.48e4·76-s + 4.34e3·79-s − 1.76e3·80-s + 4.45e3·81-s + 1.39e3·89-s − 740·95-s + 1.02e4·99-s + ⋯ |
| L(s) = 1 | − 5·4-s − 0.178·5-s − 3.33·9-s − 3.12·11-s + 55/4·16-s + 4.46·19-s + 0.894·20-s − 0.343·25-s + 2.35·29-s − 3.59·31-s + 50/3·36-s − 3.32·41-s + 15.6·44-s + 0.596·45-s + 11.9·49-s + 0.558·55-s + 7.12·59-s − 5.46·61-s − 27.5·64-s − 3.82·71-s − 22.3·76-s + 6.18·79-s − 2.45·80-s + 55/9·81-s + 1.65·89-s − 0.799·95-s + 10.4·99-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 31^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 31^{20}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 31^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.12322\times 10^{34}\) |
| Root analytic conductor: | \(7.40754\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{20} \cdot 3^{20} \cdot 5^{20} \cdot 31^{20} ,\ ( \ : [3/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.005721567908\) |
| \(L(\frac12)\) | \(\approx\) | \(0.005721567908\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{10} \) |
| 3 | \( ( 1 + p^{2} T^{2} )^{10} \) | |
| 5 | \( 1 + 2 T + 47 T^{2} + 742 T^{3} - 12543 T^{4} - 292244 T^{5} - 440596 p T^{6} - 133756 p^{2} T^{7} - 392418 p^{3} T^{8} + 751744 p^{5} T^{9} + 27435874 p^{5} T^{10} + 751744 p^{8} T^{11} - 392418 p^{9} T^{12} - 133756 p^{11} T^{13} - 440596 p^{13} T^{14} - 292244 p^{15} T^{15} - 12543 p^{18} T^{16} + 742 p^{21} T^{17} + 47 p^{24} T^{18} + 2 p^{27} T^{19} + p^{30} T^{20} \) | |
| 31 | \( ( 1 + p T )^{20} \) | |
| good | 7 | \( 1 - 4097 T^{2} + 7947447 T^{4} - 9665980976 T^{6} + 8203473145979 T^{8} - 5090800854728681 T^{10} + 2331296988174962016 T^{12} - \)\(75\!\cdots\!71\)\( T^{14} + \)\(15\!\cdots\!48\)\( T^{16} - \)\(42\!\cdots\!27\)\( T^{18} - \)\(59\!\cdots\!82\)\( T^{20} - \)\(42\!\cdots\!27\)\( p^{6} T^{22} + \)\(15\!\cdots\!48\)\( p^{12} T^{24} - \)\(75\!\cdots\!71\)\( p^{18} T^{26} + 2331296988174962016 p^{24} T^{28} - 5090800854728681 p^{30} T^{30} + 8203473145979 p^{36} T^{32} - 9665980976 p^{42} T^{34} + 7947447 p^{48} T^{36} - 4097 p^{54} T^{38} + p^{60} T^{40} \) |
| 11 | \( ( 1 + 57 T + 886 p T^{2} + 498195 T^{3} + 47196409 T^{4} + 2124865500 T^{5} + 145119248104 T^{6} + 5725634307588 T^{7} + 311013851732334 T^{8} + 10663038874111230 T^{9} + 483340729856013452 T^{10} + 10663038874111230 p^{3} T^{11} + 311013851732334 p^{6} T^{12} + 5725634307588 p^{9} T^{13} + 145119248104 p^{12} T^{14} + 2124865500 p^{15} T^{15} + 47196409 p^{18} T^{16} + 498195 p^{21} T^{17} + 886 p^{25} T^{18} + 57 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 13 | \( 1 - 21948 T^{2} + 250455982 T^{4} - 1963934892364 T^{6} + 11800425046523421 T^{8} - 57489117252337026908 T^{10} + \)\(18\!