Dirichlet series
| L(s) = 1 | − 32·7-s + 624·9-s + 24·11-s + 216·17-s + 596·19-s − 576·23-s − 5.54e3·25-s + 1.25e3·43-s + 3.76e3·47-s − 2.48e4·49-s + 352·61-s − 1.99e4·63-s + 1.35e3·73-s − 768·77-s + 1.86e5·81-s + 1.61e4·83-s + 1.49e4·99-s − 2.76e4·101-s − 6.91e3·119-s − 1.39e5·121-s − 1.26e4·125-s + 127-s + 131-s − 1.90e4·133-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 0.653·7-s + 7.70·9-s + 0.198·11-s + 0.747·17-s + 1.65·19-s − 1.08·23-s − 8.87·25-s + 0.679·43-s + 1.70·47-s − 10.3·49-s + 0.0945·61-s − 5.03·63-s + 0.253·73-s − 0.129·77-s + 28.3·81-s + 2.33·83-s + 1.52·99-s − 2.71·101-s − 0.488·119-s − 9.50·121-s − 0.812·125-s + 6.20e−5·127-s + 5.82e−5·131-s − 1.07·133-s + 5.32e−5·137-s + 5.17e−5·139-s + 4.50e−5·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{80} \cdot 19^{20}\right)^{s/2} \, \Gamma_{\C}(s+2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(2^{80} \cdot 19^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.81777\times 10^{29}\) |
| Root analytic conductor: | \(5.60575\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 2^{80} \cdot 19^{20} ,\ ( \ : [2]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.4269083130\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4269083130\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 596 T - 4874 T^{2} + 61177004 T^{3} + 296051727 p T^{4} - 54043589488 p^{2} T^{5} + 853557961592 p^{3} T^{6} + 14083576428176 p^{4} T^{7} - 385068940608810 p^{5} T^{8} - 153672322794536 p^{7} T^{9} + 24883972937124 p^{10} T^{10} - 153672322794536 p^{11} T^{11} - 385068940608810 p^{13} T^{12} + 14083576428176 p^{16} T^{13} + 853557961592 p^{19} T^{14} - 54043589488 p^{22} T^{15} + 296051727 p^{25} T^{16} + 61177004 p^{28} T^{17} - 4874 p^{32} T^{18} - 596 p^{36} T^{19} + p^{40} T^{20} \) | |
| good | 3 | \( 1 - 208 p T^{2} + 203276 T^{4} - 46238332 T^{6} + 8277303254 T^{8} - 1241549861824 T^{10} + 17957293241074 p^{2} T^{12} - 230080184683000 p^{4} T^{14} + 293826540564473 p^{8} T^{16} - 27479536768114396 p^{8} T^{18} + 9604763979413612 p^{13} T^{20} - 27479536768114396 p^{16} T^{22} + 293826540564473 p^{24} T^{24} - 230080184683000 p^{28} T^{26} + 17957293241074 p^{34} T^{28} - 1241549861824 p^{40} T^{30} + 8277303254 p^{48} T^{32} - 46238332 p^{56} T^{34} + 203276 p^{64} T^{36} - 208 p^{73} T^{38} + p^{80} T^{40} \) |
