Dirichlet series
| L(s) = 1 | + 78·3-s + 858·4-s − 1.74e3·7-s − 1.26e4·9-s + 3.49e3·11-s + 6.69e4·12-s + 3.50e5·16-s − 1.36e5·21-s + 7.27e5·25-s − 1.30e6·27-s − 1.49e6·28-s + 2.72e5·33-s − 1.08e7·36-s + 1.08e5·37-s − 1.57e6·41-s + 3.00e6·44-s − 1.51e6·47-s + 2.73e7·48-s − 5.07e6·49-s + 2.99e6·53-s + 2.20e7·63-s + 8.87e7·64-s + 3.56e6·67-s − 1.52e7·71-s + 1.10e7·73-s + 5.67e7·75-s − 6.10e6·77-s + ⋯ |
| L(s) = 1 | + 1.66·3-s + 6.70·4-s − 1.92·7-s − 5.78·9-s + 0.792·11-s + 11.1·12-s + 21.3·16-s − 3.20·21-s + 9.30·25-s − 12.7·27-s − 12.8·28-s + 1.32·33-s − 38.7·36-s + 0.352·37-s − 3.57·41-s + 5.31·44-s − 2.12·47-s + 35.6·48-s − 6.16·49-s + 2.76·53-s + 11.1·63-s + 42.3·64-s + 1.44·67-s − 5.05·71-s + 3.33·73-s + 15.5·75-s − 1.52·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{20}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(37^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.81066\times 10^{21}\) |
| Root analytic conductor: | \(3.39974\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 37^{20} ,\ ( \ : [7/2]^{20} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.2870662514\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2870662514\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 37 | \( 1 - 108732 T + 10292619926 p T^{2} - 37655339698284 p^{2} T^{3} + 1442046101398897664 p^{3} T^{4} - \)\(17\!\cdots\!28\)\( p^{5} T^{5} + \)\(95\!\cdots\!16\)\( p^{7} T^{6} - \)\(15\!\cdots\!12\)\( p^{9} T^{7} + \)\(12\!\cdots\!44\)\( p^{12} T^{8} - \)\(75\!\cdots\!60\)\( p^{15} T^{9} + \)\(45\!\cdots\!24\)\( p^{18} T^{10} - \)\(75\!\cdots\!60\)\( p^{22} T^{11} + \)\(12\!\cdots\!44\)\( p^{26} T^{12} - \)\(15\!\cdots\!12\)\( p^{30} T^{13} + \)\(95\!\cdots\!16\)\( p^{35} T^{14} - \)\(17\!\cdots\!28\)\( p^{40} T^{15} + 1442046101398897664 p^{45} T^{16} - 37655339698284 p^{51} T^{17} + 10292619926 p^{57} T^{18} - 108732 p^{63} T^{19} + p^{70} T^{20} \) |
| good | 2 | \( 1 - 429 p T^{2} + 386061 T^{4} - 29904073 p^{2} T^{6} + 7122970849 p^{2} T^{8} - 346559288787 p^{4} T^{10} + 7183441729441 p^{7} T^{12} - 263209606457759 p^{9} T^{14} + 17711556010908601 p^{10} T^{16} - 570078570895404715 p^{12} T^{18} + 1136776573793268275 p^{18} T^{20} - 570078570895404715 p^{26} T^{22} + 17711556010908601 p^{38} T^{24} - 263209606457759 p^{51} T^{26} + 7183441729441 p^{63} T^{28} - 346559288787 p^{74} T^{30} + 7122970849 p^{86} T^{32} - 29904073 p^{100} T^{34} + 386061 p^{112} T^{36} - 429 p^{127} T^{38} + p^{140} T^{40} \) |
