Dirichlet series
| L(s) = 1 | − 1.42e15·3-s + 1.05e20·4-s + 6.82e26·7-s − 1.35e29·9-s − 1.50e35·12-s − 7.51e35·13-s + 4.94e39·16-s + 1.09e41·19-s − 9.69e41·21-s + 4.71e45·25-s + 2.85e45·27-s + 7.21e46·28-s + 4.58e47·31-s − 1.43e49·36-s − 1.67e50·37-s + 1.06e51·39-s + 1.03e52·43-s − 7.02e54·48-s − 1.09e55·49-s − 7.94e55·52-s − 1.55e56·57-s + 4.55e57·61-s − 9.23e55·63-s + 1.35e59·64-s + 1.08e59·67-s − 5.69e59·73-s − 6.70e60·75-s + ⋯ |
| L(s) = 1 | − 0.766·3-s + 5.73·4-s + 0.617·7-s − 0.0394·9-s − 4.39·12-s − 1.69·13-s + 14.5·16-s + 1.31·19-s − 0.473·21-s + 8.70·25-s + 0.449·27-s + 3.53·28-s + 0.865·31-s − 0.226·36-s − 1.09·37-s + 1.30·39-s + 0.554·43-s − 11.1·48-s − 8.97·49-s − 9.72·52-s − 1.00·57-s + 3.37·61-s − 0.0243·63-s + 21.5·64-s + 4.00·67-s − 1.34·73-s − 6.67·75-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(65-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{20}\right)^{s/2} \, \Gamma_{\C}(s+32)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(40\) |
| Conductor: | \(3^{20}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.63645\times 10^{37}\) |
| Root analytic conductor: | \(8.82162\) |
| Motivic weight: | \(64\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((40,\ 3^{20} ,\ ( \ : [32]^{20} ),\ 1 )\) |
Particular Values
| \(L(\frac{65}{2})\) | \(\approx\) | \(0.01948931901\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01948931901\) |
| \(L(33)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 + 52620665539196 p^{3} T + \)\(36\!\cdots\!52\)\( p^{10} T^{2} + \)\(11\!\cdots\!92\)\( p^{20} T^{3} - \)\(52\!\cdots\!60\)\( p^{32} T^{4} - \)\(18\!\cdots\!40\)\( p^{49} T^{5} - \)\(87\!\cdots\!12\)\( p^{67} T^{6} - \)\(46\!\cdots\!96\)\( p^{86} T^{7} + \)\(19\!\cdots\!72\)\( p^{110} T^{8} + \)\(77\!\cdots\!28\)\( p^{136} T^{9} + \)\(75\!\cdots\!52\)\( p^{166} T^{10} + \)\(77\!\cdots\!28\)\( p^{200} T^{11} + \)\(19\!\cdots\!72\)\( p^{238} T^{12} - \)\(46\!\cdots\!96\)\( p^{278} T^{13} - \)\(87\!\cdots\!12\)\( p^{323} T^{14} - \)\(18\!\cdots\!40\)\( p^{369} T^{15} - \)\(52\!\cdots\!60\)\( p^{416} T^{16} + \)\(11\!\cdots\!92\)\( p^{468} T^{17} + \)\(36\!\cdots\!52\)\( p^{522} T^{18} + 52620665539196 p^{579} T^{19} + p^{640} T^{20} \) |
| good | 2 | \( 1 - 13214568786825290125 p^{3} T^{2} + \)\(60\!\cdots\!85\)\( p^{10} T^{4} - \)\(50\!\cdots\!75\)\( p^{29} T^{6} + \)\(27\!\cdots\!35\)\( p^{45} T^{8} - \)\(85\!\cdots\!25\)\( p^{55} T^{10} + \)\(17\!\cdots\!45\)\( p^{82} T^{12} - \)\(52\!\cdots\!25\)\( p^{115} T^{14} + \)\(55\!\cdots\!65\)\( p^{146} T^{16} - \)\(21\!\cdots\!25\)\( p^{175} T^{18} + \)\(12\!\cdots\!77\)\( p^{210} T^{20} - \)\(21\!\cdots\!25\)\( p^{303} T^{22} + \)\(55\!\cdots\!65\)\( p^{402} T^{24} - \)\(52\!\cdots\!25\)\( p^{499} T^{26} + \)\(17\!\cdots\!45\)\( p^{594} T^{28} - \)\(85\!\cdots\!25\)\( p^{695} T^{30} + \)\(27\!\cdots\!35\)\( p^{813} T^{32} - \)\(50\!\cdots\!75\)\( p^{925} T^{34} + \)\(60\!\cdots\!85\)\( p^{1034} T^{36} - 13214568786825290125 p^{1155} T^{38} + p^{1280} T^{40} \) |
