Dirichlet series
| L(s) = 1 | + 476·3-s − 5.15e3·4-s + 9.76e3·5-s + 8.50e4·7-s + 4.21e4·8-s − 4.40e5·9-s + 3.98e5·11-s − 2.45e6·12-s + 2.27e6·13-s + 4.64e6·15-s + 8.63e6·16-s + 5.62e6·17-s + 2.98e7·19-s − 5.03e7·20-s + 4.04e7·21-s + 5.26e7·23-s + 2.00e7·24-s − 1.96e8·25-s − 1.41e8·27-s − 4.38e8·28-s − 2.87e8·29-s + 6.34e8·31-s − 1.73e7·32-s + 1.89e8·33-s + 8.29e8·35-s + 2.27e9·36-s + 4.88e8·37-s + ⋯ |
| L(s) = 1 | + 1.13·3-s − 2.51·4-s + 1.39·5-s + 1.91·7-s + 0.455·8-s − 2.48·9-s + 0.745·11-s − 2.84·12-s + 1.69·13-s + 1.57·15-s + 2.05·16-s + 0.960·17-s + 2.76·19-s − 3.51·20-s + 2.16·21-s + 1.70·23-s + 0.514·24-s − 4.02·25-s − 1.89·27-s − 4.81·28-s − 2.59·29-s + 3.97·31-s − 0.0912·32-s + 0.842·33-s + 2.67·35-s + 6.26·36-s + 1.15·37-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(29^{14}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(29^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.43610\times 10^{18}\) |
| Root analytic conductor: | \(4.72037\) |
| Motivic weight: | \(11\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 29^{14} ,\ ( \ : [11/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(6)\) | \(\approx\) | \(12.02792973\) |
| \(L(\frac12)\) | \(\approx\) | \(12.02792973\) |
| \(L(\frac{13}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 29 | \( ( 1 + p^{5} T )^{14} \) |
| good | 2 | \( 1 + 5155 T^{2} - 10549 p^{2} T^{3} + 4483643 p^{2} T^{4} - 26108007 p^{4} T^{5} + 1616152585 p^{5} T^{6} - 1320300449 p^{10} T^{7} + 496198237105 p^{8} T^{8} - 3518978558827 p^{10} T^{9} + 139500271687357 p^{11} T^{10} - 134631477534839 p^{16} T^{11} + 2082219209449383 p^{18} T^{12} - 2383874624234763 p^{23} T^{13} + 127131187894936973 p^{23} T^{14} - 2383874624234763 p^{34} T^{15} + 2082219209449383 p^{40} T^{16} - 134631477534839 p^{49} T^{17} + 139500271687357 p^{55} T^{18} - 3518978558827 p^{65} T^{19} + 496198237105 p^{74} T^{20} - 1320300449 p^{87} T^{21} + 1616152585 p^{93} T^{22} - 26108007 p^{103} T^{23} + 4483643 p^{112} T^{24} - 10549 p^{123} T^{25} + 5155 p^{132} T^{26} + p^{154} T^{28} \) |
| 3 | \( 1 - 476 T + 667244 T^{2} - 386000180 T^{3} + 286118673526 T^{4} - 147212822233324 T^{5} + 31666077177189014 p T^{6} - 4649316996759442244 p^{2} T^{7} + \)\(31\!\cdots\!03\)\( p^{4} T^{8} - \)\(14\!\cdots\!56\)\( p^{6} T^{9} + \)\(89\!\cdots\!72\)\( p^{8} T^{10} - \)\(42\!\cdots\!24\)\( p^{10} T^{11} + \)\(23\!\cdots\!95\)\( p^{12} T^{12} - \)\(34\!\cdots\!08\)\( p^{15} T^{13} + \)\(56\!\cdots\!46\)\( p^{16} T^{14} - \)\(34\!\cdots\!08\)\( p^{26} T^{15} + \)\(23\!\cdots\!95\)\( p^{34} T^{16} - \)\(42\!\cdots\!24\)\( p^{43} T^{17} + \)\(89\!\cdots\!72\)\( p^{52} T^{18} - \)\(14\!\cdots\!56\)\( p^{61} T^{19} + \)\(31\!\cdots\!03\)\( p^{70} T^{20} - 4649316996759442244 p^{79} T^{21} + 31666077177189014 p^{89} T^{22} - 147212822233324 p^{99} T^{23} + 286118673526 p^{110} T^{24} - 386000180 p^{121} T^{25} + 667244 p^{132} T^{26} - 476 p^{143} T^{27} + p^{154} T^{28} \) | |