| 3 | \( ( 1 - 13 p T + 8605 T^{2} - 78902 p T^{3} + 39344785 T^{4} - 95796269 p^{2} T^{5} + 5102674045 p^{3} T^{6} - 32981139578 p^{4} T^{7} + 530775893407 p^{6} T^{8} - 8716931975531 p^{6} T^{9} + 1239906364186418 p^{6} T^{10} - 8716931975531 p^{13} T^{11} + 530775893407 p^{20} T^{12} - 32981139578 p^{25} T^{13} + 5102674045 p^{31} T^{14} - 95796269 p^{37} T^{15} + 39344785 p^{42} T^{16} - 78902 p^{50} T^{17} + 8605 p^{56} T^{18} - 13 p^{64} T^{19} + p^{70} T^{20} )^{2} \) | |
| 5 | \( 1 - 727194 T^{2} + 271995316341 T^{4} - 69491678850817048 T^{6} + \)\(13\!\cdots\!34\)\( T^{8} - \)\(87\!\cdots\!94\)\( p^{2} T^{10} + \)\(47\!\cdots\!06\)\( p^{4} T^{12} - \)\(22\!\cdots\!26\)\( p^{6} T^{14} + \)\(73\!\cdots\!77\)\( p^{11} T^{16} - \)\(34\!\cdots\!34\)\( p^{10} T^{18} + \)\(11\!\cdots\!86\)\( p^{12} T^{20} - \)\(34\!\cdots\!34\)\( p^{24} T^{22} + \)\(73\!\cdots\!77\)\( p^{39} T^{24} - \)\(22\!\cdots\!26\)\( p^{48} T^{26} + \)\(47\!\cdots\!06\)\( p^{60} T^{28} - \)\(87\!\cdots\!94\)\( p^{72} T^{30} + \)\(13\!\cdots\!34\)\( p^{84} T^{32} - 69491678850817048 p^{98} T^{34} + 271995316341 p^{112} T^{36} - 727194 p^{126} T^{38} + p^{140} T^{40} \) | |
| 7 | \( ( 1 + 873 T + 75132 p^{2} T^{2} + 3152830157 T^{3} + 7376077440862 T^{4} + 5763779305277667 T^{5} + 10445401189113325826 T^{6} + \)\(73\!\cdots\!89\)\( T^{7} + \)\(11\!\cdots\!97\)\( T^{8} + \)\(73\!\cdots\!54\)\( T^{9} + \)\(10\!\cdots\!76\)\( T^{10} + \)\(73\!\cdots\!54\)\( p^{7} T^{11} + \)\(11\!\cdots\!97\)\( p^{14} T^{12} + \)\(73\!\cdots\!89\)\( p^{21} T^{13} + 10445401189113325826 p^{28} T^{14} + 5763779305277667 p^{35} T^{15} + 7376077440862 p^{42} T^{16} + 3152830157 p^{49} T^{17} + 75132 p^{58} T^{18} + 873 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 11 | \( ( 1 - 159 p T + 103161017 T^{2} - 196085676978 T^{3} + 527402296843443 p T^{4} - 11200610048769313083 T^{5} + \)\(22\!\cdots\!39\)\( T^{6} - \)\(41\!\cdots\!42\)\( T^{7} + \)\(63\!\cdots\!99\)\( T^{8} - \)\(10\!\cdots\!97\)\( T^{9} + \)\(14\!\cdots\!10\)\( T^{10} - \)\(10\!\cdots\!97\)\( p^{7} T^{11} + \)\(63\!\cdots\!99\)\( p^{14} T^{12} - \)\(41\!\cdots\!42\)\( p^{21} T^{13} + \)\(22\!\cdots\!39\)\( p^{28} T^{14} - 11200610048769313083 p^{35} T^{15} + 527402296843443 p^{43} T^{16} - 196085676978 p^{49} T^{17} + 103161017 p^{56} T^{18} - 159 p^{64} T^{19} + p^{70} T^{20} )^{2} \) | |