| 5 | \( 1 - \)\(37\!\cdots\!32\)\( p^{3} T^{2} + \)\(70\!\cdots\!38\)\( p^{6} T^{4} - \)\(86\!\cdots\!04\)\( p^{9} T^{6} + \)\(25\!\cdots\!77\)\( p^{17} T^{8} - \)\(22\!\cdots\!88\)\( p^{27} T^{10} + \)\(34\!\cdots\!24\)\( p^{38} T^{12} - \)\(34\!\cdots\!84\)\( p^{52} T^{14} + \)\(25\!\cdots\!38\)\( p^{69} T^{16} - \)\(35\!\cdots\!72\)\( p^{87} T^{18} + \)\(19\!\cdots\!04\)\( p^{107} T^{20} - \)\(35\!\cdots\!72\)\( p^{215} T^{22} + \)\(25\!\cdots\!38\)\( p^{325} T^{24} - \)\(34\!\cdots\!84\)\( p^{436} T^{26} + \)\(34\!\cdots\!24\)\( p^{550} T^{28} - \)\(22\!\cdots\!88\)\( p^{667} T^{30} + \)\(25\!\cdots\!77\)\( p^{785} T^{32} - \)\(86\!\cdots\!04\)\( p^{905} T^{34} + \)\(70\!\cdots\!38\)\( p^{1030} T^{36} - \)\(37\!\cdots\!32\)\( p^{1155} T^{38} + p^{1280} T^{40} \) | |
| 7 | \( ( 1 - \)\(48\!\cdots\!84\)\( p T + \)\(16\!\cdots\!54\)\( p^{3} T^{2} - \)\(86\!\cdots\!48\)\( p^{4} T^{3} + \)\(22\!\cdots\!39\)\( p^{7} T^{4} - \)\(48\!\cdots\!28\)\( p^{12} T^{5} + \)\(26\!\cdots\!28\)\( p^{18} T^{6} - \)\(52\!\cdots\!56\)\( p^{23} T^{7} + \)\(23\!\cdots\!46\)\( p^{29} T^{8} - \)\(86\!\cdots\!28\)\( p^{36} T^{9} + \)\(67\!\cdots\!12\)\( p^{44} T^{10} - \)\(86\!\cdots\!28\)\( p^{100} T^{11} + \)\(23\!\cdots\!46\)\( p^{157} T^{12} - \)\(52\!\cdots\!56\)\( p^{215} T^{13} + \)\(26\!\cdots\!28\)\( p^{274} T^{14} - \)\(48\!\cdots\!28\)\( p^{332} T^{15} + \)\(22\!\cdots\!39\)\( p^{391} T^{16} - \)\(86\!\cdots\!48\)\( p^{452} T^{17} + \)\(16\!\cdots\!54\)\( p^{515} T^{18} - \)\(48\!\cdots\!84\)\( p^{577} T^{19} + p^{640} T^{20} )^{2} \) | |
| 11 | \( 1 - \)\(37\!\cdots\!20\)\( p^{2} T^{2} + \)\(72\!\cdots\!90\)\( p^{4} T^{4} - \)\(86\!\cdots\!40\)\( p^{7} T^{6} + \)\(78\!\cdots\!45\)\( p^{10} T^{8} - \)\(57\!\cdots\!84\)\( p^{13} T^{10} + \)\(32\!\cdots\!60\)\( p^{17} T^{12} - \)\(15\!\cdots\!20\)\( p^{21} T^{14} + \)\(44\!\cdots\!70\)\( p^{29} T^{16} - \)\(10\!\cdots\!60\)\( p^{38} T^{18} + \)\(18\!\cdots\!76\)\( p^{48} T^{20} - \)\(10\!\cdots\!60\)\( p^{166} T^{22} + \)\(44\!\cdots\!70\)\( p^{285} T^{24} - \)\(15\!\cdots\!20\)\( p^{405} T^{26} + \)\(32\!\cdots\!60\)\( p^{529} T^{28} - \)\(57\!\cdots\!84\)\( p^{653} T^{30} + \)\(78\!\cdots\!45\)\( p^{778} T^{32} - \)\(86\!\cdots\!40\)\( p^{903} T^{34} + \)\(72\!\cdots\!90\)\( p^{1028} T^{36} - \)\(37\!\cdots\!20\)\( p^{1154} T^{38} + p^{1280} T^{40} \) | |