\cdots\!52\)\( p T^{12} - \)\(82\!\cdots\!56\)\( T^{14} + \)\(19\!\cdots\!10\)\( p T^{16} - \)\(66\!\cdots\!12\)\( T^{18} + \)\(15\!\cdots\!40\)\( T^{20} - \)\(66\!\cdots\!12\)\( p^{6} T^{22} + \)\(19\!\cdots\!10\)\( p^{13} T^{24} - \)\(82\!\cdots\!56\)\( p^{18} T^{26} + \)\(18\!\cdots\!52\)\( p^{25} T^{28} - 57489117252337026908 p^{30} T^{30} + 11800425046523421 p^{36} T^{32} - 1963934892364 p^{42} T^{34} + 250455982 p^{48} T^{36} - 21948 p^{54} T^{38} + p^{60} T^{40} \) | |
| 17 | \( 1 - 53676 T^{2} + 1420432970 T^{4} - 24432257684412 T^{6} + 17907073758158093 p T^{8} - \)\(29\!\cdots\!08\)\( T^{10} + \)\(22\!\cdots\!64\)\( T^{12} - \)\(13\!\cdots\!32\)\( T^{14} + \)\(71\!\cdots\!50\)\( T^{16} - \)\(34\!\cdots\!76\)\( T^{18} + \)\(16\!\cdots\!08\)\( T^{20} - \)\(34\!\cdots\!76\)\( p^{6} T^{22} + \)\(71\!\cdots\!50\)\( p^{12} T^{24} - \)\(13\!\cdots\!32\)\( p^{18} T^{26} + \)\(22\!\cdots\!64\)\( p^{24} T^{28} - \)\(29\!\cdots\!08\)\( p^{30} T^{30} + 17907073758158093 p^{37} T^{32} - 24432257684412 p^{42} T^{34} + 1420432970 p^{48} T^{36} - 53676 p^{54} T^{38} + p^{60} T^{40} \) | |
| 19 | \( ( 1 - 185 T + 49633 T^{2} - 6237654 T^{3} + 1025034405 T^{4} - 105747646171 T^{5} + 13870129955860 T^{6} - 1253768228009715 T^{7} + 139507635679305762 T^{8} - 11091935736338999873 T^{9} + \)\(10\!\cdots\!90\)\( T^{10} - 11091935736338999873 p^{3} T^{11} + 139507635679305762 p^{6} T^{12} - 1253768228009715 p^{9} T^{13} + 13870129955860 p^{12} T^{14} - 105747646171 p^{15} T^{15} + 1025034405 p^{18} T^{16} - 6237654 p^{21} T^{17} + 49633 p^{24} T^{18} - 185 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 23 | \( 1 - 115411 T^{2} + 265082588 p T^{4} - 191023385801479 T^{6} + 3797479102331188197 T^{8} - \)\(45\!\cdots\!32\)\( T^{10} + \)\(22\!\cdots\!16\)\( T^{12} + \)\(16\!\cdots\!60\)\( T^{14} - \)\(19\!\cdots\!06\)\( T^{16} - \)\(40\!\cdots\!18\)\( T^{18} + \)\(97\!\cdots\!76\)\( T^{20} - \)\(40\!\cdots\!18\)\( p^{6} T^{22} - \)\(19\!\cdots\!06\)\( p^{12} T^{24} + \)\(16\!\cdots\!60\)\( p^{18} T^{26} + \)\(22\!\cdots\!16\)\( p^{24} T^{28} - \)\(45\!\cdots\!32\)\( p^{30} T^{30} + 3797479102331188197 p^{36} T^{32} - 191023385801479 p^{42} T^{34} + 265082588 p^{49} T^{36} - 115411 p^{54} T^{38} + p^{60} T^{40} \) | |