| 13 | \( 1 - 622831610 T^{2} + 188648247377265685 T^{4} - \)\(37\!\cdots\!60\)\( T^{6} + \)\(54\!\cdots\!10\)\( T^{8} - \)\(64\!\cdots\!10\)\( T^{10} + \)\(64\!\cdots\!90\)\( T^{12} - \)\(56\!\cdots\!10\)\( T^{14} + \)\(44\!\cdots\!65\)\( T^{16} - \)\(32\!\cdots\!10\)\( T^{18} + \)\(21\!\cdots\!98\)\( T^{20} - \)\(32\!\cdots\!10\)\( p^{14} T^{22} + \)\(44\!\cdots\!65\)\( p^{28} T^{24} - \)\(56\!\cdots\!10\)\( p^{42} T^{26} + \)\(64\!\cdots\!90\)\( p^{56} T^{28} - \)\(64\!\cdots\!10\)\( p^{70} T^{30} + \)\(54\!\cdots\!10\)\( p^{84} T^{32} - \)\(37\!\cdots\!60\)\( p^{98} T^{34} + 188648247377265685 p^{112} T^{36} - 622831610 p^{126} T^{38} + p^{140} T^{40} \) | |
| 17 | \( 1 - 4885177968 T^{2} + 11858722448117676198 T^{4} - \)\(19\!\cdots\!36\)\( T^{6} + \)\(22\!\cdots\!49\)\( T^{8} - \)\(21\!\cdots\!20\)\( T^{10} + \)\(99\!\cdots\!44\)\( p T^{12} - \)\(65\!\cdots\!28\)\( p T^{14} + \)\(37\!\cdots\!30\)\( p T^{16} - \)\(31\!\cdots\!08\)\( T^{18} + \)\(13\!\cdots\!68\)\( T^{20} - \)\(31\!\cdots\!08\)\( p^{14} T^{22} + \)\(37\!\cdots\!30\)\( p^{29} T^{24} - \)\(65\!\cdots\!28\)\( p^{43} T^{26} + \)\(99\!\cdots\!44\)\( p^{57} T^{28} - \)\(21\!\cdots\!20\)\( p^{70} T^{30} + \)\(22\!\cdots\!49\)\( p^{84} T^{32} - \)\(19\!\cdots\!36\)\( p^{98} T^{34} + 11858722448117676198 p^{112} T^{36} - 4885177968 p^{126} T^{38} + p^{140} T^{40} \) | |
| 19 | \( 1 - 11617365380 T^{2} + 66710808174213133822 T^{4} - \)\(25\!\cdots\!88\)\( T^{6} + \)\(70\!\cdots\!29\)\( T^{8} - \)\(15\!\cdots\!16\)\( T^{10} + \)\(14\!\cdots\!48\)\( p T^{12} - \)\(41\!\cdots\!68\)\( T^{14} + \)\(52\!\cdots\!98\)\( T^{16} - \)\(58\!\cdots\!92\)\( T^{18} + \)\(55\!\cdots\!44\)\( T^{20} - \)\(58\!\cdots\!92\)\( p^{14} T^{22} + \)\(52\!\cdots\!98\)\( p^{28} T^{24} - \)\(41\!\cdots\!68\)\( p^{42} T^{26} + \)\(14\!\cdots\!48\)\( p^{57} T^{28} - \)\(15\!\cdots\!16\)\( p^{70} T^{30} + \)\(70\!\cdots\!29\)\( p^{84} T^{32} - \)\(25\!\cdots\!88\)\( p^{98} T^{34} + 66710808174213133822 p^{112} T^{36} - 11617365380 p^{126} T^{38} + p^{140} T^{40} \) | |
| 23 | \( 1 - 40049051130 T^{2} + 34679389639583098875 p T^{4} - \)\(10\!\cdots\!48\)\( T^{6} + \)\(10\!\cdots\!38\)\( T^{8} - \)\(80\!\cdots\!94\)\( T^{10} + \)\(51\!\cdots\!98\)\( T^{12} - \)\(28\!\cdots\!50\)\( T^{14} + \)\(13\!\cdots\!77\)\( T^{16} - \)\(54\!\cdots\!34\)\( T^{18} + \)\(19\!\cdots\!62\)\( T^{20} - \)\(54\!\cdots\!34\)\( p^{14} T^{22} + \)\(13\!\cdots\!77\)\( p^{28} T^{24} - \)\(28\!\cdots\!50\)\( p^{42} T^{26} + \)\(51\!\cdots\!98\)\( p^{56} T^{28} - \)\(80\!\cdots\!94\)\( p^{70} T^{30} + \)\(10\!\cdots\!38\)\( p^{84} T^{32} - \)\(10\!\cdots\!48\)\( p^{98} T^{34} + 34679389639583098875 p^{113} T^{36} - 40049051130 p^{126} T^{38} + p^{140} T^{40} \) | |
