Dirichlet series
| L(s) = 1 | + 3.58e3·2-s + 8.83e3·3-s + 6.88e6·4-s + 1.17e6·5-s + 3.16e7·6-s + 2.06e7·7-s + 9.39e9·8-s − 4.29e8·9-s + 4.21e9·10-s + 6.65e8·11-s + 6.08e10·12-s + 3.98e9·13-s + 7.38e10·14-s + 1.04e10·15-s + 1.02e13·16-s + 3.62e9·17-s − 1.53e12·18-s + 3.10e11·19-s + 8.10e12·20-s + 1.82e11·21-s + 2.38e12·22-s + 1.15e12·23-s + 8.30e13·24-s − 3.21e12·25-s + 1.42e13·26-s − 5.47e12·27-s + 1.41e14·28-s + ⋯ |
| L(s) = 1 | + 9.89·2-s + 0.777·3-s + 52.5·4-s + 1.34·5-s + 7.69·6-s + 1.35·7-s + 197.·8-s − 3.32·9-s + 13.3·10-s + 0.936·11-s + 40.8·12-s + 1.35·13-s + 13.3·14-s + 1.04·15-s + 595·16-s + 0.125·17-s − 32.9·18-s + 4.19·19-s + 70.7·20-s + 1.05·21-s + 9.26·22-s + 3.08·23-s + 153.·24-s − 4.21·25-s + 13.4·26-s − 3.73·27-s + 70.9·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+17/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{14} \cdot 37^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.09469\times 10^{29}\) |
| Root analytic conductor: | \(11.6440\) |
| Motivic weight: | \(17\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{14} \cdot 37^{14} ,\ ( \ : [17/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(9)\) | \(\approx\) | \(8.336810135\times10^{6}\) |
| \(L(\frac12)\) | \(\approx\) | \(8.336810135\times10^{6}\) |
| \(L(\frac{19}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{8} T )^{14} \) |
| 37 | \( ( 1 + p^{8} T )^{14} \) | |
| good | 3 | \( 1 - 8839 T + 169261921 p T^{2} - 2811574528831 T^{3} + 45190158888556382 p T^{4} - 37483001014721430179 p^{2} T^{5} + \)\(34\!\cdots\!94\)\( p^{4} T^{6} - \)\(58\!\cdots\!22\)\( p^{6} T^{7} + \)\(28\!\cdots\!93\)\( p^{9} T^{8} - \)\(57\!\cdots\!14\)\( p^{11} T^{9} + \)\(19\!\cdots\!32\)\( p^{12} T^{10} - \)\(32\!\cdots\!03\)\( p^{14} T^{11} + \)\(34\!\cdots\!15\)\( p^{16} T^{12} - \)\(33\!\cdots\!76\)\( p^{20} T^{13} + \)\(69\!\cdots\!28\)\( p^{24} T^{14} - \)\(33\!\cdots\!76\)\( p^{37} T^{15} + \)\(34\!\cdots\!15\)\( p^{50} T^{16} - \)\(32\!\cdots\!03\)\( p^{65} T^{17} + \)\(19\!\cdots\!32\)\( p^{80} T^{18} - \)\(57\!\cdots\!14\)\( p^{96} T^{19} + \)\(28\!\cdots\!93\)\( p^{111} T^{20} - \)\(58\!\cdots\!22\)\( p^{125} T^{21} + \)\(34\!\cdots\!94\)\( p^{140} T^{22} - 37483001014721430179 p^{155} T^{23} + 45190158888556382 p^{171} T^{24} - 2811574528831 p^{187} T^{25} + 169261921 p^{205} T^{26} - 8839 p^{221} T^{27} + p^{238} T^{28} \) |