| 29 | \( ( 1 - 184 T + 156584 T^{2} - 19822748 T^{3} + 9972862235 T^{4} - 23250921050 p T^{5} + 328039439505088 T^{6} + 3610487930710958 T^{7} + 6082959787231824844 T^{8} + \)\(84\!\cdots\!08\)\( T^{9} + \)\(10\!\cdots\!32\)\( T^{10} + \)\(84\!\cdots\!08\)\( p^{3} T^{11} + 6082959787231824844 p^{6} T^{12} + 3610487930710958 p^{9} T^{13} + 328039439505088 p^{12} T^{14} - 23250921050 p^{16} T^{15} + 9972862235 p^{18} T^{16} - 19822748 p^{21} T^{17} + 156584 p^{24} T^{18} - 184 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 37 | \( 1 - 720824 T^{2} + 253704498094 T^{4} - 58132683427193356 T^{6} + \)\(97\!\cdots\!37\)\( T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(13\!\cdots\!36\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{14} + \)\(88\!\cdots\!94\)\( T^{16} - \)\(56\!\cdots\!52\)\( T^{18} + \)\(30\!\cdots\!36\)\( T^{20} - \)\(56\!\cdots\!52\)\( p^{6} T^{22} + \)\(88\!\cdots\!94\)\( p^{12} T^{24} - \)\(11\!\cdots\!00\)\( p^{18} T^{26} + \)\(13\!\cdots\!36\)\( p^{24} T^{28} - \)\(12\!\cdots\!08\)\( p^{30} T^{30} + \)\(97\!\cdots\!37\)\( p^{36} T^{32} - 58132683427193356 p^{42} T^{34} + 253704498094 p^{48} T^{36} - 720824 p^{54} T^{38} + p^{60} T^{40} \) | |
| 41 | \( ( 1 + 436 T + 485147 T^{2} + 198909566 T^{3} + 119695891121 T^{4} + 43866255007378 T^{5} + 18936554435056204 T^{6} + 6110931343292881462 T^{7} + \)\(20\!\cdots\!10\)\( T^{8} + \)\(58\!\cdots\!14\)\( T^{9} + \)\(16\!\cdots\!42\)\( T^{10} + \)\(58\!\cdots\!14\)\( p^{3} T^{11} + \)\(20\!\cdots\!10\)\( p^{6} T^{12} + 6110931343292881462 p^{9} T^{13} + 18936554435056204 p^{12} T^{14} + 43866255007378 p^{15} T^{15} + 119695891121 p^{18} T^{16} + 198909566 p^{21} T^{17} + 485147 p^{24} T^{18} + 436 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 43 | \( 1 - 18001 p T^{2} + 298981121592 T^{4} - 76203840600576683 T^{6} + \)\(14\!\cdots\!93\)\( T^{8} - \)\(21\!\cdots\!04\)\( T^{10} + \)\(25\!\cdots\!72\)\( T^{12} - \)\(26\!\cdots\!28\)\( T^{14} + \)\(24\!\cdots\!78\)\( T^{16} - \)\(20\!\cdots\!78\)\( T^{18} + \)\(16\!\cdots\!28\)\( T^{20} - \)\(20\!\cdots\!78\)\( p^{6} T^{22} + \)\(24\!\cdots\!78\)\( p^{12} T^{24} - \)\(26\!\cdots\!28\)\( p^{18} T^{26} + \)\(25\!\cdots\!72\)\( p^{24} T^{28} - \)\(21\!\cdots\!04\)\( p^{30} T^{30} + \)\(14\!\cdots\!93\)\( p^{36} T^{32} - 76203840600576683 p^{42} T^{34} + 298981121592 p^{48} T^{36} - 18001 p^{55} T^{38} + p^{60} T^{40} \) | |
| 47 | \( 1 - 821088 T^{2} + 327107317218 T^{4} - 84207960636999520 T^{6} + \)\(15\!\cdots\!49\)\( T^{8} - \)\(23\!\cdots\!36\)\( T^{10} + \)\(28\!\cdots\!80\)\( T^{12} - \)\(31\!\cdots\!24\)\( T^{14} + \)\(31\!\cdots\!86\)\( T^{16} - \)\(30\!\cdots\!16\)\( T^{18} + \)\(30\!\cdots\!32\)\( T^{20} - \)\(30\!\cdots\!16\)\( p^{6} T^{22} + \)\(31\!\cdots\!86\)\( p^{12} T^{24} - \)\(31\!\cdots\!24\)\( p^{18} T^{26} + \)\(28\!\cdots\!80\)\( p^{24} T^{28} - \)\(23\!\cdots\!36\)\( p^{30} T^{30} + \)\(15\!\cdots\!49\)\( p^{36} T^{32} - 84207960636999520 p^{42} T^{34} + 327107317218 p^{48} T^{36} - 821088 p^{54} T^{38} + p^{60} T^{40} \) | |