| 5 | \( 1 - 1952 p T + 291940444 T^{2} - 2606254059276 T^{3} + 46109310497802898 T^{4} - \)\(37\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!06\)\( p T^{6} - \)\(15\!\cdots\!12\)\( p^{2} T^{7} + \)\(35\!\cdots\!43\)\( p^{3} T^{8} - \)\(49\!\cdots\!88\)\( p^{4} T^{9} + \)\(99\!\cdots\!36\)\( p^{5} T^{10} - \)\(13\!\cdots\!48\)\( p^{6} T^{11} + \)\(24\!\cdots\!31\)\( p^{7} T^{12} - \)\(29\!\cdots\!08\)\( p^{8} T^{13} + \)\(50\!\cdots\!54\)\( p^{9} T^{14} - \)\(29\!\cdots\!08\)\( p^{19} T^{15} + \)\(24\!\cdots\!31\)\( p^{29} T^{16} - \)\(13\!\cdots\!48\)\( p^{39} T^{17} + \)\(99\!\cdots\!36\)\( p^{49} T^{18} - \)\(49\!\cdots\!88\)\( p^{59} T^{19} + \)\(35\!\cdots\!43\)\( p^{69} T^{20} - \)\(15\!\cdots\!12\)\( p^{79} T^{21} + \)\(10\!\cdots\!06\)\( p^{89} T^{22} - \)\(37\!\cdots\!04\)\( p^{99} T^{23} + 46109310497802898 p^{110} T^{24} - 2606254059276 p^{121} T^{25} + 291940444 p^{132} T^{26} - 1952 p^{144} T^{27} + p^{154} T^{28} \) | |
| 7 | \( 1 - 85024 T + 16903216258 T^{2} - 1307491225037376 T^{3} + \)\(14\!\cdots\!87\)\( T^{4} - \)\(13\!\cdots\!56\)\( p T^{5} + \)\(16\!\cdots\!12\)\( p^{2} T^{6} - \)\(13\!\cdots\!88\)\( p^{3} T^{7} + \)\(13\!\cdots\!77\)\( p^{4} T^{8} - \)\(10\!\cdots\!12\)\( p^{5} T^{9} + \)\(83\!\cdots\!86\)\( p^{6} T^{10} - \)\(58\!\cdots\!80\)\( p^{7} T^{11} + \)\(43\!\cdots\!71\)\( p^{8} T^{12} - \)\(27\!\cdots\!28\)\( p^{9} T^{13} + \)\(19\!\cdots\!44\)\( p^{10} T^{14} - \)\(27\!\cdots\!28\)\( p^{20} T^{15} + \)\(43\!\cdots\!71\)\( p^{30} T^{16} - \)\(58\!\cdots\!80\)\( p^{40} T^{17} + \)\(83\!\cdots\!86\)\( p^{50} T^{18} - \)\(10\!\cdots\!12\)\( p^{60} T^{19} + \)\(13\!\cdots\!77\)\( p^{70} T^{20} - \)\(13\!\cdots\!88\)\( p^{80} T^{21} + \)\(16\!\cdots\!12\)\( p^{90} T^{22} - \)\(13\!\cdots\!56\)\( p^{100} T^{23} + \)\(14\!\cdots\!87\)\( p^{110} T^{24} - 1307491225037376 p^{121} T^{25} + 16903216258 p^{132} T^{26} - 85024 p^{143} T^{27} + p^{154} T^{28} \) | |
| 11 | \( 1 - 398020 T + 2018685128228 T^{2} - 688754514117877796 T^{3} + \)\(18\!\cdots\!54\)\( T^{4} - \)\(59\!\cdots\!60\)\( T^{5} + \)\(90\!\cdots\!10\)\( p^{2} T^{6} - \)\(38\!\cdots\!76\)\( T^{7} + \)\(48\!\cdots\!19\)\( T^{8} - \)\(20\!\cdots\!60\)\( T^{9} + \)\(17\!\cdots\!20\)\( T^{10} - \)\(89\!\cdots\!12\)\( T^{11} + \)\(56\!\cdots\!07\)\( T^{12} - \)\(31\!\cdots\!40\)\( T^{13} + \)\(16\!\cdots\!10\)\( T^{14} - \)\(31\!\cdots\!40\)\( p^{11} T^{15} + \)\(56\!\cdots\!07\)\( p^{22} T^{16} - \)\(89\!\cdots\!12\)\( p^{33} T^{17} + \)\(17\!\cdots\!20\)\( p^{44} T^{18} - \)\(20\!\cdots\!60\)\( p^{55} T^{19} + \)\(48\!\cdots\!19\)\( p^{66} T^{20} - \)\(38\!\cdots\!76\)\( p^{77} T^{21} + \)\(90\!\cdots\!10\)\( p^{90} T^{22} - \)\(59\!\cdots\!60\)\( p^{99} T^{23} + \)\(18\!\cdots\!54\)\( p^{110} T^{24} - 688754514117877796 p^{121} T^{25} + 2018685128228 p^{132} T^{26} - 398020 p^{143} T^{27} + p^{154} T^{28} \) | |
| 13 | \( 1 - 2272440 T + 11798739327812 T^{2} - 20352867323870606548 T^{3} + \)\(62\!\cdots\!34\)\( T^{4} - \)\(86\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!98\)\( T^{6} - \)\(22\!\cdots\!52\)\( T^{7} + \)\(47\!\cdots\!55\)\( T^{8} - \)\(26\!\cdots\!44\)\( p T^{9} + \)\(74\!\cdots\!92\)\( T^{10} - \)\(18\!\cdots\!24\)\( T^{11} + \)\(85\!\cdots\!11\)\( T^{12} + \)\(39\!\cdots\!60\)\( T^{13} + \)\(10\!\cdots\!94\)\( T^{14} + \)\(39\!\cdots\!60\)\( p^{11} T^{15} + \)\(85\!\cdots\!11\)\( p^{22} T^{16} - \)\(18\!\cdots\!24\)\( p^{33} T^{17} + \)\(74\!\cdots\!92\)\( p^{44} T^{18} - \)\(26\!\cdots\!44\)\( p^{56} T^{19} + \)\(47\!\cdots\!55\)\( p^{66} T^{20} - \)\(22\!\cdots\!52\)\( p^{77} T^{21} + \)\(20\!\cdots\!98\)\( p^{88} T^{22} - \)\(86\!\cdots\!00\)\( p^{99} T^{23} + \)\(62\!\cdots\!34\)\( p^{110} T^{24} - 20352867323870606548 p^{121} T^{25} + 11798739327812 p^{132} T^{26} - 2272440 p^{143} T^{27} + p^{154} T^{28} \) | |