| 13 | \( ( 1 + \)\(28\!\cdots\!24\)\( p T + \)\(66\!\cdots\!38\)\( p^{2} T^{2} + \)\(63\!\cdots\!92\)\( p^{4} T^{3} + \)\(14\!\cdots\!29\)\( p^{5} T^{4} - \)\(54\!\cdots\!04\)\( p^{7} T^{5} + \)\(21\!\cdots\!32\)\( p^{8} T^{6} - \)\(13\!\cdots\!92\)\( p^{10} T^{7} + \)\(20\!\cdots\!62\)\( p^{12} T^{8} - \)\(99\!\cdots\!28\)\( p^{16} T^{9} + \)\(53\!\cdots\!72\)\( p^{20} T^{10} - \)\(99\!\cdots\!28\)\( p^{80} T^{11} + \)\(20\!\cdots\!62\)\( p^{140} T^{12} - \)\(13\!\cdots\!92\)\( p^{202} T^{13} + \)\(21\!\cdots\!32\)\( p^{264} T^{14} - \)\(54\!\cdots\!04\)\( p^{327} T^{15} + \)\(14\!\cdots\!29\)\( p^{389} T^{16} + \)\(63\!\cdots\!92\)\( p^{452} T^{17} + \)\(66\!\cdots\!38\)\( p^{514} T^{18} + \)\(28\!\cdots\!24\)\( p^{577} T^{19} + p^{640} T^{20} )^{2} \) | |
| 17 | \( 1 - \)\(52\!\cdots\!00\)\( T^{2} + \)\(13\!\cdots\!90\)\( T^{4} - \)\(76\!\cdots\!00\)\( p^{2} T^{6} + \)\(32\!\cdots\!45\)\( p^{4} T^{8} - \)\(10\!\cdots\!00\)\( p^{6} T^{10} + \)\(99\!\cdots\!20\)\( p^{10} T^{12} - \)\(81\!\cdots\!00\)\( p^{14} T^{14} + \)\(35\!\cdots\!70\)\( p^{19} T^{16} - \)\(85\!\cdots\!00\)\( p^{25} T^{18} + \)\(20\!\cdots\!56\)\( p^{31} T^{20} - \)\(85\!\cdots\!00\)\( p^{153} T^{22} + \)\(35\!\cdots\!70\)\( p^{275} T^{24} - \)\(81\!\cdots\!00\)\( p^{398} T^{26} + \)\(99\!\cdots\!20\)\( p^{522} T^{28} - \)\(10\!\cdots\!00\)\( p^{646} T^{30} + \)\(32\!\cdots\!45\)\( p^{772} T^{32} - \)\(76\!\cdots\!00\)\( p^{898} T^{34} + \)\(13\!\cdots\!90\)\( p^{1024} T^{36} - \)\(52\!\cdots\!00\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 19 | \( ( 1 - \)\(54\!\cdots\!00\)\( T + \)\(24\!\cdots\!10\)\( p T^{2} - \)\(72\!\cdots\!00\)\( p^{2} T^{3} + \)\(15\!\cdots\!55\)\( p^{3} T^{4} - \)\(23\!\cdots\!00\)\( p^{5} T^{5} + \)\(18\!\cdots\!20\)\( p^{7} T^{6} - \)\(25\!\cdots\!00\)\( p^{9} T^{7} + \)\(78\!\cdots\!10\)\( p^{12} T^{8} - \)\(51\!\cdots\!00\)\( p^{15} T^{9} + \)\(13\!\cdots\!28\)\( p^{18} T^{10} - \)\(51\!\cdots\!00\)\( p^{79} T^{11} + \)\(78\!\cdots\!10\)\( p^{140} T^{12} - \)\(25\!\cdots\!00\)\( p^{201} T^{13} + \)\(18\!\cdots\!20\)\( p^{263} T^{14} - \)\(23\!\cdots\!00\)\( p^{325} T^{15} + \)\(15\!\cdots\!55\)\( p^{387} T^{16} - \)\(72\!\cdots\!00\)\( p^{450} T^{17} + \)\(24\!\cdots\!10\)\( p^{513} T^{18} - \)\(54\!\cdots\!00\)\( p^{576} T^{19} + p^{640} T^{20} )^{2} \) | |
| 23 | \( 1 - \)\(12\!\cdots\!00\)\( T^{2} + \)\(79\!\cdots\!90\)\( T^{4} - \)\(33\!\cdots\!00\)\( T^{6} + \)\(20\!\cdots\!05\)\( p^{2} T^{8} - \)\(10\!\cdots\!00\)\( p^{4} T^{10} + \)\(42\!\cdots\!20\)\( p^{6} T^{12} - \)\(68\!\cdots\!00\)\( p^{9} T^{14} + \)\(43\!\cdots\!70\)\( p^{13} T^{16} - \)\(25\!\cdots\!00\)\( p^{17} T^{18} + \)\(13\!\cdots\!76\)\( p^{21} T^{20} - \)\(25\!\cdots\!00\)\( p^{145} T^{22} + \)\(43\!\cdots\!70\)\( p^{269} T^{24} - \)\(68\!\cdots\!00\)\( p^{393} T^{26} + \)\(42\!\cdots\!20\)\( p^{518} T^{28} - \)\(10\!\cdots\!00\)\( p^{644} T^{30} + \)\(20\!\cdots\!05\)\( p^{770} T^{32} - \)\(33\!\cdots\!00\)\( p^{896} T^{34} + \)\(79\!\cdots\!90\)\( p^{1024} T^{36} - \)\(12\!