| 53 | \( 1 - 1471827 T^{2} + 1110875507040 T^{4} - 569292739535710927 T^{6} + \)\(22\!\cdots\!41\)\( T^{8} - \)\(70\!\cdots\!04\)\( T^{10} + \)\(18\!\cdots\!84\)\( T^{12} - \)\(42\!\cdots\!68\)\( T^{14} + \)\(84\!\cdots\!90\)\( T^{16} - \)\(15\!\cdots\!42\)\( T^{18} + \)\(23\!\cdots\!88\)\( T^{20} - \)\(15\!\cdots\!42\)\( p^{6} T^{22} + \)\(84\!\cdots\!90\)\( p^{12} T^{24} - \)\(42\!\cdots\!68\)\( p^{18} T^{26} + \)\(18\!\cdots\!84\)\( p^{24} T^{28} - \)\(70\!\cdots\!04\)\( p^{30} T^{30} + \)\(22\!\cdots\!41\)\( p^{36} T^{32} - 569292739535710927 p^{42} T^{34} + 1110875507040 p^{48} T^{36} - 1471827 p^{54} T^{38} + p^{60} T^{40} \) | |
| 59 | \( ( 1 - 1614 T + 2139629 T^{2} - 1803523290 T^{3} + 1329567904213 T^{4} - 739853958706050 T^{5} + 376442212966554814 T^{6} - \)\(15\!\cdots\!96\)\( T^{7} + \)\(61\!\cdots\!62\)\( T^{8} - \)\(20\!\cdots\!18\)\( T^{9} + \)\(94\!\cdots\!62\)\( T^{10} - \)\(20\!\cdots\!18\)\( p^{3} T^{11} + \)\(61\!\cdots\!62\)\( p^{6} T^{12} - \)\(15\!\cdots\!96\)\( p^{9} T^{13} + 376442212966554814 p^{12} T^{14} - 739853958706050 p^{15} T^{15} + 1329567904213 p^{18} T^{16} - 1803523290 p^{21} T^{17} + 2139629 p^{24} T^{18} - 1614 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 61 | \( ( 1 + 1302 T + 2143738 T^{2} + 1914512670 T^{3} + 1881950452965 T^{4} + 1302113050059112 T^{5} + 964730693331666168 T^{6} + \)\(55\!\cdots\!88\)\( T^{7} + \)\(33\!\cdots\!38\)\( T^{8} + \)\(27\!\cdots\!88\)\( p T^{9} + \)\(87\!\cdots\!00\)\( T^{10} + \)\(27\!\cdots\!88\)\( p^{4} T^{11} + \)\(33\!\cdots\!38\)\( p^{6} T^{12} + \)\(55\!\cdots\!88\)\( p^{9} T^{13} + 964730693331666168 p^{12} T^{14} + 1302113050059112 p^{15} T^{15} + 1881950452965 p^{18} T^{16} + 1914512670 p^{21} T^{17} + 2143738 p^{24} T^{18} + 1302 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 67 | \( 1 - 2975540 T^{2} + 4594212432758 T^{4} - 4886493412344175940 T^{6} + \)\(39\!\cdots\!57\)\( T^{8} - \)\(26\!\cdots\!92\)\( T^{10} + \)\(14\!\cdots\!68\)\( T^{12} - \)\(70\!\cdots\!76\)\( T^{14} + \)\(29\!\cdots\!22\)\( T^{16} - \)\(10\!\cdots\!44\)\( T^{18} + \)\(34\!\cdots\!08\)\( T^{20} - \)\(10\!\cdots\!44\)\( p^{6} T^{22} + \)\(29\!\cdots\!22\)\( p^{12} T^{24} - \)\(70\!\cdots\!76\)\( p^{18} T^{26} + \)\(14\!\cdots\!68\)\( p^{24} T^{28} - \)\(26\!\cdots\!92\)\( p^{30} T^{30} + \)\(39\!\cdots\!57\)\( p^{36} T^{32} - 4886493412344175940 p^{42} T^{34} + 4594212432758 p^{48} T^{36} - 2975540 p^{54} T^{38} + p^{60} T^{40} \) | |