| 5 | \( ( 1 + 2772 T^{2} + 6348 T^{3} + 3500342 T^{4} + 5962512 T^{5} + 2663051978 T^{6} - 5622127896 T^{7} + 1414897258729 T^{8} - 12031053829188 T^{9} + 743437575270276 T^{10} - 12031053829188 p^{4} T^{11} + 1414897258729 p^{8} T^{12} - 5622127896 p^{12} T^{13} + 2663051978 p^{16} T^{14} + 5962512 p^{20} T^{15} + 3500342 p^{24} T^{16} + 6348 p^{28} T^{17} + 2772 p^{32} T^{18} + p^{40} T^{20} )^{2} \) | |
| 7 | \( ( 1 + 16 T + 12818 T^{2} + 282860 T^{3} + 83015315 T^{4} + 2040710456 T^{5} + 367081823016 T^{6} + 8762268868820 T^{7} + 1240267712002189 T^{8} + 26738241700983588 T^{9} + 3329540538909777314 T^{10} + 26738241700983588 p^{4} T^{11} + 1240267712002189 p^{8} T^{12} + 8762268868820 p^{12} T^{13} + 367081823016 p^{16} T^{14} + 2040710456 p^{20} T^{15} + 83015315 p^{24} T^{16} + 282860 p^{28} T^{17} + 12818 p^{32} T^{18} + 16 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 11 | \( ( 1 - 12 T + 6348 p T^{2} + 116148 p T^{3} + 2280205190 T^{4} + 108600211116 T^{5} + 4405449730646 p T^{6} + 3666887782320204 T^{7} + 72432993520638875 p T^{8} + 77402102336302625496 T^{9} + \)\(11\!\cdots\!12\)\( T^{10} + 77402102336302625496 p^{4} T^{11} + 72432993520638875 p^{9} T^{12} + 3666887782320204 p^{12} T^{13} + 4405449730646 p^{17} T^{14} + 108600211116 p^{20} T^{15} + 2280205190 p^{24} T^{16} + 116148 p^{29} T^{17} + 6348 p^{33} T^{18} - 12 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 13 | \( 1 - 301952 T^{2} + 45758159020 T^{4} - 4637321592403724 T^{6} + \)\(35\!\cdots\!18\)\( T^{8} - \)\(21\!\cdots\!08\)\( T^{10} + \)\(11\!\cdots\!74\)\( T^{12} - \)\(37\!\cdots\!20\)\( p T^{14} + \)\(14\!\cdots\!97\)\( p T^{16} - \)\(63\!\cdots\!84\)\( T^{18} + \)\(19\!\cdots\!64\)\( T^{20} - \)\(63\!\cdots\!84\)\( p^{8} T^{22} + \)\(14\!\cdots\!97\)\( p^{17} T^{24} - \)\(37\!\cdots\!20\)\( p^{25} T^{26} + \)\(11\!\cdots\!74\)\( p^{32} T^{28} - \)\(21\!\cdots\!08\)\( p^{40} T^{30} + \)\(35\!\cdots\!18\)\( p^{48} T^{32} - 4637321592403724 p^{56} T^{34} + 45758159020 p^{64} T^{36} - 301952 p^{72} T^{38} + p^{80} T^{40} \) | |
| 17 | \( ( 1 - 108 T + 379674 T^{2} - 33764820 T^{3} + 74603899091 T^{4} - 4697089720632 T^{5} + 10230421621548768 T^{6} - 433635093644868480 T^{7} + \)\(11\!\cdots\!09\)\( T^{8} - \)\(35\!\cdots\!96\)\( T^{9} + \)\(10\!\cdots\!14\)\( T^{10} - \)\(35\!\cdots\!96\)\( p^{4} T^{11} + \)\(11\!\cdots\!09\)\( p^{8} T^{12} - 433635093644868480 p^{12} T^{13} + 10230421621548768 p^{16} T^{14} - 4697089720632 p^{20} T^{15} + 74603899091 p^{24} T^{16} - 33764820 p^{28} T^{17} + 379674 p^{32} T^{18} - 108 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 23 | \( ( 1 + 288 T + 937516 T^{2} + 377818428 T^{3} + 514505105398 T^{4} + 260114220464400 T^{5} + 223475374628074962 T^{6} + \)\(12\!