| 5 | \( 1 - 1177383 T + 4603472491684 T^{2} - 874743614140924246 p T^{3} + \)\(10\!\cdots\!61\)\( T^{4} - \)\(78\!\cdots\!06\)\( T^{5} + \)\(29\!\cdots\!97\)\( p T^{6} - \)\(35\!\cdots\!58\)\( p^{2} T^{7} + \)\(13\!\cdots\!39\)\( p^{3} T^{8} - \)\(23\!\cdots\!84\)\( p^{5} T^{9} + \)\(37\!\cdots\!96\)\( p^{8} T^{10} - \)\(25\!\cdots\!23\)\( p^{9} T^{11} + \)\(24\!\cdots\!59\)\( p^{11} T^{12} - \)\(27\!\cdots\!28\)\( p^{13} T^{13} + \)\(30\!\cdots\!22\)\( p^{15} T^{14} - \)\(27\!\cdots\!28\)\( p^{30} T^{15} + \)\(24\!\cdots\!59\)\( p^{45} T^{16} - \)\(25\!\cdots\!23\)\( p^{60} T^{17} + \)\(37\!\cdots\!96\)\( p^{76} T^{18} - \)\(23\!\cdots\!84\)\( p^{90} T^{19} + \)\(13\!\cdots\!39\)\( p^{105} T^{20} - \)\(35\!\cdots\!58\)\( p^{121} T^{21} + \)\(29\!\cdots\!97\)\( p^{137} T^{22} - \)\(78\!\cdots\!06\)\( p^{153} T^{23} + \)\(10\!\cdots\!61\)\( p^{170} T^{24} - 874743614140924246 p^{188} T^{25} + 4603472491684 p^{204} T^{26} - 1177383 p^{221} T^{27} + p^{238} T^{28} \) | |
| 7 | \( 1 - 20618754 T + 1546828174964659 T^{2} - \)\(37\!\cdots\!48\)\( p T^{3} + \)\(11\!\cdots\!90\)\( T^{4} - \)\(17\!\cdots\!36\)\( T^{5} + \)\(90\!\cdots\!13\)\( p T^{6} - \)\(18\!\cdots\!02\)\( p^{2} T^{7} + \)\(15\!\cdots\!04\)\( p^{5} T^{8} - \)\(30\!\cdots\!78\)\( p^{6} T^{9} + \)\(15\!\cdots\!65\)\( p^{8} T^{10} - \)\(41\!\cdots\!52\)\( p^{10} T^{11} + \)\(18\!\cdots\!09\)\( p^{12} T^{12} - \)\(47\!\cdots\!06\)\( p^{14} T^{13} + \)\(19\!\cdots\!66\)\( p^{16} T^{14} - \)\(47\!\cdots\!06\)\( p^{31} T^{15} + \)\(18\!\cdots\!09\)\( p^{46} T^{16} - \)\(41\!\cdots\!52\)\( p^{61} T^{17} + \)\(15\!\cdots\!65\)\( p^{76} T^{18} - \)\(30\!\cdots\!78\)\( p^{91} T^{19} + \)\(15\!\cdots\!04\)\( p^{107} T^{20} - \)\(18\!\cdots\!02\)\( p^{121} T^{21} + \)\(90\!\cdots\!13\)\( p^{137} T^{22} - \)\(17\!\cdots\!36\)\( p^{153} T^{23} + \)\(11\!\cdots\!90\)\( p^{170} T^{24} - \)\(37\!\cdots\!48\)\( p^{188} T^{25} + 1546828174964659 p^{204} T^{26} - 20618754 p^{221} T^{27} + p^{238} T^{28} \) | |