| 71 | \( ( 1 + 1145 T + 2290147 T^{2} + 1847901074 T^{3} + 2106081851315 T^{4} + 1223639700278855 T^{5} + 1018762508314471782 T^{6} + \)\(39\!\cdots\!63\)\( T^{7} + \)\(29\!\cdots\!26\)\( T^{8} + \)\(67\!\cdots\!77\)\( T^{9} + \)\(78\!\cdots\!06\)\( T^{10} + \)\(67\!\cdots\!77\)\( p^{3} T^{11} + \)\(29\!\cdots\!26\)\( p^{6} T^{12} + \)\(39\!\cdots\!63\)\( p^{9} T^{13} + 1018762508314471782 p^{12} T^{14} + 1223639700278855 p^{15} T^{15} + 2106081851315 p^{18} T^{16} + 1847901074 p^{21} T^{17} + 2290147 p^{24} T^{18} + 1145 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 73 | \( 1 - 3553051 T^{2} + 6416248770372 T^{4} - 7814850811746107527 T^{6} + \)\(72\!\cdots\!05\)\( T^{8} - \)\(54\!\cdots\!64\)\( T^{10} + \)\(34\!\cdots\!12\)\( T^{12} - \)\(19\!\cdots\!56\)\( T^{14} + \)\(97\!\cdots\!74\)\( T^{16} - \)\(44\!\cdots\!34\)\( T^{18} + \)\(18\!\cdots\!52\)\( T^{20} - \)\(44\!\cdots\!34\)\( p^{6} T^{22} + \)\(97\!\cdots\!74\)\( p^{12} T^{24} - \)\(19\!\cdots\!56\)\( p^{18} T^{26} + \)\(34\!\cdots\!12\)\( p^{24} T^{28} - \)\(54\!\cdots\!64\)\( p^{30} T^{30} + \)\(72\!\cdots\!05\)\( p^{36} T^{32} - 7814850811746107527 p^{42} T^{34} + 6416248770372 p^{48} T^{36} - 3553051 p^{54} T^{38} + p^{60} T^{40} \) | |
| 79 | \( ( 1 - 2171 T + 5074966 T^{2} - 7319350921 T^{3} + 10386776571653 T^{4} - 11578691157742328 T^{5} + 12514066660493320068 T^{6} - \)\(11\!\cdots\!44\)\( T^{7} + \)\(10\!\cdots\!70\)\( T^{8} - \)\(78\!\cdots\!02\)\( T^{9} + \)\(58\!\cdots\!04\)\( T^{10} - \)\(78\!\cdots\!02\)\( p^{3} T^{11} + \)\(10\!\cdots\!70\)\( p^{6} T^{12} - \)\(11\!\cdots\!44\)\( p^{9} T^{13} + 12514066660493320068 p^{12} T^{14} - 11578691157742328 p^{15} T^{15} + 10386776571653 p^{18} T^{16} - 7319350921 p^{21} T^{17} + 5074966 p^{24} T^{18} - 2171 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 83 | \( 1 - 5420300 T^{2} + 15015050307250 T^{4} - 27896698749977827756 T^{6} + \)\(38\!\cdots\!37\)\( T^{8} - \)\(42\!\cdots\!00\)\( T^{10} + \)\(38\!\cdots\!56\)\( T^{12} - \)\(29\!\cdots\!72\)\( T^{14} + \)\(19\!\cdots\!18\)\( T^{16} - \)\(12\!\cdots\!32\)\( T^{18} + \)\(70\!\cdots\!16\)\( T^{20} - \)\(12\!\cdots\!32\)\( p^{6} T^{22} + \)\(19\!\cdots\!18\)\( p^{12} T^{24} - \)\(29\!\cdots\!72\)\( p^{18} T^{26} + \)\(38\!\cdots\!56\)\( p^{24} T^{28} - \)\(42\!\cdots\!00\)\( p^{30} T^{30} + \)\(38\!\cdots\!37\)\( p^{36} T^{32} - 27896698749977827756 p^{42} T^{34} + 15015050307250 p^{48} T^{36} - 5420300 p^{54} T^{38} + p^{60} T^{40} \) | |
| 89 | \( ( 1 - 695 T + 4564136 T^{2} - 2668620535 T^{3} + 10146846601625 T^{4} - 5131307481375896 T^{5} + 14741569843846202206 T^{6} - \)\(65\!\cdots\!64\)\( T^{7} + \)\(15\!\cdots\!74\)\( T^{8} - \)\(61\!\cdots\!98\)\( T^{9} + \)\(12\!\cdots\!16\)\( T^{10} - \)\(61\!\cdots\!98\)\( p^{3} T^{11} + \)\(15\!\cdots\!74\)\( p^{6} T^{12} - \)\(65\!\cdots\!64\)\( p^{9} T^{13} + 14741569843846202206 p^{12} T^{14} - 5131307481375896 p^{15} T^{15} + 10146846601625 p^{18} T^{16} - 2668620535 p^{21} T^{17} + 4564136 p^{24} T^{18} - 695 p^{27} T^{19} + p^{30} T^{20} )^{2} \) | |