\cdots\!00\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 29 | \( 1 - \)\(39\!\cdots\!20\)\( T^{2} + \)\(93\!\cdots\!90\)\( p^{2} T^{4} - \)\(51\!\cdots\!60\)\( p^{5} T^{6} + \)\(18\!\cdots\!45\)\( p^{6} T^{8} - \)\(18\!\cdots\!64\)\( p^{8} T^{10} + \)\(52\!\cdots\!40\)\( p^{11} T^{12} - \)\(44\!\cdots\!80\)\( p^{15} T^{14} + \)\(33\!\cdots\!30\)\( p^{19} T^{16} - \)\(22\!\cdots\!40\)\( p^{23} T^{18} + \)\(12\!\cdots\!84\)\( p^{27} T^{20} - \)\(22\!\cdots\!40\)\( p^{151} T^{22} + \)\(33\!\cdots\!30\)\( p^{275} T^{24} - \)\(44\!\cdots\!80\)\( p^{399} T^{26} + \)\(52\!\cdots\!40\)\( p^{523} T^{28} - \)\(18\!\cdots\!64\)\( p^{648} T^{30} + \)\(18\!\cdots\!45\)\( p^{774} T^{32} - \)\(51\!\cdots\!60\)\( p^{901} T^{34} + \)\(93\!\cdots\!90\)\( p^{1026} T^{36} - \)\(39\!\cdots\!20\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 31 | \( ( 1 - \)\(73\!\cdots\!20\)\( p T + \)\(13\!\cdots\!90\)\( p^{2} T^{2} - \)\(69\!\cdots\!40\)\( p^{3} T^{3} + \)\(94\!\cdots\!45\)\( p^{4} T^{4} - \)\(10\!\cdots\!04\)\( p^{5} T^{5} + \)\(12\!\cdots\!60\)\( p^{7} T^{6} + \)\(16\!\cdots\!80\)\( p^{9} T^{7} + \)\(39\!\cdots\!70\)\( p^{11} T^{8} + \)\(11\!\cdots\!40\)\( p^{13} T^{9} + \)\(11\!\cdots\!56\)\( p^{15} T^{10} + \)\(11\!\cdots\!40\)\( p^{77} T^{11} + \)\(39\!\cdots\!70\)\( p^{139} T^{12} + \)\(16\!\cdots\!80\)\( p^{201} T^{13} + \)\(12\!\cdots\!60\)\( p^{263} T^{14} - \)\(10\!\cdots\!04\)\( p^{325} T^{15} + \)\(94\!\cdots\!45\)\( p^{388} T^{16} - \)\(69\!\cdots\!40\)\( p^{451} T^{17} + \)\(13\!\cdots\!90\)\( p^{514} T^{18} - \)\(73\!\cdots\!20\)\( p^{577} T^{19} + p^{640} T^{20} )^{2} \) | |
| 37 | \( ( 1 + \)\(83\!\cdots\!52\)\( T + \)\(11\!\cdots\!02\)\( T^{2} + \)\(46\!\cdots\!52\)\( T^{3} + \)\(64\!\cdots\!97\)\( T^{4} - \)\(69\!\cdots\!44\)\( p T^{5} + \)\(16\!\cdots\!08\)\( p^{2} T^{6} - \)\(19\!\cdots\!56\)\( p^{3} T^{7} + \)\(32\!\cdots\!02\)\( p^{4} T^{8} - \)\(65\!\cdots\!44\)\( p^{5} T^{9} + \)\(15\!\cdots\!44\)\( p^{7} T^{10} - \)\(65\!\cdots\!44\)\( p^{69} T^{11} + \)\(32\!\cdots\!02\)\( p^{132} T^{12} - \)\(19\!\cdots\!56\)\( p^{195} T^{13} + \)\(16\!\cdots\!08\)\( p^{258} T^{14} - \)\(69\!\cdots\!44\)\( p^{321} T^{15} + \)\(64\!\cdots\!97\)\( p^{384} T^{16} + \)\(46\!\cdots\!52\)\( p^{448} T^{17} + \)\(11\!\cdots\!02\)\( p^{512} T^{18} + \)\(83\!\cdots\!52\)\( p^{576} T^{19} + p^{640} T^{20} )^{2} \) | |