\cdots\!96\)\( T^{7} + \)\(81\!\cdots\!89\)\( T^{8} + \)\(42\!\cdots\!92\)\( T^{9} + \)\(25\!\cdots\!44\)\( T^{10} + \)\(42\!\cdots\!92\)\( p^{4} T^{11} + \)\(81\!\cdots\!89\)\( p^{8} T^{12} + \)\(12\!\cdots\!96\)\( p^{12} T^{13} + 223475374628074962 p^{16} T^{14} + 260114220464400 p^{20} T^{15} + 514505105398 p^{24} T^{16} + 377818428 p^{28} T^{17} + 937516 p^{32} T^{18} + 288 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 29 | \( 1 - 7153392 T^{2} + 24574321075884 T^{4} - 54351182917079072444 T^{6} + \)\(30\!\cdots\!98\)\( p T^{8} - \)\(11\!\cdots\!16\)\( T^{10} + \)\(11\!\cdots\!06\)\( T^{12} - \)\(10\!\cdots\!04\)\( T^{14} + \)\(85\!\cdots\!25\)\( T^{16} - \)\(63\!\cdots\!76\)\( T^{18} + \)\(45\!\cdots\!44\)\( T^{20} - \)\(63\!\cdots\!76\)\( p^{8} T^{22} + \)\(85\!\cdots\!25\)\( p^{16} T^{24} - \)\(10\!\cdots\!04\)\( p^{24} T^{26} + \)\(11\!\cdots\!06\)\( p^{32} T^{28} - \)\(11\!\cdots\!16\)\( p^{40} T^{30} + \)\(30\!\cdots\!98\)\( p^{49} T^{32} - 54351182917079072444 p^{56} T^{34} + 24574321075884 p^{64} T^{36} - 7153392 p^{72} T^{38} + p^{80} T^{40} \) | |
| 31 | \( 1 - 7893844 T^{2} + 31792808382430 T^{4} - 88028946811392639156 T^{6} + \)\(18\!\cdots\!45\)\( T^{8} - \)\(33\!\cdots\!28\)\( T^{10} + \)\(51\!\cdots\!72\)\( T^{12} - \)\(69\!\cdots\!68\)\( T^{14} + \)\(82\!\cdots\!70\)\( T^{16} - \)\(89\!\cdots\!80\)\( T^{18} + \)\(86\!\cdots\!80\)\( T^{20} - \)\(89\!\cdots\!80\)\( p^{8} T^{22} + \)\(82\!\cdots\!70\)\( p^{16} T^{24} - \)\(69\!\cdots\!68\)\( p^{24} T^{26} + \)\(51\!\cdots\!72\)\( p^{32} T^{28} - \)\(33\!\cdots\!28\)\( p^{40} T^{30} + \)\(18\!\cdots\!45\)\( p^{48} T^{32} - 88028946811392639156 p^{56} T^{34} + 31792808382430 p^{64} T^{36} - 7893844 p^{72} T^{38} + p^{80} T^{40} \) | |
| 37 | \( 1 - 12744308 T^{2} + 92798355500510 T^{4} - \)\(48\!\cdots\!28\)\( T^{6} + \)\(20\!\cdots\!69\)\( T^{8} - \)\(70\!\cdots\!72\)\( T^{10} + \)\(21\!\cdots\!20\)\( T^{12} - \)\(57\!\cdots\!96\)\( T^{14} + \)\(13\!\cdots\!58\)\( T^{16} - \)\(29\!\cdots\!80\)\( T^{18} + \)\(57\!\cdots\!88\)\( T^{20} - \)\(29\!\cdots\!80\)\( p^{8} T^{22} + \)\(13\!\cdots\!58\)\( p^{16} T^{24} - \)\(57\!\cdots\!96\)\( p^{24} T^{26} + \)\(21\!\cdots\!20\)\( p^{32} T^{28} - \)\(70\!\cdots\!72\)\( p^{40} T^{30} + \)\(20\!\cdots\!69\)\( p^{48} T^{32} - \)\(48\!\cdots\!28\)\( p^{56} T^{34} + 92798355500510 p^{64} T^{36} - 12744308 p^{72} T^{38} + p^{80} T^{40} \) | |