| 29 | \( 1 - 120142903734 T^{2} + \)\(73\!\cdots\!37\)\( T^{4} - \)\(30\!\cdots\!40\)\( T^{6} + \)\(95\!\cdots\!26\)\( T^{8} - \)\(24\!\cdots\!18\)\( T^{10} + \)\(56\!\cdots\!30\)\( T^{12} - \)\(11\!\cdots\!74\)\( T^{14} + \)\(24\!\cdots\!81\)\( T^{16} - \)\(47\!\cdots\!18\)\( T^{18} + \)\(85\!\cdots\!78\)\( T^{20} - \)\(47\!\cdots\!18\)\( p^{14} T^{22} + \)\(24\!\cdots\!81\)\( p^{28} T^{24} - \)\(11\!\cdots\!74\)\( p^{42} T^{26} + \)\(56\!\cdots\!30\)\( p^{56} T^{28} - \)\(24\!\cdots\!18\)\( p^{70} T^{30} + \)\(95\!\cdots\!26\)\( p^{84} T^{32} - \)\(30\!\cdots\!40\)\( p^{98} T^{34} + \)\(73\!\cdots\!37\)\( p^{112} T^{36} - 120142903734 p^{126} T^{38} + p^{140} T^{40} \) | |
| 31 | \( 1 - 286512913010 T^{2} + \)\(41\!\cdots\!65\)\( T^{4} - \)\(41\!\cdots\!68\)\( T^{6} + \)\(31\!\cdots\!50\)\( T^{8} - \)\(18\!\cdots\!78\)\( T^{10} + \)\(95\!\cdots\!06\)\( T^{12} - \)\(13\!\cdots\!98\)\( p T^{14} + \)\(15\!\cdots\!17\)\( T^{16} - \)\(51\!\cdots\!50\)\( T^{18} + \)\(15\!\cdots\!50\)\( T^{20} - \)\(51\!\cdots\!50\)\( p^{14} T^{22} + \)\(15\!\cdots\!17\)\( p^{28} T^{24} - \)\(13\!\cdots\!98\)\( p^{43} T^{26} + \)\(95\!\cdots\!06\)\( p^{56} T^{28} - \)\(18\!\cdots\!78\)\( p^{70} T^{30} + \)\(31\!\cdots\!50\)\( p^{84} T^{32} - \)\(41\!\cdots\!68\)\( p^{98} T^{34} + \)\(41\!\cdots\!65\)\( p^{112} T^{36} - 286512913010 p^{126} T^{38} + p^{140} T^{40} \) | |
| 41 | \( ( 1 + 788871 T + 1433871498551 T^{2} + 947485497405808386 T^{3} + \)\(99\!\cdots\!21\)\( T^{4} + \)\(13\!\cdots\!55\)\( p T^{5} + \)\(43\!\cdots\!95\)\( T^{6} + \)\(21\!\cdots\!80\)\( T^{7} + \)\(13\!\cdots\!27\)\( T^{8} + \)\(57\!\cdots\!17\)\( T^{9} + \)\(30\!\cdots\!92\)\( T^{10} + \)\(57\!\cdots\!17\)\( p^{7} T^{11} + \)\(13\!\cdots\!27\)\( p^{14} T^{12} + \)\(21\!\cdots\!80\)\( p^{21} T^{13} + \)\(43\!\cdots\!95\)\( p^{28} T^{14} + \)\(13\!\cdots\!55\)\( p^{36} T^{15} + \)\(99\!\cdots\!21\)\( p^{42} T^{16} + 947485497405808386 p^{49} T^{17} + 1433871498551 p^{56} T^{18} + 788871 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 43 | \( 1 - 2657080100492 T^{2} + \)\(35\!\cdots\!46\)\( T^{4} - \)\(32\!\cdots\!72\)\( T^{6} + \)\(22\!\cdots\!65\)\( T^{8} - \)\(12\!\cdots\!40\)\( T^{10} + \)\(60\!\cdots\!60\)\( T^{12} - \)\(24\!