| 11 | \( 1 - 665624451 T + 266916704638001431 p T^{2} - \)\(13\!\cdots\!99\)\( T^{3} + \)\(38\!\cdots\!34\)\( p T^{4} - \)\(12\!\cdots\!89\)\( p T^{5} + \)\(42\!\cdots\!76\)\( T^{6} - \)\(83\!\cdots\!94\)\( p T^{7} + \)\(22\!\cdots\!53\)\( p^{4} T^{8} - \)\(35\!\cdots\!94\)\( p^{3} T^{9} + \)\(15\!\cdots\!94\)\( p^{4} T^{10} - \)\(12\!\cdots\!25\)\( p^{5} T^{11} + \)\(73\!\cdots\!27\)\( p^{6} T^{12} - \)\(43\!\cdots\!88\)\( p^{7} T^{13} + \)\(32\!\cdots\!08\)\( p^{8} T^{14} - \)\(43\!\cdots\!88\)\( p^{24} T^{15} + \)\(73\!\cdots\!27\)\( p^{40} T^{16} - \)\(12\!\cdots\!25\)\( p^{56} T^{17} + \)\(15\!\cdots\!94\)\( p^{72} T^{18} - \)\(35\!\cdots\!94\)\( p^{88} T^{19} + \)\(22\!\cdots\!53\)\( p^{106} T^{20} - \)\(83\!\cdots\!94\)\( p^{120} T^{21} + \)\(42\!\cdots\!76\)\( p^{136} T^{22} - \)\(12\!\cdots\!89\)\( p^{154} T^{23} + \)\(38\!\cdots\!34\)\( p^{171} T^{24} - \)\(13\!\cdots\!99\)\( p^{187} T^{25} + 266916704638001431 p^{205} T^{26} - 665624451 p^{221} T^{27} + p^{238} T^{28} \) | |
| 13 | \( 1 - 3986363599 T + 34933950166900287532 T^{2} - \)\(78\!\cdots\!54\)\( p T^{3} + \)\(59\!\cdots\!61\)\( T^{4} - \)\(14\!\cdots\!74\)\( T^{5} + \)\(55\!\cdots\!93\)\( p T^{6} - \)\(96\!\cdots\!54\)\( p^{2} T^{7} + \)\(35\!\cdots\!03\)\( p^{3} T^{8} - \)\(57\!\cdots\!28\)\( p^{4} T^{9} + \)\(21\!\cdots\!52\)\( p^{5} T^{10} - \)\(33\!\cdots\!07\)\( p^{6} T^{11} + \)\(12\!\cdots\!55\)\( p^{7} T^{12} - \)\(19\!\cdots\!52\)\( p^{8} T^{13} + \)\(69\!\cdots\!82\)\( p^{9} T^{14} - \)\(19\!\cdots\!52\)\( p^{25} T^{15} + \)\(12\!\cdots\!55\)\( p^{41} T^{16} - \)\(33\!\cdots\!07\)\( p^{57} T^{17} + \)\(21\!\cdots\!52\)\( p^{73} T^{18} - \)\(57\!\cdots\!28\)\( p^{89} T^{19} + \)\(35\!\cdots\!03\)\( p^{105} T^{20} - \)\(96\!\cdots\!54\)\( p^{121} T^{21} + \)\(55\!\cdots\!93\)\( p^{137} T^{22} - \)\(14\!\cdots\!74\)\( p^{153} T^{23} + \)\(59\!\cdots\!61\)\( p^{170} T^{24} - \)\(78\!\cdots\!54\)\( p^{188} T^{25} + 34933950166900287532 p^{204} T^{26} - 3986363599 p^{221} T^{27} + p^{238} T^{28} \) | |
| 17 | \( 1 - 3620739192 T + \)\(59\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!87\)\( T^{4} - \)\(15\!\cdots\!12\)\( T^{5} + \)\(41\!\cdots\!28\)\( T^{6} + \)\(11\!\cdots\!16\)\( T^{7} + \)\(69\!\cdots\!05\)\( T^{8} + \)\(43\!\cdots\!20\)\( T^{9} + \)\(91\!\cdots\!38\)\( T^{10} + \)\(90\!\cdots\!40\)\( T^{11} + \)\(99\!\cdots\!59\)\( T^{12} + \)\(10\!\cdots\!52\)\( T^{13} + \)\(90\!\cdots\!12\)\( T^{14} + \)\(10\!\cdots\!52\)\( p^{17} T^{15} + \)\(99\!\cdots\!59\)\( p^{34} T^{16} + \)\(90\!\cdots\!40\)\( p^{51} T^{17} + \)\(91\!\cdots\!38\)\( p^{68} T^{18} + \)\(43\!\cdots\!20\)\( p^{85} T^{19} + \)\(69\!\cdots\!05\)\( p^{102} T^{20} + \)\(11\!\cdots\!16\)\( p^{119} T^{21} + \)\(41\!\cdots\!28\)\( p^{136} T^{22} - \)\(15\!\cdots\!12\)\( p^{153} T^{23} + \)\(19\!\cdots\!87\)\( p^{170} T^{24} - \)\(11\!\cdots\!28\)\( p^{187} T^{25} + \)\(59\!\cdots\!26\)\( p^{204} T^{26} - 3620739192 p^{221} T^{27} + p^{238} T^{28} \) | |