| 17 | \( 1 - 5623508 T + 204420928735370 T^{2} - \)\(11\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!51\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!56\)\( T^{6} - \)\(74\!\cdots\!92\)\( T^{7} + \)\(83\!\cdots\!01\)\( T^{8} - \)\(39\!\cdots\!72\)\( T^{9} + \)\(39\!\cdots\!70\)\( T^{10} - \)\(17\!\cdots\!48\)\( T^{11} + \)\(16\!\cdots\!35\)\( T^{12} - \)\(69\!\cdots\!60\)\( T^{13} + \)\(58\!\cdots\!16\)\( T^{14} - \)\(69\!\cdots\!60\)\( p^{11} T^{15} + \)\(16\!\cdots\!35\)\( p^{22} T^{16} - \)\(17\!\cdots\!48\)\( p^{33} T^{17} + \)\(39\!\cdots\!70\)\( p^{44} T^{18} - \)\(39\!\cdots\!72\)\( p^{55} T^{19} + \)\(83\!\cdots\!01\)\( p^{66} T^{20} - \)\(74\!\cdots\!92\)\( p^{77} T^{21} + \)\(15\!\cdots\!56\)\( p^{88} T^{22} - \)\(11\!\cdots\!40\)\( p^{99} T^{23} + \)\(21\!\cdots\!51\)\( p^{110} T^{24} - \)\(11\!\cdots\!04\)\( p^{121} T^{25} + 204420928735370 p^{132} T^{26} - 5623508 p^{143} T^{27} + p^{154} T^{28} \) | |
| 19 | \( 1 - 29803300 T + 1303289775554078 T^{2} - \)\(28\!\cdots\!28\)\( T^{3} + \)\(39\!\cdots\!01\)\( p T^{4} - \)\(69\!\cdots\!96\)\( p T^{5} + \)\(25\!\cdots\!12\)\( T^{6} - \)\(38\!\cdots\!04\)\( T^{7} + \)\(60\!\cdots\!85\)\( T^{8} - \)\(78\!\cdots\!76\)\( T^{9} + \)\(10\!\cdots\!74\)\( T^{10} - \)\(12\!\cdots\!64\)\( T^{11} + \)\(15\!\cdots\!19\)\( T^{12} - \)\(16\!\cdots\!00\)\( T^{13} + \)\(19\!\cdots\!04\)\( T^{14} - \)\(16\!\cdots\!00\)\( p^{11} T^{15} + \)\(15\!\cdots\!19\)\( p^{22} T^{16} - \)\(12\!\cdots\!64\)\( p^{33} T^{17} + \)\(10\!\cdots\!74\)\( p^{44} T^{18} - \)\(78\!\cdots\!76\)\( p^{55} T^{19} + \)\(60\!\cdots\!85\)\( p^{66} T^{20} - \)\(38\!\cdots\!04\)\( p^{77} T^{21} + \)\(25\!\cdots\!12\)\( p^{88} T^{22} - \)\(69\!\cdots\!96\)\( p^{100} T^{23} + \)\(39\!\cdots\!01\)\( p^{111} T^{24} - \)\(28\!\cdots\!28\)\( p^{121} T^{25} + 1303289775554078 p^{132} T^{26} - 29803300 p^{143} T^{27} + p^{154} T^{28} \) | |
| 23 | \( 1 - 52654304 T + 9272170034983530 T^{2} - \)\(38\!\cdots\!72\)\( T^{3} + \)\(39\!\cdots\!39\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{5} + \)\(10\!\cdots\!76\)\( T^{6} - \)\(13\!\cdots\!64\)\( p T^{7} + \)\(20\!\cdots\!33\)\( T^{8} - \)\(49\!\cdots\!60\)\( T^{9} + \)\(29\!\cdots\!50\)\( T^{10} - \)\(64\!\cdots\!64\)\( T^{11} + \)\(36\!\cdots\!83\)\( T^{12} - \)\(70\!\cdots\!80\)\( T^{13} + \)\(37\!\cdots\!16\)\( T^{14} - \)\(70\!\cdots\!80\)\( p^{11} T^{15} + \)\(36\!\cdots\!83\)\( p^{22} T^{16} - \)\(64\!\cdots\!64\)\( p^{33} T^{17} + \)\(29\!\cdots\!50\)\( p^{44} T^{18} - \)\(49\!\cdots\!60\)\( p^{55} T^{19} + \)\(20\!\cdots\!33\)\( p^{66} T^{20} - \)\(13\!\cdots\!64\)\( p^{78} T^{21} + \)\(10\!\cdots\!76\)\( p^{88} T^{22} - \)\(13\!\cdots\!76\)\( p^{99} T^{23} + \)\(39\!\cdots\!39\)\( p^{110} T^{24} - \)\(38\!\cdots\!72\)\( p^{121} T^{25} + 9272170034983530 p^{132} T^{26} - 52654304 p^{143} T^{27} + p^{154} T^{28} \) | |