\cdots\!72\)\( T^{14} + \)\(88\!\cdots\!30\)\( T^{16} - \)\(28\!\cdots\!92\)\( T^{18} + \)\(81\!\cdots\!96\)\( T^{20} - \)\(28\!\cdots\!92\)\( p^{14} T^{22} + \)\(88\!\cdots\!30\)\( p^{28} T^{24} - \)\(24\!\cdots\!72\)\( p^{42} T^{26} + \)\(60\!\cdots\!60\)\( p^{56} T^{28} - \)\(12\!\cdots\!40\)\( p^{70} T^{30} + \)\(22\!\cdots\!65\)\( p^{84} T^{32} - \)\(32\!\cdots\!72\)\( p^{98} T^{34} + \)\(35\!\cdots\!46\)\( p^{112} T^{36} - 2657080100492 p^{126} T^{38} + p^{140} T^{40} \) | |
| 47 | \( ( 1 + 756393 T + 1347124280 p^{2} T^{2} + 2671654184873958753 T^{3} + \)\(45\!\cdots\!02\)\( T^{4} + \)\(42\!\cdots\!43\)\( T^{5} + \)\(47\!\cdots\!70\)\( T^{6} + \)\(41\!\cdots\!05\)\( T^{7} + \)\(36\!\cdots\!33\)\( T^{8} + \)\(28\!\cdots\!62\)\( T^{9} + \)\(21\!\cdots\!16\)\( T^{10} + \)\(28\!\cdots\!62\)\( p^{7} T^{11} + \)\(36\!\cdots\!33\)\( p^{14} T^{12} + \)\(41\!\cdots\!05\)\( p^{21} T^{13} + \)\(47\!\cdots\!70\)\( p^{28} T^{14} + \)\(42\!\cdots\!43\)\( p^{35} T^{15} + \)\(45\!\cdots\!02\)\( p^{42} T^{16} + 2671654184873958753 p^{49} T^{17} + 1347124280 p^{58} T^{18} + 756393 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 53 | \( ( 1 - 1499679 T + 4700327244002 T^{2} - 5660203623968689749 T^{3} + \)\(12\!\cdots\!58\)\( T^{4} - \)\(13\!\cdots\!85\)\( T^{5} + \)\(23\!\cdots\!32\)\( T^{6} - \)\(26\!\cdots\!21\)\( T^{7} + \)\(37\!\cdots\!65\)\( T^{8} - \)\(38\!\cdots\!78\)\( T^{9} + \)\(47\!\cdots\!76\)\( T^{10} - \)\(38\!\cdots\!78\)\( p^{7} T^{11} + \)\(37\!\cdots\!65\)\( p^{14} T^{12} - \)\(26\!\cdots\!21\)\( p^{21} T^{13} + \)\(23\!\cdots\!32\)\( p^{28} T^{14} - \)\(13\!\cdots\!85\)\( p^{35} T^{15} + \)\(12\!\cdots\!58\)\( p^{42} T^{16} - 5660203623968689749 p^{49} T^{17} + 4700327244002 p^{56} T^{18} - 1499679 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 59 | \( 1 - 25992738716160 T^{2} + \)\(32\!\cdots\!50\)\( T^{4} - \)\(27\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!45\)\( T^{8} - \)\(84\!\cdots\!68\)\( T^{10} + \)\(35\!\cdots\!56\)\( T^{12} - \)\(13\!\cdots\!88\)\( T^{14} + \)\(42\!\cdots\!42\)\( T^{16} - \)\(12\!\cdots\!00\)\( T^{18} + \)\(32\!\cdots\!20\)\( T^{20} - \)\(12\!\cdots\!00\)\( p^{14} T^{22} + \)\(42\!\cdots\!42\)\( p^{28} T^{24} - \)\(13\!\cdots\!88\)\( p^{42} T^{26} + \)\(35\!\cdots\!56\)\( p^{56} T^{28} - \)\(84\!\cdots\!68\)\( p^{70} T^{30} + \)\(16\!\cdots\!45\)\( p^{84} T^{32} - \)\(27\!\cdots\!68\)\( p^{98} T^{34} + \)\(32\!\cdots\!50\)\( p^{112} T^{36} - 25992738716160 p^{126} T^{38} + p^{140} T^{40} \) | |