| 19 | \( 1 - 310184371112 T + \)\(83\!\cdots\!66\)\( T^{2} - \)\(15\!\cdots\!48\)\( T^{3} + \)\(27\!\cdots\!51\)\( T^{4} - \)\(40\!\cdots\!00\)\( T^{5} + \)\(54\!\cdots\!32\)\( T^{6} - \)\(65\!\cdots\!64\)\( T^{7} + \)\(73\!\cdots\!01\)\( T^{8} - \)\(76\!\cdots\!00\)\( T^{9} + \)\(74\!\cdots\!58\)\( T^{10} - \)\(68\!\cdots\!32\)\( T^{11} + \)\(58\!\cdots\!47\)\( T^{12} - \)\(47\!\cdots\!12\)\( T^{13} + \)\(36\!\cdots\!48\)\( T^{14} - \)\(47\!\cdots\!12\)\( p^{17} T^{15} + \)\(58\!\cdots\!47\)\( p^{34} T^{16} - \)\(68\!\cdots\!32\)\( p^{51} T^{17} + \)\(74\!\cdots\!58\)\( p^{68} T^{18} - \)\(76\!\cdots\!00\)\( p^{85} T^{19} + \)\(73\!\cdots\!01\)\( p^{102} T^{20} - \)\(65\!\cdots\!64\)\( p^{119} T^{21} + \)\(54\!\cdots\!32\)\( p^{136} T^{22} - \)\(40\!\cdots\!00\)\( p^{153} T^{23} + \)\(27\!\cdots\!51\)\( p^{170} T^{24} - \)\(15\!\cdots\!48\)\( p^{187} T^{25} + \)\(83\!\cdots\!66\)\( p^{204} T^{26} - 310184371112 p^{221} T^{27} + p^{238} T^{28} \) | |
| 23 | \( 1 - 1157337774835 T + \)\(15\!\cdots\!88\)\( T^{2} - \)\(11\!\cdots\!68\)\( T^{3} + \)\(95\!\cdots\!17\)\( T^{4} - \)\(56\!\cdots\!64\)\( T^{5} + \)\(37\!\cdots\!45\)\( T^{6} - \)\(18\!\cdots\!72\)\( T^{7} + \)\(10\!\cdots\!59\)\( T^{8} - \)\(47\!\cdots\!38\)\( T^{9} + \)\(23\!\cdots\!56\)\( T^{10} - \)\(95\!\cdots\!49\)\( T^{11} + \)\(43\!\cdots\!31\)\( T^{12} - \)\(16\!\cdots\!72\)\( T^{13} + \)\(66\!\cdots\!70\)\( T^{14} - \)\(16\!\cdots\!72\)\( p^{17} T^{15} + \)\(43\!\cdots\!31\)\( p^{34} T^{16} - \)\(95\!\cdots\!49\)\( p^{51} T^{17} + \)\(23\!\cdots\!56\)\( p^{68} T^{18} - \)\(47\!\cdots\!38\)\( p^{85} T^{19} + \)\(10\!\cdots\!59\)\( p^{102} T^{20} - \)\(18\!\cdots\!72\)\( p^{119} T^{21} + \)\(37\!\cdots\!45\)\( p^{136} T^{22} - \)\(56\!\cdots\!64\)\( p^{153} T^{23} + \)\(95\!\cdots\!17\)\( p^{170} T^{24} - \)\(11\!\cdots\!68\)\( p^{187} T^{25} + \)\(15\!\cdots\!88\)\( p^{204} T^{26} - 1157337774835 p^{221} T^{27} + p^{238} T^{28} \) | |