| 31 | \( 1 - 634041348 T + 267162327157674060 T^{2} - \)\(83\!\cdots\!28\)\( T^{3} + \)\(24\!\cdots\!10\)\( T^{4} - \)\(67\!\cdots\!24\)\( T^{5} + \)\(16\!\cdots\!94\)\( T^{6} - \)\(38\!\cdots\!12\)\( T^{7} + \)\(84\!\cdots\!27\)\( T^{8} - \)\(55\!\cdots\!40\)\( p T^{9} + \)\(34\!\cdots\!16\)\( T^{10} - \)\(63\!\cdots\!08\)\( T^{11} + \)\(11\!\cdots\!27\)\( T^{12} - \)\(19\!\cdots\!04\)\( T^{13} + \)\(31\!\cdots\!78\)\( T^{14} - \)\(19\!\cdots\!04\)\( p^{11} T^{15} + \)\(11\!\cdots\!27\)\( p^{22} T^{16} - \)\(63\!\cdots\!08\)\( p^{33} T^{17} + \)\(34\!\cdots\!16\)\( p^{44} T^{18} - \)\(55\!\cdots\!40\)\( p^{56} T^{19} + \)\(84\!\cdots\!27\)\( p^{66} T^{20} - \)\(38\!\cdots\!12\)\( p^{77} T^{21} + \)\(16\!\cdots\!94\)\( p^{88} T^{22} - \)\(67\!\cdots\!24\)\( p^{99} T^{23} + \)\(24\!\cdots\!10\)\( p^{110} T^{24} - \)\(83\!\cdots\!28\)\( p^{121} T^{25} + 267162327157674060 p^{132} T^{26} - 634041348 p^{143} T^{27} + p^{154} T^{28} \) | |
| 37 | \( 1 - 488665204 T + 1067553527408087098 T^{2} - \)\(21\!\cdots\!20\)\( T^{3} + \)\(48\!\cdots\!23\)\( T^{4} - \)\(14\!\cdots\!92\)\( T^{5} + \)\(17\!\cdots\!68\)\( T^{6} + \)\(56\!\cdots\!64\)\( T^{7} + \)\(57\!\cdots\!61\)\( T^{8} + \)\(36\!\cdots\!84\)\( T^{9} + \)\(14\!\cdots\!90\)\( T^{10} + \)\(14\!\cdots\!48\)\( T^{11} + \)\(32\!\cdots\!87\)\( T^{12} + \)\(32\!\cdots\!88\)\( T^{13} + \)\(64\!\cdots\!48\)\( T^{14} + \)\(32\!\cdots\!88\)\( p^{11} T^{15} + \)\(32\!\cdots\!87\)\( p^{22} T^{16} + \)\(14\!\cdots\!48\)\( p^{33} T^{17} + \)\(14\!\cdots\!90\)\( p^{44} T^{18} + \)\(36\!\cdots\!84\)\( p^{55} T^{19} + \)\(57\!\cdots\!61\)\( p^{66} T^{20} + \)\(56\!\cdots\!64\)\( p^{77} T^{21} + \)\(17\!\cdots\!68\)\( p^{88} T^{22} - \)\(14\!\cdots\!92\)\( p^{99} T^{23} + \)\(48\!\cdots\!23\)\( p^{110} T^{24} - \)\(21\!\cdots\!20\)\( p^{121} T^{25} + 1067553527408087098 p^{132} T^{26} - 488665204 p^{143} T^{27} + p^{154} T^{28} \) | |
| 41 | \( 1 - 198215164 T + 5686097063177381666 T^{2} - \)\(18\!\cdots\!92\)\( T^{3} + \)\(15\!\cdots\!35\)\( T^{4} - \)\(71\!\cdots\!80\)\( T^{5} + \)\(25\!\cdots\!24\)\( T^{6} - \)\(15\!\cdots\!28\)\( T^{7} + \)\(31\!\cdots\!09\)\( T^{8} - \)\(21\!\cdots\!12\)\( T^{9} + \)\(29\!\cdots\!50\)\( T^{10} - \)\(52\!\cdots\!92\)\( p T^{11} + \)\(21\!\cdots\!95\)\( T^{12} - \)\(15\!\cdots\!92\)\( T^{13} + \)\(13\!\cdots\!60\)\( T^{14} - \)\(15\!\cdots\!92\)\( p^{11} T^{15} + \)\(21\!\cdots\!95\)\( p^{22} T^{16} - \)\(52\!\cdots\!92\)\( p^{34} T^{17} + \)\(29\!\cdots\!50\)\( p^{44} T^{18} - \)\(21\!\cdots\!12\)\( p^{55} T^{19} + \)\(31\!\cdots\!09\)\( p^{66} T^{20} - \)\(15\!\cdots\!28\)\( p^{77} T^{21} + \)\(25\!\cdots\!24\)\( p^{88} T^{22} - \)\(71\!\cdots\!80\)\( p^{99} T^{23} + \)\(15\!\cdots\!35\)\( p^{110} T^{24} - \)\(18\!\cdots\!92\)\( p^{121} T^{25} + 5686097063177381666 p^{132} T^{26} - 198215164 p^{143} T^{27} + p^{154} T^{28} \) | |