| 41 | \( 1 - 18443156 T^{2} + 183431814607838 T^{4} - \)\(12\!\cdots\!56\)\( T^{6} + \)\(62\!\cdots\!97\)\( T^{8} - \)\(25\!\cdots\!32\)\( T^{10} + \)\(91\!\cdots\!48\)\( T^{12} - \)\(30\!\cdots\!00\)\( T^{14} + \)\(97\!\cdots\!22\)\( T^{16} - \)\(30\!\cdots\!56\)\( T^{18} + \)\(89\!\cdots\!64\)\( T^{20} - \)\(30\!\cdots\!56\)\( p^{8} T^{22} + \)\(97\!\cdots\!22\)\( p^{16} T^{24} - \)\(30\!\cdots\!00\)\( p^{24} T^{26} + \)\(91\!\cdots\!48\)\( p^{32} T^{28} - \)\(25\!\cdots\!32\)\( p^{40} T^{30} + \)\(62\!\cdots\!97\)\( p^{48} T^{32} - \)\(12\!\cdots\!56\)\( p^{56} T^{34} + 183431814607838 p^{64} T^{36} - 18443156 p^{72} T^{38} + p^{80} T^{40} \) | |
| 43 | \( ( 1 - 628 T + 19185244 T^{2} - 18004957452 T^{3} + 192094454008374 T^{4} - 213470902443589228 T^{5} + \)\(13\!\cdots\!38\)\( T^{6} - \)\(14\!\cdots\!84\)\( T^{7} + \)\(66\!\cdots\!93\)\( T^{8} - \)\(71\!\cdots\!88\)\( T^{9} + \)\(25\!\cdots\!08\)\( T^{10} - \)\(71\!\cdots\!88\)\( p^{4} T^{11} + \)\(66\!\cdots\!93\)\( p^{8} T^{12} - \)\(14\!\cdots\!84\)\( p^{12} T^{13} + \)\(13\!\cdots\!38\)\( p^{16} T^{14} - 213470902443589228 p^{20} T^{15} + 192094454008374 p^{24} T^{16} - 18004957452 p^{28} T^{17} + 19185244 p^{32} T^{18} - 628 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 47 | \( ( 1 - 1884 T + 26720116 T^{2} - 64590116580 T^{3} + 388056864623574 T^{4} - 979138895894739876 T^{5} + 83026953866461629166 p T^{6} - \)\(93\!\cdots\!28\)\( T^{7} + \)\(28\!\cdots\!09\)\( T^{8} - \)\(62\!\cdots\!88\)\( T^{9} + \)\(16\!\cdots\!04\)\( T^{10} - \)\(62\!\cdots\!88\)\( p^{4} T^{11} + \)\(28\!\cdots\!09\)\( p^{8} T^{12} - \)\(93\!\cdots\!28\)\( p^{12} T^{13} + 83026953866461629166 p^{17} T^{14} - 979138895894739876 p^{20} T^{15} + 388056864623574 p^{24} T^{16} - 64590116580 p^{28} T^{17} + 26720116 p^{32} T^{18} - 1884 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 53 | \( 1 - 45180464 T^{2} + 1207174144352620 T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(40\!\cdots\!86\)\( T^{8} - \)\(57\!\cdots\!56\)\( T^{10} + \)\(71\!\cdots\!22\)\( T^{12} - \)\(79\!\cdots\!72\)\( T^{14} + \)\(79\!\cdots\!69\)\( T^{16} - \)\(72\!\cdots\!96\)\( T^{18} + \)\(60\!\cdots\!84\)\( T^{20} - \)\(72\!\cdots\!96\)\( p^{8} T^{22} + \)\(79\!\cdots\!69\)\( p^{16} T^{24} - \)\(79\!\cdots\!72\)\( p^{24} T^{26} + \)\(71\!\cdots\!22\)\( p^{32} T^{28} - \)\(57\!\cdots\!56\)\( p^{40} T^{30} + \)\(40\!\cdots\!86\)\( p^{48} T^{32} - \)\(24\!\cdots\!20\)\( p^{56} T^{34} + 1207174144352620 p^{64} T^{36} - 45180464 p^{72} T^{38} + p^{80} T^{40} \) | |