| 61 | \( 1 - 43203018329942 T^{2} + \)\(92\!\cdots\!37\)\( T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(13\!\cdots\!14\)\( T^{8} - \)\(11\!\cdots\!14\)\( T^{10} + \)\(73\!\cdots\!34\)\( T^{12} - \)\(40\!\cdots\!10\)\( T^{14} + \)\(18\!\cdots\!89\)\( T^{16} - \)\(74\!\cdots\!70\)\( T^{18} + \)\(25\!\cdots\!62\)\( T^{20} - \)\(74\!\cdots\!70\)\( p^{14} T^{22} + \)\(18\!\cdots\!89\)\( p^{28} T^{24} - \)\(40\!\cdots\!10\)\( p^{42} T^{26} + \)\(73\!\cdots\!34\)\( p^{56} T^{28} - \)\(11\!\cdots\!14\)\( p^{70} T^{30} + \)\(13\!\cdots\!14\)\( p^{84} T^{32} - \)\(13\!\cdots\!40\)\( p^{98} T^{34} + \)\(92\!\cdots\!37\)\( p^{112} T^{36} - 43203018329942 p^{126} T^{38} + p^{140} T^{40} \) | |
| 67 | \( ( 1 - 1781112 T + 26601960072807 T^{2} - 46228955987509245736 T^{3} + \)\(38\!\cdots\!74\)\( T^{4} - \)\(73\!\cdots\!40\)\( T^{5} + \)\(39\!\cdots\!26\)\( T^{6} - \)\(80\!\cdots\!12\)\( T^{7} + \)\(31\!\cdots\!77\)\( T^{8} - \)\(65\!\cdots\!36\)\( T^{9} + \)\(20\!\cdots\!38\)\( T^{10} - \)\(65\!\cdots\!36\)\( p^{7} T^{11} + \)\(31\!\cdots\!77\)\( p^{14} T^{12} - \)\(80\!\cdots\!12\)\( p^{21} T^{13} + \)\(39\!\cdots\!26\)\( p^{28} T^{14} - \)\(73\!\cdots\!40\)\( p^{35} T^{15} + \)\(38\!\cdots\!74\)\( p^{42} T^{16} - 46228955987509245736 p^{49} T^{17} + 26601960072807 p^{56} T^{18} - 1781112 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 71 | \( ( 1 + 7629543 T + 64326869669996 T^{2} + \)\(30\!\cdots\!71\)\( T^{3} + \)\(15\!\cdots\!54\)\( T^{4} + \)\(54\!\cdots\!53\)\( T^{5} + \)\(22\!\cdots\!46\)\( T^{6} + \)\(64\!\cdots\!11\)\( T^{7} + \)\(22\!\cdots\!93\)\( T^{8} + \)\(59\!\cdots\!38\)\( T^{9} + \)\(21\!\cdots\!08\)\( T^{10} + \)\(59\!\cdots\!38\)\( p^{7} T^{11} + \)\(22\!\cdots\!93\)\( p^{14} T^{12} + \)\(64\!\cdots\!11\)\( p^{21} T^{13} + \)\(22\!\cdots\!46\)\( p^{28} T^{14} + \)\(54\!\cdots\!53\)\( p^{35} T^{15} + \)\(15\!\cdots\!54\)\( p^{42} T^{16} + \)\(30\!\cdots\!71\)\( p^{49} T^{17} + 64326869669996 p^{56} T^{18} + 7629543 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 73 | \( ( 1 - 5544009 T + 54423815408043 T^{2} - \)\(22\!\cdots\!58\)\( T^{3} + \)\(15\!\cdots\!13\)\( T^{4} - \)\(54\!\cdots\!61\)\( T^{5} + \)\(29\!\cdots\!71\)\( T^{6} - \)\(97\!\cdots\!48\)\( T^{7} + \)\(46\!\cdots\!99\)\( T^{8} - \)\(13\!\cdots\!71\)\( T^{9} + \)\(56\!\cdots\!80\)\( T^{10} - \)\(13\!\cdots\!71\)\( p^{7} T^{11} + \)\(46\!