| 43 | \( 1 - 2193188100 T + 6618233916243790212 T^{2} - \)\(11\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!06\)\( T^{4} - \)\(30\!\cdots\!80\)\( T^{5} + \)\(43\!\cdots\!14\)\( T^{6} - \)\(58\!\cdots\!80\)\( T^{7} + \)\(73\!\cdots\!19\)\( T^{8} - \)\(88\!\cdots\!00\)\( T^{9} + \)\(10\!\cdots\!48\)\( T^{10} - \)\(11\!\cdots\!36\)\( T^{11} + \)\(12\!\cdots\!91\)\( T^{12} - \)\(12\!\cdots\!92\)\( T^{13} + \)\(12\!\cdots\!78\)\( T^{14} - \)\(12\!\cdots\!92\)\( p^{11} T^{15} + \)\(12\!\cdots\!91\)\( p^{22} T^{16} - \)\(11\!\cdots\!36\)\( p^{33} T^{17} + \)\(10\!\cdots\!48\)\( p^{44} T^{18} - \)\(88\!\cdots\!00\)\( p^{55} T^{19} + \)\(73\!\cdots\!19\)\( p^{66} T^{20} - \)\(58\!\cdots\!80\)\( p^{77} T^{21} + \)\(43\!\cdots\!14\)\( p^{88} T^{22} - \)\(30\!\cdots\!80\)\( p^{99} T^{23} + \)\(20\!\cdots\!06\)\( p^{110} T^{24} - \)\(11\!\cdots\!76\)\( p^{121} T^{25} + 6618233916243790212 p^{132} T^{26} - 2193188100 p^{143} T^{27} + p^{154} T^{28} \) | |
| 47 | \( 1 + 4175934476 T + 25252789570542429284 T^{2} + \)\(74\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!62\)\( T^{4} + \)\(69\!\cdots\!24\)\( T^{5} + \)\(20\!\cdots\!18\)\( T^{6} + \)\(44\!\cdots\!64\)\( T^{7} + \)\(11\!\cdots\!35\)\( T^{8} + \)\(21\!\cdots\!16\)\( T^{9} + \)\(46\!\cdots\!04\)\( T^{10} + \)\(17\!\cdots\!36\)\( p T^{11} + \)\(15\!\cdots\!63\)\( T^{12} + \)\(24\!\cdots\!44\)\( T^{13} + \)\(42\!\cdots\!66\)\( T^{14} + \)\(24\!\cdots\!44\)\( p^{11} T^{15} + \)\(15\!\cdots\!63\)\( p^{22} T^{16} + \)\(17\!\cdots\!36\)\( p^{34} T^{17} + \)\(46\!\cdots\!04\)\( p^{44} T^{18} + \)\(21\!\cdots\!16\)\( p^{55} T^{19} + \)\(11\!\cdots\!35\)\( p^{66} T^{20} + \)\(44\!\cdots\!64\)\( p^{77} T^{21} + \)\(20\!\cdots\!18\)\( p^{88} T^{22} + \)\(69\!\cdots\!24\)\( p^{99} T^{23} + \)\(28\!\cdots\!62\)\( p^{110} T^{24} + \)\(74\!\cdots\!20\)\( p^{121} T^{25} + 25252789570542429284 p^{132} T^{26} + 4175934476 p^{143} T^{27} + p^{154} T^{28} \) | |
| 53 | \( 1 + 13223081840 T + \)\(10\!\cdots\!04\)\( T^{2} + \)\(59\!\cdots\!08\)\( T^{3} + \)\(29\!\cdots\!10\)\( T^{4} + \)\(13\!\cdots\!64\)\( T^{5} + \)\(57\!\cdots\!62\)\( T^{6} + \)\(23\!\cdots\!96\)\( T^{7} + \)\(89\!\cdots\!63\)\( T^{8} + \)\(32\!\cdots\!64\)\( T^{9} + \)\(11\!\cdots\!28\)\( T^{10} + \)\(39\!\cdots\!36\)\( T^{11} + \)\(13\!\cdots\!11\)\( T^{12} + \)\(41\!\cdots\!80\)\( T^{13} + \)\(12\!\cdots\!86\)\( T^{14} + \)\(41\!\cdots\!80\)\( p^{11} T^{15} + \)\(13\!\cdots\!11\)\( p^{22} T^{16} + \)\(39\!\cdots\!36\)\( p^{33} T^{17} + \)\(11\!\cdots\!28\)\( p^{44} T^{18} + \)\(32\!\cdots\!64\)\( p^{55} T^{19} + \)\(89\!\cdots\!63\)\( p^{66} T^{20} + \)\(23\!\cdots\!96\)\( p^{77} T^{21} + \)\(57\!\cdots\!62\)\( p^{88} T^{22} + \)\(13\!\cdots\!64\)\( p^{99} T^{23} + \)\(29\!\cdots\!10\)\( p^{110} T^{24} + \)\(59\!\cdots\!08\)\( p^{121} T^{25} + \)\(10\!\cdots\!04\)\( p^{132} T^{26} + 13223081840 p^{143} T^{27} + p^{154} T^{28} \) | |
| 59 | \( 1 - 352219640 T + \)\(21\!\cdots\!02\)\( T^{2} - \)\(16\!\cdots\!88\)\( T^{3} + \)\(23\!\cdots\!31\)\( T^{4} - \)\(25\!\cdots\!04\)\( T^{5} + \)\(17\!\cdots\!40\)\( T^{6} - \)\(22\!\cdots\!24\)\( T^{7} + \)\(10\!\cdots\!57\)\( T^{8} - \)\(13\!\cdots\!88\)\( T^{9} + \)\(48\!\cdots\!22\)\( T^{10} - \)\(64\!\cdots\!44\)\( T^{11} + \)\(18\!\cdots\!43\)\( T^{12} - \)\(24\!\cdots\!36\)\( T^{13} + \)\(61\!\cdots\!48\)\( T^{14} - \)\(24\!\cdots\!36\)\( p^{11} T^{15} + \)\(18\!\cdots\!43\)\( p^{22} T^{16} - \)\(64\!\cdots\!44\)\( p^{33} T^{17} + \)\(48\!\cdots\!22\)\( p^{44} T^{18} - \)\(13\!\cdots\!88\)\( p^{55} T^{19} + \)\(10\!\cdots\!57\)\( p^{66} T^{20} - \)\(22\!\cdots\!24\)\( p^{77} T^{21} + \)\(17\!\cdots\!40\)\( p^{88} T^{22} - \)\(25\!\cdots\!04\)\( p^{99} T^{23} + \)\(23\!\cdots\!31\)\( p^{110} T^{24} - \)\(16\!\cdots\!88\)\( p^{121} T^{25} + \)\(21\!\cdots\!02\)\( p^{132} T^{26} - 352219640 p^{143} T^{27} + p^{154} T^{28} \) | |