| 59 | \( 1 - 90496128 T^{2} + 4550297691727244 T^{4} - \)\(16\!\cdots\!68\)\( T^{6} + \)\(46\!\cdots\!94\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{10} + \)\(22\!\cdots\!82\)\( T^{12} - \)\(39\!\cdots\!16\)\( T^{14} + \)\(63\!\cdots\!05\)\( T^{16} - \)\(89\!\cdots\!00\)\( T^{18} + \)\(11\!\cdots\!44\)\( T^{20} - \)\(89\!\cdots\!00\)\( p^{8} T^{22} + \)\(63\!\cdots\!05\)\( p^{16} T^{24} - \)\(39\!\cdots\!16\)\( p^{24} T^{26} + \)\(22\!\cdots\!82\)\( p^{32} T^{28} - \)\(10\!\cdots\!24\)\( p^{40} T^{30} + \)\(46\!\cdots\!94\)\( p^{48} T^{32} - \)\(16\!\cdots\!68\)\( p^{56} T^{34} + 4550297691727244 p^{64} T^{36} - 90496128 p^{72} T^{38} + p^{80} T^{40} \) | |
| 61 | \( ( 1 - 176 T + 91450676 T^{2} - 40305406180 T^{3} + 4121565308315798 T^{4} - 2565934153063042112 T^{5} + \)\(12\!\cdots\!22\)\( T^{6} - \)\(86\!\cdots\!32\)\( T^{7} + \)\(25\!\cdots\!49\)\( T^{8} - \)\(18\!\cdots\!68\)\( T^{9} + \)\(40\!\cdots\!16\)\( T^{10} - \)\(18\!\cdots\!68\)\( p^{4} T^{11} + \)\(25\!\cdots\!49\)\( p^{8} T^{12} - \)\(86\!\cdots\!32\)\( p^{12} T^{13} + \)\(12\!\cdots\!22\)\( p^{16} T^{14} - 2565934153063042112 p^{20} T^{15} + 4121565308315798 p^{24} T^{16} - 40305406180 p^{28} T^{17} + 91450676 p^{32} T^{18} - 176 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 67 | \( 1 - 2272304 p T^{2} + 12075825760538476 T^{4} - \)\(65\!\cdots\!00\)\( T^{6} + \)\(27\!\cdots\!18\)\( T^{8} - \)\(91\!\cdots\!64\)\( T^{10} + \)\(26\!\cdots\!46\)\( T^{12} - \)\(65\!\cdots\!96\)\( T^{14} + \)\(15\!\cdots\!01\)\( T^{16} - \)\(31\!\cdots\!12\)\( T^{18} + \)\(65\!\cdots\!92\)\( T^{20} - \)\(31\!\cdots\!12\)\( p^{8} T^{22} + \)\(15\!\cdots\!01\)\( p^{16} T^{24} - \)\(65\!\cdots\!96\)\( p^{24} T^{26} + \)\(26\!\cdots\!46\)\( p^{32} T^{28} - \)\(91\!\cdots\!64\)\( p^{40} T^{30} + \)\(27\!\cdots\!18\)\( p^{48} T^{32} - \)\(65\!\cdots\!00\)\( p^{56} T^{34} + 12075825760538476 p^{64} T^{36} - 2272304 p^{73} T^{38} + p^{80} T^{40} \) | |
| 71 | \( 1 - 200042628 T^{2} + 21635693559256702 T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(95\!\cdots\!81\)\( T^{8} - \)\(45\!\cdots\!64\)\( T^{10} + \)\(18\!\cdots\!68\)\( T^{12} - \)\(64\!\cdots\!76\)\( T^{14} + \)\(20\!\cdots\!30\)\( T^{16} - \)\(57\!\cdots\!24\)\( T^{18} + \)\(15\!\cdots\!36\)\( T^{20} - \)\(57\!\cdots\!24\)\( p^{8} T^{22} + \)\(20\!\cdots\!30\)\( p^{16} T^{24} - \)\(64\!\cdots\!76\)\( p^{24} T^{26} + \)\(18\!\cdots\!68\)\( p^{32} T^{28} - \)\(45\!\cdots\!64\)\( p^{40} T^{30} + \)\(95\!\cdots\!81\)\( p^{48} T^{32} - \)\(16\!\cdots\!80\)\( p^{56} T^{34} + 21635693559256702 p^{64} T^{36} - 200042628 p^{72} T^{38} + p^{80} T^{40} \) | |