| 29 | \( 1 - 2625324116903 T + \)\(42\!\cdots\!40\)\( T^{2} - \)\(14\!\cdots\!14\)\( T^{3} + \)\(34\!\cdots\!49\)\( p T^{4} - \)\(35\!\cdots\!78\)\( T^{5} + \)\(17\!\cdots\!65\)\( T^{6} - \)\(54\!\cdots\!90\)\( T^{7} + \)\(22\!\cdots\!75\)\( T^{8} - \)\(63\!\cdots\!80\)\( T^{9} + \)\(22\!\cdots\!08\)\( T^{10} - \)\(60\!\cdots\!11\)\( T^{11} + \)\(19\!\cdots\!59\)\( T^{12} - \)\(48\!\cdots\!40\)\( T^{13} + \)\(15\!\cdots\!62\)\( T^{14} - \)\(48\!\cdots\!40\)\( p^{17} T^{15} + \)\(19\!\cdots\!59\)\( p^{34} T^{16} - \)\(60\!\cdots\!11\)\( p^{51} T^{17} + \)\(22\!\cdots\!08\)\( p^{68} T^{18} - \)\(63\!\cdots\!80\)\( p^{85} T^{19} + \)\(22\!\cdots\!75\)\( p^{102} T^{20} - \)\(54\!\cdots\!90\)\( p^{119} T^{21} + \)\(17\!\cdots\!65\)\( p^{136} T^{22} - \)\(35\!\cdots\!78\)\( p^{153} T^{23} + \)\(34\!\cdots\!49\)\( p^{171} T^{24} - \)\(14\!\cdots\!14\)\( p^{187} T^{25} + \)\(42\!\cdots\!40\)\( p^{204} T^{26} - 2625324116903 p^{221} T^{27} + p^{238} T^{28} \) | |
| 31 | \( 1 + 3876877351243 T + \)\(16\!\cdots\!78\)\( T^{2} + \)\(63\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} + \)\(51\!\cdots\!12\)\( T^{5} + \)\(24\!\cdots\!13\)\( p T^{6} + \)\(27\!\cdots\!60\)\( T^{7} + \)\(30\!\cdots\!09\)\( T^{8} + \)\(10\!\cdots\!98\)\( T^{9} + \)\(10\!\cdots\!98\)\( T^{10} + \)\(33\!\cdots\!05\)\( T^{11} + \)\(28\!\cdots\!11\)\( T^{12} + \)\(88\!\cdots\!64\)\( T^{13} + \)\(68\!\cdots\!90\)\( T^{14} + \)\(88\!\cdots\!64\)\( p^{17} T^{15} + \)\(28\!\cdots\!11\)\( p^{34} T^{16} + \)\(33\!\cdots\!05\)\( p^{51} T^{17} + \)\(10\!\cdots\!98\)\( p^{68} T^{18} + \)\(10\!\cdots\!98\)\( p^{85} T^{19} + \)\(30\!\cdots\!09\)\( p^{102} T^{20} + \)\(27\!\cdots\!60\)\( p^{119} T^{21} + \)\(24\!\cdots\!13\)\( p^{137} T^{22} + \)\(51\!\cdots\!12\)\( p^{153} T^{23} + \)\(13\!\cdots\!63\)\( p^{170} T^{24} + \)\(63\!\cdots\!76\)\( p^{187} T^{25} + \)\(16\!\cdots\!78\)\( p^{204} T^{26} + 3876877351243 p^{221} T^{27} + p^{238} T^{28} \) | |
| 41 | \( 1 - 158886863281289 T + \)\(26\!\cdots\!91\)\( T^{2} - \)\(30\!\cdots\!89\)\( T^{3} + \)\(33\!\cdots\!16\)\( T^{4} - \)\(30\!\cdots\!73\)\( T^{5} + \)\(27\!\cdots\!88\)\( T^{6} - \)\(21\!\cdots\!72\)\( T^{7} + \)\(15\!\cdots\!97\)\( T^{8} - \)\(10\!\cdots\!44\)\( T^{9} + \)\(71\!\cdots\!66\)\( T^{10} - \)\(43\!\cdots\!99\)\( T^{11} + \)\(25\!\cdots\!01\)\( T^{12} - \)\(14\!\cdots\!14\)\( T^{13} + \)\(74\!\cdots\!40\)\( T^{14} - \)\(14\!\cdots\!14\)\( p^{17} T^{15} + \)\(25\!\cdots\!01\)\( p^{34} T^{16} - \)\(43\!