| 61 | \( 1 + 7658546476 T + \)\(29\!\cdots\!82\)\( T^{2} + \)\(18\!\cdots\!52\)\( T^{3} + \)\(39\!\cdots\!51\)\( T^{4} + \)\(21\!\cdots\!20\)\( T^{5} + \)\(32\!\cdots\!84\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(19\!\cdots\!69\)\( T^{8} + \)\(91\!\cdots\!28\)\( T^{9} + \)\(94\!\cdots\!90\)\( T^{10} + \)\(49\!\cdots\!72\)\( T^{11} + \)\(44\!\cdots\!95\)\( T^{12} + \)\(25\!\cdots\!68\)\( T^{13} + \)\(20\!\cdots\!04\)\( T^{14} + \)\(25\!\cdots\!68\)\( p^{11} T^{15} + \)\(44\!\cdots\!95\)\( p^{22} T^{16} + \)\(49\!\cdots\!72\)\( p^{33} T^{17} + \)\(94\!\cdots\!90\)\( p^{44} T^{18} + \)\(91\!\cdots\!28\)\( p^{55} T^{19} + \)\(19\!\cdots\!69\)\( p^{66} T^{20} + \)\(15\!\cdots\!20\)\( p^{77} T^{21} + \)\(32\!\cdots\!84\)\( p^{88} T^{22} + \)\(21\!\cdots\!20\)\( p^{99} T^{23} + \)\(39\!\cdots\!51\)\( p^{110} T^{24} + \)\(18\!\cdots\!52\)\( p^{121} T^{25} + \)\(29\!\cdots\!82\)\( p^{132} T^{26} + 7658546476 p^{143} T^{27} + p^{154} T^{28} \) | |
| 67 | \( 1 - 21781534280 T + \)\(10\!\cdots\!10\)\( T^{2} - \)\(18\!\cdots\!40\)\( T^{3} + \)\(52\!\cdots\!23\)\( T^{4} - \)\(70\!\cdots\!20\)\( T^{5} + \)\(15\!\cdots\!28\)\( T^{6} - \)\(16\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!33\)\( T^{8} - \)\(23\!\cdots\!80\)\( T^{9} + \)\(39\!\cdots\!26\)\( T^{10} - \)\(22\!\cdots\!00\)\( T^{11} + \)\(42\!\cdots\!75\)\( T^{12} - \)\(16\!\cdots\!20\)\( T^{13} + \)\(47\!\cdots\!72\)\( T^{14} - \)\(16\!\cdots\!20\)\( p^{11} T^{15} + \)\(42\!\cdots\!75\)\( p^{22} T^{16} - \)\(22\!\cdots\!00\)\( p^{33} T^{17} + \)\(39\!\cdots\!26\)\( p^{44} T^{18} - \)\(23\!\cdots\!80\)\( p^{55} T^{19} + \)\(28\!\cdots\!33\)\( p^{66} T^{20} - \)\(16\!\cdots\!00\)\( p^{77} T^{21} + \)\(15\!\cdots\!28\)\( p^{88} T^{22} - \)\(70\!\cdots\!20\)\( p^{99} T^{23} + \)\(52\!\cdots\!23\)\( p^{110} T^{24} - \)\(18\!\cdots\!40\)\( p^{121} T^{25} + \)\(10\!\cdots\!10\)\( p^{132} T^{26} - 21781534280 p^{143} T^{27} + p^{154} T^{28} \) | |
| 71 | \( 1 + 5573287168 T + \)\(21\!\cdots\!78\)\( T^{2} + \)\(14\!\cdots\!08\)\( T^{3} + \)\(21\!\cdots\!59\)\( T^{4} + \)\(18\!\cdots\!48\)\( T^{5} + \)\(14\!\cdots\!64\)\( T^{6} + \)\(14\!\cdots\!44\)\( T^{7} + \)\(71\!\cdots\!61\)\( T^{8} + \)\(75\!\cdots\!60\)\( T^{9} + \)\(26\!\cdots\!18\)\( T^{10} + \)\(29\!\cdots\!68\)\( T^{11} + \)\(82\!\cdots\!03\)\( T^{12} + \)\(89\!\cdots\!04\)\( T^{13} + \)\(20\!\cdots\!12\)\( T^{14} + \)\(89\!\cdots\!04\)\( p^{11} T^{15} + \)\(82\!\cdots\!03\)\( p^{22} T^{16} + \)\(29\!\cdots\!68\)\( p^{33} T^{17} + \)\(26\!\cdots\!18\)\( p^{44} T^{18} + \)\(75\!\cdots\!60\)\( p^{55} T^{19} + \)\(71\!\cdots\!61\)\( p^{66} T^{20} + \)\(14\!\cdots\!44\)\( p^{77} T^{21} + \)\(14\!\cdots\!64\)\( p^{88} T^{22} + \)\(18\!\cdots\!48\)\( p^{99} T^{23} + \)\(21\!\cdots\!59\)\( p^{110} T^{24} + \)\(14\!\cdots\!08\)\( p^{121} T^{25} + \)\(21\!\cdots\!78\)\( p^{132} T^{26} + 5573287168 p^{143} T^{27} + p^{154} T^{28} \) | |