\cdots\!99\)\( p^{14} T^{12} - \)\(97\!\cdots\!48\)\( p^{21} T^{13} + \)\(29\!\cdots\!71\)\( p^{28} T^{14} - \)\(54\!\cdots\!61\)\( p^{35} T^{15} + \)\(15\!\cdots\!13\)\( p^{42} T^{16} - \)\(22\!\cdots\!58\)\( p^{49} T^{17} + 54423815408043 p^{56} T^{18} - 5544009 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 79 | \( 1 - 213443963915246 T^{2} + \)\(27\!\cdots\!83\)\( p T^{4} - \)\(14\!\cdots\!04\)\( T^{6} + \)\(65\!\cdots\!98\)\( T^{8} - \)\(21\!\cdots\!98\)\( T^{10} + \)\(50\!\cdots\!66\)\( T^{12} - \)\(73\!\cdots\!14\)\( T^{14} + \)\(57\!\cdots\!37\)\( T^{16} + \)\(28\!\cdots\!98\)\( T^{18} - \)\(78\!\cdots\!50\)\( T^{20} + \)\(28\!\cdots\!98\)\( p^{14} T^{22} + \)\(57\!\cdots\!37\)\( p^{28} T^{24} - \)\(73\!\cdots\!14\)\( p^{42} T^{26} + \)\(50\!\cdots\!66\)\( p^{56} T^{28} - \)\(21\!\cdots\!98\)\( p^{70} T^{30} + \)\(65\!\cdots\!98\)\( p^{84} T^{32} - \)\(14\!\cdots\!04\)\( p^{98} T^{34} + \)\(27\!\cdots\!83\)\( p^{113} T^{36} - 213443963915246 p^{126} T^{38} + p^{140} T^{40} \) | |
| 83 | \( ( 1 + 6436911 T + 125315437807208 T^{2} + \)\(74\!\cdots\!67\)\( T^{3} + \)\(78\!\cdots\!10\)\( T^{4} + \)\(46\!\cdots\!21\)\( T^{5} + \)\(34\!\cdots\!98\)\( T^{6} + \)\(20\!\cdots\!27\)\( T^{7} + \)\(11\!\cdots\!13\)\( T^{8} + \)\(70\!\cdots\!14\)\( T^{9} + \)\(35\!\cdots\!52\)\( T^{10} + \)\(70\!\cdots\!14\)\( p^{7} T^{11} + \)\(11\!\cdots\!13\)\( p^{14} T^{12} + \)\(20\!\cdots\!27\)\( p^{21} T^{13} + \)\(34\!\cdots\!98\)\( p^{28} T^{14} + \)\(46\!\cdots\!21\)\( p^{35} T^{15} + \)\(78\!\cdots\!10\)\( p^{42} T^{16} + \)\(74\!\cdots\!67\)\( p^{49} T^{17} + 125315437807208 p^{56} T^{18} + 6436911 p^{63} T^{19} + p^{70} T^{20} )^{2} \) | |
| 89 | \( 1 - 623542198254264 T^{2} + \)\(18\!\cdots\!86\)\( T^{4} - \)\(37\!\cdots\!00\)\( T^{6} + \)\(53\!\cdots\!17\)\( T^{8} - \)\(60\!\cdots\!00\)\( T^{10} + \)\(54\!\cdots\!64\)\( T^{12} - \)\(41\!\cdots\!52\)\( T^{14} + \)\(26\!\cdots\!50\)\( T^{16} - \)\(14\!\cdots\!68\)\( T^{18} + \)\(69\!\cdots\!12\)\( T^{20} - \)\(14\!\cdots\!68\)\( p^{14} T^{22} + \)\(26\!\cdots\!50\)\( p^{28} T^{24} - \)\(41\!\cdots\!52\)\( p^{42} T^{26} + \)\(54\!\cdots\!64\)\( p^{56} T^{28} - \)\(60\!\cdots\!00\)\( p^{70} T^{30} + \)\(53\!\cdots\!17\)\( p^{84} T^{32} - \)\(37\!\cdots\!00\)\( p^{98} T^{34} + \)\(18\!\cdots\!86\)\( p^{112} T^{36} - 623542198254264 p^{126} T^{38} + p^{140} T^{40} \) | |