| 73 | \( ( 1 - 676 T + 209661770 T^{2} - 141748691828 T^{3} + 20828030928413603 T^{4} - 15031324419466925304 T^{5} + \)\(13\!\cdots\!96\)\( T^{6} - \)\(98\!\cdots\!40\)\( T^{7} + \)\(57\!\cdots\!41\)\( T^{8} - \)\(41\!\cdots\!84\)\( T^{9} + \)\(18\!\cdots\!30\)\( T^{10} - \)\(41\!\cdots\!84\)\( p^{4} T^{11} + \)\(57\!\cdots\!41\)\( p^{8} T^{12} - \)\(98\!\cdots\!40\)\( p^{12} T^{13} + \)\(13\!\cdots\!96\)\( p^{16} T^{14} - 15031324419466925304 p^{20} T^{15} + 20828030928413603 p^{24} T^{16} - 141748691828 p^{28} T^{17} + 209661770 p^{32} T^{18} - 676 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 79 | \( 1 - 216329108 T^{2} + 27615318214687838 T^{4} - \)\(26\!\cdots\!60\)\( T^{6} + \)\(21\!\cdots\!41\)\( T^{8} - \)\(15\!\cdots\!36\)\( T^{10} + \)\(92\!\cdots\!12\)\( T^{12} - \)\(50\!\cdots\!20\)\( T^{14} + \)\(24\!\cdots\!90\)\( T^{16} - \)\(11\!\cdots\!04\)\( T^{18} + \)\(45\!\cdots\!16\)\( T^{20} - \)\(11\!\cdots\!04\)\( p^{8} T^{22} + \)\(24\!\cdots\!90\)\( p^{16} T^{24} - \)\(50\!\cdots\!20\)\( p^{24} T^{26} + \)\(92\!\cdots\!12\)\( p^{32} T^{28} - \)\(15\!\cdots\!36\)\( p^{40} T^{30} + \)\(21\!\cdots\!41\)\( p^{48} T^{32} - \)\(26\!\cdots\!60\)\( p^{56} T^{34} + 27615318214687838 p^{64} T^{36} - 216329108 p^{72} T^{38} + p^{80} T^{40} \) | |
| 83 | \( ( 1 - 8052 T + 278936086 T^{2} - 2125596719412 T^{3} + 40015977446491421 T^{4} - \)\(28\!\cdots\!04\)\( T^{5} + \)\(38\!\cdots\!84\)\( T^{6} - \)\(24\!\cdots\!88\)\( T^{7} + \)\(27\!\cdots\!42\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(14\!\cdots\!08\)\( T^{10} - \)\(15\!\cdots\!00\)\( p^{4} T^{11} + \)\(27\!\cdots\!42\)\( p^{8} T^{12} - \)\(24\!\cdots\!88\)\( p^{12} T^{13} + \)\(38\!\cdots\!84\)\( p^{16} T^{14} - \)\(28\!\cdots\!04\)\( p^{20} T^{15} + 40015977446491421 p^{24} T^{16} - 2125596719412 p^{28} T^{17} + 278936086 p^{32} T^{18} - 8052 p^{36} T^{19} + p^{40} T^{20} )^{2} \) | |
| 89 | \( 1 - 725619172 T^{2} + 264760502328520990 T^{4} - \)\(64\!\cdots\!64\)\( T^{6} + \)\(11\!\cdots\!05\)\( T^{8} - \)\(19\!\cdots\!44\)\( p T^{10} + \)\(21\!\cdots\!68\)\( T^{12} - \)\(21\!\cdots\!04\)\( T^{14} + \)\(19\!\cdots\!66\)\( T^{16} - \)\(14\!\cdots\!00\)\( T^{18} + \)\(98\!\cdots\!96\)\( T^{20} - \)\(14\!\cdots\!00\)\( p^{8} T^{22} + \)\(19\!\cdots\!66\)\( p^{16} T^{24} - \)\(21\!\cdots\!04\)\( p^{24} T^{26} + \)\(21\!\cdots\!68\)\( p^{32} T^{28} - \)\(19\!\cdots\!44\)\( p^{41} T^{30} + \)\(11\!\cdots\!05\)\( p^{48} T^{32} - \)\(64\!\cdots\!64\)\( p^{56} T^{34} + 264760502328520990 p^{64} T^{36} - 725619172 p^{72} T^{38} + p^{80} T^{40} \) | |