| 73 | \( 1 - 39661511924 T + \)\(30\!\cdots\!26\)\( T^{2} - \)\(97\!\cdots\!80\)\( T^{3} + \)\(43\!\cdots\!75\)\( T^{4} - \)\(11\!\cdots\!72\)\( T^{5} + \)\(37\!\cdots\!40\)\( T^{6} - \)\(84\!\cdots\!24\)\( T^{7} + \)\(22\!\cdots\!33\)\( T^{8} - \)\(45\!\cdots\!32\)\( T^{9} + \)\(10\!\cdots\!02\)\( T^{10} - \)\(19\!\cdots\!00\)\( T^{11} + \)\(41\!\cdots\!55\)\( T^{12} - \)\(68\!\cdots\!88\)\( T^{13} + \)\(13\!\cdots\!80\)\( T^{14} - \)\(68\!\cdots\!88\)\( p^{11} T^{15} + \)\(41\!\cdots\!55\)\( p^{22} T^{16} - \)\(19\!\cdots\!00\)\( p^{33} T^{17} + \)\(10\!\cdots\!02\)\( p^{44} T^{18} - \)\(45\!\cdots\!32\)\( p^{55} T^{19} + \)\(22\!\cdots\!33\)\( p^{66} T^{20} - \)\(84\!\cdots\!24\)\( p^{77} T^{21} + \)\(37\!\cdots\!40\)\( p^{88} T^{22} - \)\(11\!\cdots\!72\)\( p^{99} T^{23} + \)\(43\!\cdots\!75\)\( p^{110} T^{24} - \)\(97\!\cdots\!80\)\( p^{121} T^{25} + \)\(30\!\cdots\!26\)\( p^{132} T^{26} - 39661511924 p^{143} T^{27} + p^{154} T^{28} \) | |
| 79 | \( 1 - 105565209020 T + \)\(10\!\cdots\!60\)\( T^{2} - \)\(66\!\cdots\!16\)\( T^{3} + \)\(39\!\cdots\!42\)\( T^{4} - \)\(19\!\cdots\!92\)\( T^{5} + \)\(87\!\cdots\!54\)\( T^{6} - \)\(34\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!83\)\( T^{8} - \)\(47\!\cdots\!68\)\( T^{9} + \)\(16\!\cdots\!32\)\( T^{10} - \)\(51\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!95\)\( T^{12} - \)\(46\!\cdots\!04\)\( T^{13} + \)\(13\!\cdots\!26\)\( T^{14} - \)\(46\!\cdots\!04\)\( p^{11} T^{15} + \)\(15\!\cdots\!95\)\( p^{22} T^{16} - \)\(51\!\cdots\!52\)\( p^{33} T^{17} + \)\(16\!\cdots\!32\)\( p^{44} T^{18} - \)\(47\!\cdots\!68\)\( p^{55} T^{19} + \)\(13\!\cdots\!83\)\( p^{66} T^{20} - \)\(34\!\cdots\!48\)\( p^{77} T^{21} + \)\(87\!\cdots\!54\)\( p^{88} T^{22} - \)\(19\!\cdots\!92\)\( p^{99} T^{23} + \)\(39\!\cdots\!42\)\( p^{110} T^{24} - \)\(66\!\cdots\!16\)\( p^{121} T^{25} + \)\(10\!\cdots\!60\)\( p^{132} T^{26} - 105565209020 p^{143} T^{27} + p^{154} T^{28} \) | |
| 83 | \( 1 - 127846064024 T + \)\(21\!\cdots\!30\)\( T^{2} - \)\(20\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!15\)\( T^{4} - \)\(14\!\cdots\!60\)\( T^{5} + \)\(11\!\cdots\!12\)\( T^{6} - \)\(67\!\cdots\!92\)\( T^{7} + \)\(40\!\cdots\!81\)\( T^{8} - \)\(21\!\cdots\!08\)\( T^{9} + \)\(10\!\cdots\!14\)\( T^{10} - \)\(48\!\cdots\!88\)\( T^{11} + \)\(20\!\cdots\!11\)\( T^{12} - \)\(81\!\cdots\!76\)\( T^{13} + \)\(30\!\cdots\!88\)\( T^{14} - \)\(81\!\cdots\!76\)\( p^{11} T^{15} + \)\(20\!\cdots\!11\)\( p^{22} T^{16} - \)\(48\!\cdots\!88\)\( p^{33} T^{17} + \)\(10\!\cdots\!14\)\( p^{44} T^{18} - \)\(21\!\cdots\!08\)\( p^{55} T^{19} + \)\(40\!\cdots\!81\)\( p^{66} T^{20} - \)\(67\!\cdots\!92\)\( p^{77} T^{21} + \)\(11\!\cdots\!12\)\( p^{88} T^{22} - \)\(14\!\cdots\!60\)\( p^{99} T^{23} + \)\(20\!\cdots\!15\)\( p^{110} T^{24} - \)\(20\!\cdots\!04\)\( p^{121} T^{25} + \)\(21\!\cdots\!30\)\( p^{132} T^{26} - 127846064024 p^{143} T^{27} + p^{154} T^{28} \) | |