| 97 | \( 1 - 827863769782700 T^{2} + \)\(34\!\cdots\!34\)\( T^{4} - \)\(94\!\cdots\!20\)\( T^{6} + \)\(19\!\cdots\!13\)\( T^{8} - \)\(32\!\cdots\!40\)\( T^{10} + \)\(44\!\cdots\!04\)\( T^{12} - \)\(53\!\cdots\!60\)\( T^{14} + \)\(57\!\cdots\!22\)\( T^{16} - \)\(54\!\cdots\!00\)\( T^{18} + \)\(46\!\cdots\!72\)\( T^{20} - \)\(54\!\cdots\!00\)\( p^{14} T^{22} + \)\(57\!\cdots\!22\)\( p^{28} T^{24} - \)\(53\!\cdots\!60\)\( p^{42} T^{26} + \)\(44\!\cdots\!04\)\( p^{56} T^{28} - \)\(32\!\cdots\!40\)\( p^{70} T^{30} + \)\(19\!\cdots\!13\)\( p^{84} T^{32} - \)\(94\!\cdots\!20\)\( p^{98} T^{34} + \)\(34\!\cdots\!34\)\( p^{112} T^{36} - 827863769782700 p^{126} T^{38} + p^{140} T^{40} \) | |
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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
−2.81898674213608796271804244912, −2.80041081897152907062868583088, −2.78751676481750629709516192414, −2.72418582649725531332081018477, −2.59505982484574304184004601267, −2.58458486910005267560684361807, −2.36864845340846375173578919762, −2.29945900302183669457931221076, −2.22357189971039424985398751398, −2.04266041336555557581132146346, −1.87539648857833985703224770927, −1.75340019821923220130719309388, −1.56059226140633776907760191077, −1.53944211701094299517509770700, −1.53626374310236814417939768956, −1.31431081985530182717620872660, −1.25812920343296755785578791512, −1.19851139802019810175509621863, −0.925502751766746630100518064361, −0.67235719119190520254172320125, −0.61728557581428384507123112451, −0.37293719705110555720332956572, −0.32363272696509855186877692738, −0.04890060272000120581103048580, −0.04858083825840207998894761084, 0.04858083825840207998894761084, 0.04890060272000120581103048580, 0.32363272696509855186877692738, 0.37293719705110555720332956572, 0.61728557581428384507123112451, 0.67235719119190520254172320125, 0.925502751766746630100518064361, 1.19851139802019810175509621863, 1.25812920343296755785578791512, 1.31431081985530182717620872660, 1.53626374310236814417939768956, 1.53944211701094299517509770700, 1.56059226140633776907760191077, 1.75340019821923220130719309388, 1.87539648857833985703224770927, 2.04266041336555557581132146346, 2.22357189971039424985398751398, 2.29945900302183669457931221076, 2.36864845340846375173578919762, 2.58458486910005267560684361807, 2.59505982484574304184004601267, 2.72418582649725531332081018477, 2.78751676481750629709516192414, 2.80041081897152907062868583088, 2.81898674213608796271804244912
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.