| 41 | \( 1 - \)\(91\!\cdots\!20\)\( T^{2} + \)\(51\!\cdots\!90\)\( T^{4} - \)\(21\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!45\)\( p T^{8} - \)\(31\!\cdots\!24\)\( p^{3} T^{10} + \)\(48\!\cdots\!60\)\( p^{5} T^{12} - \)\(66\!\cdots\!20\)\( p^{7} T^{14} + \)\(82\!\cdots\!70\)\( p^{9} T^{16} - \)\(92\!\cdots\!60\)\( p^{11} T^{18} + \)\(95\!\cdots\!36\)\( p^{13} T^{20} - \)\(92\!\cdots\!60\)\( p^{139} T^{22} + \)\(82\!\cdots\!70\)\( p^{265} T^{24} - \)\(66\!\cdots\!20\)\( p^{391} T^{26} + \)\(48\!\cdots\!60\)\( p^{517} T^{28} - \)\(31\!\cdots\!24\)\( p^{643} T^{30} + \)\(18\!\cdots\!45\)\( p^{769} T^{32} - \)\(21\!\cdots\!40\)\( p^{896} T^{34} + \)\(51\!\cdots\!90\)\( p^{1024} T^{36} - \)\(91\!\cdots\!20\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 43 | \( ( 1 - \)\(51\!\cdots\!08\)\( T + \)\(12\!\cdots\!82\)\( T^{2} - \)\(32\!\cdots\!48\)\( T^{3} + \)\(18\!\cdots\!19\)\( p T^{4} + \)\(11\!\cdots\!88\)\( p^{2} T^{5} + \)\(33\!\cdots\!16\)\( p^{3} T^{6} + \)\(12\!\cdots\!92\)\( p^{4} T^{7} + \)\(37\!\cdots\!74\)\( p^{5} T^{8} + \)\(44\!\cdots\!48\)\( p^{6} T^{9} + \)\(43\!\cdots\!16\)\( p^{7} T^{10} + \)\(44\!\cdots\!48\)\( p^{70} T^{11} + \)\(37\!\cdots\!74\)\( p^{133} T^{12} + \)\(12\!\cdots\!92\)\( p^{196} T^{13} + \)\(33\!\cdots\!16\)\( p^{259} T^{14} + \)\(11\!\cdots\!88\)\( p^{322} T^{15} + \)\(18\!\cdots\!19\)\( p^{385} T^{16} - \)\(32\!\cdots\!48\)\( p^{448} T^{17} + \)\(12\!\cdots\!82\)\( p^{512} T^{18} - \)\(51\!\cdots\!08\)\( p^{576} T^{19} + p^{640} T^{20} )^{2} \) | |
| 47 | \( 1 - \)\(85\!\cdots\!00\)\( T^{2} + \)\(37\!\cdots\!90\)\( T^{4} - \)\(23\!\cdots\!00\)\( p T^{6} + \)\(24\!\cdots\!15\)\( p^{3} T^{8} - \)\(19\!\cdots\!00\)\( p^{5} T^{10} + \)\(13\!\cdots\!60\)\( p^{7} T^{12} - \)\(84\!\cdots\!00\)\( p^{9} T^{14} + \)\(47\!\cdots\!70\)\( p^{11} T^{16} - \)\(24\!\cdots\!00\)\( p^{13} T^{18} + \)\(11\!\cdots\!36\)\( p^{15} T^{20} - \)\(24\!\cdots\!00\)\( p^{141} T^{22} + \)\(47\!\cdots\!70\)\( p^{267} T^{24} - \)\(84\!\cdots\!00\)\( p^{393} T^{26} + \)\(13\!\cdots\!60\)\( p^{519} T^{28} - \)\(19\!\cdots\!00\)\( p^{645} T^{30} + \)\(24\!\cdots\!15\)\( p^{771} T^{32} - \)\(23\!\cdots\!00\)\( p^{897} T^{34} + \)\(37\!\cdots\!90\)\( p^{1024} T^{36} - \)\(85\!\cdots\!00\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 53 | \( 1 - \)\(72\!\cdots\!00\)\( T^{2} + \)\(96\!\cdots\!30\)\( p T^{4} - \)\(15\!\cdots\!00\)\( p^{3} T^{6} + \)\(24\!\cdots\!65\)\( p^{5} T^{8} - \)\(29\!\cdots\!00\)\( p^{7} T^{10} + \)\(36\!\cdots\!60\)\( p^{9} T^{12} - \)\(37\!\cdots\!00\)\( p^{11} T^{14} + \)\(38\!\cdots\!70\)\( p^{13} T^{16} - \)\(33\!\cdots\!00\)\( p^{15} T^{18} + \)\(28\!\cdots\!96\)\( p^{17} T^{20} - \)\(33\!\cdots\!00\)\( p^{143} T^{22} + \)\(38\!\cdots\!70\)\( p^{269} T^{24} - \)\(37\!\cdots\!00\)\( p^{395} T^{26} + \)\(36\!\cdots\!60\)\( p^{521} T^{28} - \)\(29\!\cdots\!00\)\( p^{647} T^{30} + \)\(24\!\cdots\!65\)\( p^{773} T^{32} - \)\(15\!\cdots\!00\)\( p^{899} T^{34} + \)\(96\!\cdots\!30\)\( p^{1025} T^{36} - \)\(72\!\cdots\!00\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 59 | \( 1 - \)\(39\!\cdots\!80\)\( p T^{2} + \)\(13\!\cdots\!10\)\( p^{3} T^{4} - \)\(54\!