| 89 | \( 1 - 187826099404 T + \)\(43\!\cdots\!78\)\( T^{2} - \)\(56\!\cdots\!04\)\( T^{3} + \)\(77\!\cdots\!67\)\( T^{4} - \)\(79\!\cdots\!84\)\( T^{5} + \)\(82\!\cdots\!28\)\( T^{6} - \)\(69\!\cdots\!60\)\( T^{7} + \)\(59\!\cdots\!21\)\( T^{8} - \)\(42\!\cdots\!40\)\( T^{9} + \)\(31\!\cdots\!50\)\( T^{10} - \)\(19\!\cdots\!84\)\( T^{11} + \)\(12\!\cdots\!15\)\( T^{12} - \)\(70\!\cdots\!72\)\( T^{13} + \)\(39\!\cdots\!20\)\( T^{14} - \)\(70\!\cdots\!72\)\( p^{11} T^{15} + \)\(12\!\cdots\!15\)\( p^{22} T^{16} - \)\(19\!\cdots\!84\)\( p^{33} T^{17} + \)\(31\!\cdots\!50\)\( p^{44} T^{18} - \)\(42\!\cdots\!40\)\( p^{55} T^{19} + \)\(59\!\cdots\!21\)\( p^{66} T^{20} - \)\(69\!\cdots\!60\)\( p^{77} T^{21} + \)\(82\!\cdots\!28\)\( p^{88} T^{22} - \)\(79\!\cdots\!84\)\( p^{99} T^{23} + \)\(77\!\cdots\!67\)\( p^{110} T^{24} - \)\(56\!\cdots\!04\)\( p^{121} T^{25} + \)\(43\!\cdots\!78\)\( p^{132} T^{26} - 187826099404 p^{143} T^{27} + p^{154} T^{28} \) | |
| 97 | \( 1 - 137285937500 T + \)\(59\!\cdots\!86\)\( T^{2} - \)\(67\!\cdots\!60\)\( T^{3} + \)\(17\!\cdots\!35\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} + \)\(34\!\cdots\!44\)\( T^{6} - \)\(30\!\cdots\!00\)\( T^{7} + \)\(50\!\cdots\!89\)\( T^{8} - \)\(40\!\cdots\!80\)\( T^{9} + \)\(57\!\cdots\!10\)\( T^{10} - \)\(43\!\cdots\!80\)\( T^{11} + \)\(54\!\cdots\!39\)\( T^{12} - \)\(37\!\cdots\!60\)\( T^{13} + \)\(42\!\cdots\!80\)\( T^{14} - \)\(37\!\cdots\!60\)\( p^{11} T^{15} + \)\(54\!\cdots\!39\)\( p^{22} T^{16} - \)\(43\!\cdots\!80\)\( p^{33} T^{17} + \)\(57\!\cdots\!10\)\( p^{44} T^{18} - \)\(40\!\cdots\!80\)\( p^{55} T^{19} + \)\(50\!\cdots\!89\)\( p^{66} T^{20} - \)\(30\!\cdots\!00\)\( p^{77} T^{21} + \)\(34\!\cdots\!44\)\( p^{88} T^{22} - \)\(17\!\cdots\!00\)\( p^{99} T^{23} + \)\(17\!\cdots\!35\)\( p^{110} T^{24} - \)\(67\!\cdots\!60\)\( p^{121} T^{25} + \)\(59\!\cdots\!86\)\( p^{132} T^{26} - 137285937500 p^{143} T^{27} + p^{154} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.41653995621524462158674126371, −3.36469385193989194082147309328, −3.34187228697509942475865957520, −3.04455894450206644634964624506, −2.99459450740985886325423691545, −2.91279082303331839620494981245, −2.63218757103835017593466310460, −2.40812047487201066030140808107, −2.26881026006887646156827688382, −2.05492918681392311932962914601, −2.00755675516921083306868449032, −1.94878314706909351388013020181, −1.92015759979425527366174244325, −1.65531104080260233065011278421, −1.53143630658665196145714176876, −1.36773128678383788277628436436, −1.12325024033703143997688539267, −1.10434044309483971952719157582, −0.878272588675919680788110847456, −0.849797176202482144847415364981, −0.69743778295074663274327102426, −0.51085574732587490690510805815, −0.48752427849774458153580351622, −0.24466279364403572985299529769, −0.11083006039831654946292181041, 0.11083006039831654946292181041, 0.24466279364403572985299529769, 0.48752427849774458153580351622, 0.51085574732587490690510805815, 0.69743778295074663274327102426, 0.849797176202482144847415364981, 0.878272588675919680788110847456, 1.10434044309483971952719157582, 1.12325024033703143997688539267, 1.36773128678383788277628436436, 1.53143630658665196145714176876, 1.65531104080260233065011278421, 1.92015759979425527366174244325, 1.94878314706909351388013020181, 2.00755675516921083306868449032, 2.05492918681392311932962914601, 2.26881026006887646156827688382, 2.40812047487201066030140808107, 2.63218757103835017593466310460, 2.91279082303331839620494981245, 2.99459450740985886325423691545, 3.04455894450206644634964624506, 3.34187228697509942475865957520, 3.36469385193989194082147309328, 3.41653995621524462158674126371
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.