\cdots\!99\)\( p^{51} T^{17} + \)\(71\!\cdots\!66\)\( p^{68} T^{18} - \)\(10\!\cdots\!44\)\( p^{85} T^{19} + \)\(15\!\cdots\!97\)\( p^{102} T^{20} - \)\(21\!\cdots\!72\)\( p^{119} T^{21} + \)\(27\!\cdots\!88\)\( p^{136} T^{22} - \)\(30\!\cdots\!73\)\( p^{153} T^{23} + \)\(33\!\cdots\!16\)\( p^{170} T^{24} - \)\(30\!\cdots\!89\)\( p^{187} T^{25} + \)\(26\!\cdots\!91\)\( p^{204} T^{26} - 158886863281289 p^{221} T^{27} + p^{238} T^{28} \) | |
| 43 | \( 1 - 148816544603066 T + \)\(50\!\cdots\!46\)\( T^{2} - \)\(63\!\cdots\!86\)\( T^{3} + \)\(12\!\cdots\!91\)\( T^{4} - \)\(13\!\cdots\!96\)\( T^{5} + \)\(20\!\cdots\!60\)\( T^{6} - \)\(20\!\cdots\!72\)\( T^{7} + \)\(25\!\cdots\!13\)\( T^{8} - \)\(22\!\cdots\!46\)\( T^{9} + \)\(24\!\cdots\!46\)\( T^{10} - \)\(19\!\cdots\!70\)\( T^{11} + \)\(19\!\cdots\!11\)\( T^{12} - \)\(13\!\cdots\!12\)\( T^{13} + \)\(12\!\cdots\!92\)\( T^{14} - \)\(13\!\cdots\!12\)\( p^{17} T^{15} + \)\(19\!\cdots\!11\)\( p^{34} T^{16} - \)\(19\!\cdots\!70\)\( p^{51} T^{17} + \)\(24\!\cdots\!46\)\( p^{68} T^{18} - \)\(22\!\cdots\!46\)\( p^{85} T^{19} + \)\(25\!\cdots\!13\)\( p^{102} T^{20} - \)\(20\!\cdots\!72\)\( p^{119} T^{21} + \)\(20\!\cdots\!60\)\( p^{136} T^{22} - \)\(13\!\cdots\!96\)\( p^{153} T^{23} + \)\(12\!\cdots\!91\)\( p^{170} T^{24} - \)\(63\!\cdots\!86\)\( p^{187} T^{25} + \)\(50\!\cdots\!46\)\( p^{204} T^{26} - 148816544603066 p^{221} T^{27} + p^{238} T^{28} \) | |
| 47 | \( 1 - 609286669235070 T + \)\(43\!\cdots\!31\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(78\!\cdots\!58\)\( T^{4} - \)\(26\!\cdots\!00\)\( T^{5} + \)\(84\!\cdots\!51\)\( T^{6} - \)\(23\!\cdots\!06\)\( T^{7} + \)\(61\!\cdots\!36\)\( T^{8} - \)\(14\!\cdots\!78\)\( T^{9} + \)\(32\!\cdots\!93\)\( T^{10} - \)\(66\!\cdots\!16\)\( T^{11} + \)\(12\!\cdots\!05\)\( T^{12} - \)\(22\!\cdots\!90\)\( T^{13} + \)\(38\!\cdots\!22\)\( T^{14} - \)\(22\!\cdots\!90\)\( p^{17} T^{15} + \)\(12\!\cdots\!05\)\( p^{34} T^{16} - \)\(66\!\cdots\!16\)\( p^{51} T^{17} + \)\(32\!\cdots\!93\)\( p^{68} T^{18} - \)\(14\!\cdots\!78\)\( p^{85} T^{19} + \)\(61\!\cdots\!36\)\( p^{102} T^{20} - \)\(23\!\cdots\!06\)\( p^{119} T^{21} + \)\(84\!\cdots\!51\)\( p^{136} T^{22} - \)\(26\!\cdots\!00\)\( p^{153} T^{23} + \)\(78\!\cdots\!58\)\( p^{170} T^{24} - \)\(18\!\cdots\!20\)\( p^{187} T^{25} + \)\(43\!\cdots\!31\)\( p^{204} T^{26} - 609286669235070 p^{221} T^{27} + p^{238} T^{28} \) | |