\cdots\!40\)\( p^{6} T^{6} + \)\(56\!\cdots\!55\)\( p^{7} T^{8} - \)\(80\!\cdots\!36\)\( p^{9} T^{10} + \)\(95\!\cdots\!40\)\( p^{11} T^{12} - \)\(96\!\cdots\!80\)\( p^{13} T^{14} + \)\(83\!\cdots\!30\)\( p^{15} T^{16} - \)\(63\!\cdots\!40\)\( p^{17} T^{18} + \)\(41\!\cdots\!04\)\( p^{19} T^{20} - \)\(63\!\cdots\!40\)\( p^{145} T^{22} + \)\(83\!\cdots\!30\)\( p^{271} T^{24} - \)\(96\!\cdots\!80\)\( p^{397} T^{26} + \)\(95\!\cdots\!40\)\( p^{523} T^{28} - \)\(80\!\cdots\!36\)\( p^{649} T^{30} + \)\(56\!\cdots\!55\)\( p^{775} T^{32} - \)\(54\!\cdots\!40\)\( p^{902} T^{34} + \)\(13\!\cdots\!10\)\( p^{1027} T^{36} - \)\(39\!\cdots\!80\)\( p^{1153} T^{38} + p^{1280} T^{40} \) | |
| 61 | \( ( 1 - \)\(37\!\cdots\!20\)\( p T + \)\(23\!\cdots\!90\)\( p^{2} T^{2} - \)\(86\!\cdots\!40\)\( p^{3} T^{3} + \)\(34\!\cdots\!45\)\( p^{4} T^{4} - \)\(10\!\cdots\!04\)\( p^{5} T^{5} + \)\(32\!\cdots\!60\)\( p^{6} T^{6} - \)\(83\!\cdots\!20\)\( p^{7} T^{7} + \)\(22\!\cdots\!70\)\( p^{8} T^{8} - \)\(51\!\cdots\!60\)\( p^{9} T^{9} + \)\(12\!\cdots\!56\)\( p^{10} T^{10} - \)\(51\!\cdots\!60\)\( p^{73} T^{11} + \)\(22\!\cdots\!70\)\( p^{136} T^{12} - \)\(83\!\cdots\!20\)\( p^{199} T^{13} + \)\(32\!\cdots\!60\)\( p^{262} T^{14} - \)\(10\!\cdots\!04\)\( p^{325} T^{15} + \)\(34\!\cdots\!45\)\( p^{388} T^{16} - \)\(86\!\cdots\!40\)\( p^{451} T^{17} + \)\(23\!\cdots\!90\)\( p^{514} T^{18} - \)\(37\!\cdots\!20\)\( p^{577} T^{19} + p^{640} T^{20} )^{2} \) | |
| 67 | \( ( 1 - \)\(54\!\cdots\!08\)\( T + \)\(43\!\cdots\!82\)\( T^{2} - \)\(18\!\cdots\!48\)\( T^{3} + \)\(88\!\cdots\!17\)\( T^{4} - \)\(30\!\cdots\!28\)\( T^{5} + \)\(11\!\cdots\!32\)\( T^{6} - \)\(35\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!22\)\( T^{8} - \)\(31\!\cdots\!88\)\( T^{9} + \)\(95\!\cdots\!92\)\( T^{10} - \)\(31\!\cdots\!88\)\( p^{64} T^{11} + \)\(11\!\cdots\!22\)\( p^{128} T^{12} - \)\(35\!\cdots\!28\)\( p^{192} T^{13} + \)\(11\!\cdots\!32\)\( p^{256} T^{14} - \)\(30\!\cdots\!28\)\( p^{320} T^{15} + \)\(88\!\cdots\!17\)\( p^{384} T^{16} - \)\(18\!\cdots\!48\)\( p^{448} T^{17} + \)\(43\!\cdots\!82\)\( p^{512} T^{18} - \)\(54\!\cdots\!08\)\( p^{576} T^{19} + p^{640} T^{20} )^{2} \) | |
| 71 | \( 1 - \)\(42\!\cdots\!20\)\( T^{2} + \)\(87\!\cdots\!90\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!45\)\( T^{8} - \)\(70\!\cdots\!04\)\( T^{10} + \)\(37\!\cdots\!60\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{14} + \)\(60\!\cdots\!70\)\( T^{16} - \)\(19\!\cdots\!60\)\( T^{18} + \)\(60\!\cdots\!56\)\( T^{20} - \)\(19\!\cdots\!60\)\( p^{128} T^{22} + \)\(60\!\cdots\!70\)\( p^{256} T^{24} - \)\(16\!\cdots\!20\)\( p^{384} T^{26} + \)\(37\!\cdots\!60\)\( p^{512} T^{28} - \)\(70\!\cdots\!04\)\( p^{640} T^{30} + \)\(10\!\cdots\!45\)\( p^{768} T^{32} - \)\(11\!\cdots\!40\)\( p^{896} T^{34} + \)\(87\!