| 97 | \( 1 - 640763716 T^{2} + 216874716895343230 T^{4} - \)\(50\!\cdots\!24\)\( T^{6} + \)\(92\!\cdots\!81\)\( T^{8} - \)\(13\!\cdots\!72\)\( T^{10} + \)\(18\!\cdots\!40\)\( T^{12} - \)\(20\!\cdots\!84\)\( T^{14} + \)\(22\!\cdots\!50\)\( T^{16} - \)\(21\!\cdots\!84\)\( T^{18} + \)\(19\!\cdots\!00\)\( T^{20} - \)\(21\!\cdots\!84\)\( p^{8} T^{22} + \)\(22\!\cdots\!50\)\( p^{16} T^{24} - \)\(20\!\cdots\!84\)\( p^{24} T^{26} + \)\(18\!\cdots\!40\)\( p^{32} T^{28} - \)\(13\!\cdots\!72\)\( p^{40} T^{30} + \)\(92\!\cdots\!81\)\( p^{48} T^{32} - \)\(50\!\cdots\!24\)\( p^{56} T^{34} + 216874716895343230 p^{64} T^{36} - 640763716 p^{72} T^{38} + p^{80} T^{40} \) | |
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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
−2.02083951936309318650136146929, −1.97060384334162262561956965382, −1.95921742380903416429946539258, −1.83343488951466989060783870713, −1.73297542811278471945651094472, −1.64350722882731450068243962333, −1.50048287938005592241260992336, −1.44380290178304948632532470303, −1.42327243812986356473225739464, −1.42066764849123971167024584508, −1.37699169812107880353188465769, −1.37056151709655317818985439741, −1.15011979838501128754420549949, −1.10126203708777905154778612661, −1.04798330473150124810345572423, −0.966395800945692440362159699223, −0.902287291979255123869243762453, −0.802675427777910766307246934566, −0.46621196256573815760597180583, −0.41716438205171651993591694013, −0.37775298993572420015611913539, −0.28179561404934660535600383036, −0.20059057320650255138891229126, −0.14550586873661852575561212016, −0.02319773885020349639939474833, 0.02319773885020349639939474833, 0.14550586873661852575561212016, 0.20059057320650255138891229126, 0.28179561404934660535600383036, 0.37775298993572420015611913539, 0.41716438205171651993591694013, 0.46621196256573815760597180583, 0.802675427777910766307246934566, 0.902287291979255123869243762453, 0.966395800945692440362159699223, 1.04798330473150124810345572423, 1.10126203708777905154778612661, 1.15011979838501128754420549949, 1.37056151709655317818985439741, 1.37699169812107880353188465769, 1.42066764849123971167024584508, 1.42327243812986356473225739464, 1.44380290178304948632532470303, 1.50048287938005592241260992336, 1.64350722882731450068243962333, 1.73297542811278471945651094472, 1.83343488951466989060783870713, 1.95921742380903416429946539258, 1.97060384334162262561956965382, 2.02083951936309318650136146929
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.