| 97 | \( 1 - 7515010 T^{2} + 31151600200907 T^{4} - 91165657370856136982 T^{6} + \)\(20\!\cdots\!05\)\( T^{8} - \)\(39\!\cdots\!36\)\( T^{10} + \)\(62\!\cdots\!64\)\( T^{12} - \)\(86\!\cdots\!28\)\( T^{14} + \)\(10\!\cdots\!02\)\( T^{16} - \)\(11\!\cdots\!52\)\( T^{18} + \)\(11\!\cdots\!22\)\( T^{20} - \)\(11\!\cdots\!52\)\( p^{6} T^{22} + \)\(10\!\cdots\!02\)\( p^{12} T^{24} - \)\(86\!\cdots\!28\)\( p^{18} T^{26} + \)\(62\!\cdots\!64\)\( p^{24} T^{28} - \)\(39\!\cdots\!36\)\( p^{30} T^{30} + \)\(20\!\cdots\!05\)\( p^{36} T^{32} - 91165657370856136982 p^{42} T^{34} + 31151600200907 p^{48} T^{36} - 7515010 p^{54} T^{38} + p^{60} T^{40} \) | |
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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
−1.89653511965034861824163445947, −1.82791422947266331953848256960, −1.55765323418944905859859934425, −1.54100264902557631444581486320, −1.31230184643883891395498806611, −1.23127387107143393291733417757, −1.15713060591730730909109165720, −1.11067265271708502297206713467, −1.09877254237335493837847113757, −1.06397607911598969809347697106, −1.01220381531110687661817983363, −0.962557693420458611056782686070, −0.941984188196568303027669572236, −0.902303078572028902182789986287, −0.817083235342279949668562308920, −0.815613400531191951118960805537, −0.73152887504090409847633838249, −0.47842539147599849016956979116, −0.36273976594442804388704476972, −0.29328792259176834871224490987, −0.28895427556708047612013259443, −0.24035462582846170927954617611, −0.12456444729602490117011889682, −0.06420961086564212873280481911, −0.03086997531966733660101505266, 0.03086997531966733660101505266, 0.06420961086564212873280481911, 0.12456444729602490117011889682, 0.24035462582846170927954617611, 0.28895427556708047612013259443, 0.29328792259176834871224490987, 0.36273976594442804388704476972, 0.47842539147599849016956979116, 0.73152887504090409847633838249, 0.815613400531191951118960805537, 0.817083235342279949668562308920, 0.902303078572028902182789986287, 0.941984188196568303027669572236, 0.962557693420458611056782686070, 1.01220381531110687661817983363, 1.06397607911598969809347697106, 1.09877254237335493837847113757, 1.11067265271708502297206713467, 1.15713060591730730909109165720, 1.23127387107143393291733417757, 1.31230184643883891395498806611, 1.54100264902557631444581486320, 1.55765323418944905859859934425, 1.82791422947266331953848256960, 1.89653511965034861824163445947
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.