\cdots\!90\)\( p^{1024} T^{36} - \)\(42\!\cdots\!20\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 73 | \( ( 1 + \)\(28\!\cdots\!52\)\( T + \)\(13\!\cdots\!02\)\( T^{2} + \)\(34\!\cdots\!92\)\( T^{3} + \)\(82\!\cdots\!77\)\( T^{4} + \)\(20\!\cdots\!32\)\( T^{5} + \)\(32\!\cdots\!92\)\( T^{6} + \)\(73\!\cdots\!92\)\( T^{7} + \)\(91\!\cdots\!02\)\( T^{8} + \)\(18\!\cdots\!32\)\( T^{9} + \)\(18\!\cdots\!52\)\( T^{10} + \)\(18\!\cdots\!32\)\( p^{64} T^{11} + \)\(91\!\cdots\!02\)\( p^{128} T^{12} + \)\(73\!\cdots\!92\)\( p^{192} T^{13} + \)\(32\!\cdots\!92\)\( p^{256} T^{14} + \)\(20\!\cdots\!32\)\( p^{320} T^{15} + \)\(82\!\cdots\!77\)\( p^{384} T^{16} + \)\(34\!\cdots\!92\)\( p^{448} T^{17} + \)\(13\!\cdots\!02\)\( p^{512} T^{18} + \)\(28\!\cdots\!52\)\( p^{576} T^{19} + p^{640} T^{20} )^{2} \) | |
| 79 | \( ( 1 + \)\(11\!\cdots\!00\)\( T + \)\(14\!\cdots\!90\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(68\!\cdots\!45\)\( T^{4} + \)\(30\!\cdots\!00\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!10\)\( T^{8} - \)\(63\!\cdots\!00\)\( T^{9} - \)\(17\!\cdots\!88\)\( p T^{10} - \)\(63\!\cdots\!00\)\( p^{64} T^{11} + \)\(21\!\cdots\!10\)\( p^{128} T^{12} + \)\(16\!\cdots\!00\)\( p^{192} T^{13} + \)\(10\!\cdots\!80\)\( p^{256} T^{14} + \)\(30\!\cdots\!00\)\( p^{320} T^{15} + \)\(68\!\cdots\!45\)\( p^{384} T^{16} + \)\(10\!\cdots\!00\)\( p^{448} T^{17} + \)\(14\!\cdots\!90\)\( p^{512} T^{18} + \)\(11\!\cdots\!00\)\( p^{576} T^{19} + p^{640} T^{20} )^{2} \) | |
| 83 | \( 1 - \)\(56\!\cdots\!00\)\( T^{2} + \)\(17\!\cdots\!90\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{6} + \)\(55\!\cdots\!45\)\( T^{8} - \)\(71\!\cdots\!00\)\( T^{10} + \)\(75\!\cdots\!80\)\( T^{12} - \)\(69\!\cdots\!00\)\( T^{14} + \)\(57\!\cdots\!10\)\( T^{16} - \)\(42\!\cdots\!00\)\( T^{18} + \)\(28\!\cdots\!48\)\( T^{20} - \)\(42\!\cdots\!00\)\( p^{128} T^{22} + \)\(57\!\cdots\!10\)\( p^{256} T^{24} - \)\(69\!\cdots\!00\)\( p^{384} T^{26} + \)\(75\!\cdots\!80\)\( p^{512} T^{28} - \)\(71\!\cdots\!00\)\( p^{640} T^{30} + \)\(55\!\cdots\!45\)\( p^{768} T^{32} - \)\(35\!\cdots\!00\)\( p^{896} T^{34} + \)\(17\!\cdots\!90\)\( p^{1024} T^{36} - \)\(56\!\cdots\!00\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 89 | \( 1 - \)\(30\!\cdots\!20\)\( T^{2} + \)\(52\!\cdots\!90\)\( T^{4} - \)\(64\!\cdots\!40\)\( T^{6} + \)\(61\!\cdots\!45\)\( T^{8} - \)\(47\!\cdots\!04\)\( T^{10} + \)\(30\!\cdots\!60\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{14} + \)\(75\!\cdots\!70\)\( T^{16} - \)\(31\!\cdots\!60\)\( T^{18} + \)\(15\!\cdots\!56\)\( T^{20} - \)\(31\!\cdots\!60\)\( p^{128} T^{22} + \)\(75\!\cdots\!70\)\( p^{256} T^{24} - \)\(16\!\cdots\!20\)\( p^{384} T^{26} + \)\(30\!\cdots\!60\)\( p^{512} T^{28} - \)\(47\!\cdots\!04\)\( p^{640} T^{30} + \)\(61\!\cdots\!45\)\( p^{768} T^{32} - \)\(64\!\cdots\!40\)\( p^{896} T^{34} + \)\(52\!\cdots\!90\)\( p^{1024} T^{36} - \)\(30\!\cdots\!20\)\( p^{1152} T^{38} + p^{1280} T^{40} \) | |
| 97 | \( ( 1 - \)\(40\!\cdots\!28\)\( T + \)\(98\!\cdots\!42\)\( T^{2} - \)\(12\!\cdots\!48\)\( T^{3} + \)\(45\!\cdots\!57\)\( T^{4} - \)\(90\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!32\)\( T^{6} - \)\(31\!\cdots\!88\)\( T^{7} + \)\(28\!\cdots\!02\)\( T^{8} - \)\(66\!\cdots\!28\)\( T^{9} + \)\(47\!\cdots\!32\)\( T^{10} - \)\(66\!\cdots\!28\)\( p^{64} T^{11} + \)\(28\!\cdots\!02\)\( p^{128} T^{12} - \)\(31\!\cdots\!88\)\( p^{192} T^{13} + \)\(13\!\cdots\!32\)\( p^{256} T^{14} - \)\(90\!\cdots\!08\)\( p^{320} T^{15} + \)\(45\!\cdots\!57\)\( p^{384} T^{16} - \)\(12\!\cdots\!48\)\( p^{448} T^{17} + \)\(98\!\cdots\!42\)\( p^{512} T^{18} - \)\(40\!\cdots\!28\)\( p^{576} T^{19} + p^{640} T^{20} )^{2} \) | |
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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
−1.84409746913275247782371284123, −1.60107998100754687691625013724, −1.58350212947982379267371633952, −1.48375844900301844354014213843, −1.43753870185312100517446656189, −1.34380371420616141785707713037, −1.31085406243438378288227983283, −1.27675396795341913321111340515, −1.24417131886228033009548228129, −1.23026515415276923886777477393, −1.15786069470337618667368556060, −1.12359971822047049364671054999, −1.08414060232827837735028601714, −0.820920773988485690420255987320, −0.812723871404030884256086339850, −0.78689208089837537347389039706, −0.73127651634917206178497553654, −0.53175771890721291192394936705, −0.48556477254149072868800816286, −0.46688848508385599043374628026, −0.28885559592777601860663226805, −0.25778578096580290797123514566, −0.23374673988039799507338747074, −0.01611247201443771454824453849, −0.008047717372634555662670013002, 0.008047717372634555662670013002, 0.01611247201443771454824453849, 0.23374673988039799507338747074, 0.25778578096580290797123514566, 0.28885559592777601860663226805, 0.46688848508385599043374628026, 0.48556477254149072868800816286, 0.53175771890721291192394936705, 0.73127651634917206178497553654, 0.78689208089837537347389039706, 0.812723871404030884256086339850, 0.820920773988485690420255987320, 1.08414060232827837735028601714, 1.12359971822047049364671054999, 1.15786069470337618667368556060, 1.23026515415276923886777477393, 1.24417131886228033009548228129, 1.27675396795341913321111340515, 1.31085406243438378288227983283, 1.34380371420616141785707713037, 1.43753870185312100517446656189, 1.48375844900301844354014213843, 1.58350212947982379267371633952, 1.60107998100754687691625013724, 1.84409746913275247782371284123
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.