| 53 | \( 1 - 907172191065214 T + \)\(16\!\cdots\!41\)\( T^{2} - \)\(12\!\cdots\!90\)\( T^{3} + \)\(12\!\cdots\!50\)\( T^{4} - \)\(75\!\cdots\!26\)\( T^{5} + \)\(57\!\cdots\!93\)\( T^{6} - \)\(27\!\cdots\!02\)\( T^{7} + \)\(16\!\cdots\!96\)\( T^{8} - \)\(61\!\cdots\!34\)\( T^{9} + \)\(32\!\cdots\!91\)\( T^{10} - \)\(80\!\cdots\!70\)\( T^{11} + \)\(41\!\cdots\!97\)\( T^{12} - \)\(94\!\cdots\!92\)\( p T^{13} + \)\(55\!\cdots\!54\)\( T^{14} - \)\(94\!\cdots\!92\)\( p^{18} T^{15} + \)\(41\!\cdots\!97\)\( p^{34} T^{16} - \)\(80\!\cdots\!70\)\( p^{51} T^{17} + \)\(32\!\cdots\!91\)\( p^{68} T^{18} - \)\(61\!\cdots\!34\)\( p^{85} T^{19} + \)\(16\!\cdots\!96\)\( p^{102} T^{20} - \)\(27\!\cdots\!02\)\( p^{119} T^{21} + \)\(57\!\cdots\!93\)\( p^{136} T^{22} - \)\(75\!\cdots\!26\)\( p^{153} T^{23} + \)\(12\!\cdots\!50\)\( p^{170} T^{24} - \)\(12\!\cdots\!90\)\( p^{187} T^{25} + \)\(16\!\cdots\!41\)\( p^{204} T^{26} - 907172191065214 p^{221} T^{27} + p^{238} T^{28} \) | |
| 59 | \( 1 + 646111328250178 T + \)\(88\!\cdots\!62\)\( T^{2} + \)\(36\!\cdots\!06\)\( T^{3} + \)\(39\!\cdots\!07\)\( T^{4} + \)\(93\!\cdots\!68\)\( T^{5} + \)\(12\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!40\)\( T^{7} + \)\(28\!\cdots\!77\)\( T^{8} - \)\(39\!\cdots\!18\)\( T^{9} + \)\(55\!\cdots\!66\)\( T^{10} - \)\(35\!\cdots\!74\)\( T^{11} + \)\(90\!\cdots\!71\)\( T^{12} - \)\(81\!\cdots\!80\)\( T^{13} + \)\(12\!\cdots\!52\)\( T^{14} - \)\(81\!\cdots\!80\)\( p^{17} T^{15} + \)\(90\!\cdots\!71\)\( p^{34} T^{16} - \)\(35\!\cdots\!74\)\( p^{51} T^{17} + \)\(55\!\cdots\!66\)\( p^{68} T^{18} - \)\(39\!\cdots\!18\)\( p^{85} T^{19} + \)\(28\!\cdots\!77\)\( p^{102} T^{20} + \)\(12\!\cdots\!40\)\( p^{119} T^{21} + \)\(12\!\cdots\!00\)\( p^{136} T^{22} + \)\(93\!\cdots\!68\)\( p^{153} T^{23} + \)\(39\!\cdots\!07\)\( p^{170} T^{24} + \)\(36\!\cdots\!06\)\( p^{187} T^{25} + \)\(88\!\cdots\!62\)\( p^{204} T^{26} + 646111328250178 p^{221} T^{27} + p^{238} T^{28} \) | |
| 61 | \( 1 - 3615823349440805 T + \)\(15\!\cdots\!40\)\( T^{2} - \)\(41\!\cdots\!18\)\( T^{3} + \)\(11\!\cdots\!17\)\( T^{4} - \)\(26\!\cdots\!02\)\( T^{5} + \)\(59\!\cdots\!85\)\( T^{6} - \)\(11\!\cdots\!86\)\( T^{7} + \)\(23\!\cdots\!55\)\( T^{8} - \)\(41\!\cdots\!16\)\( T^{9} + \)\(74\!\cdots\!32\)\( T^{10} - \)\(12\!\cdots\!97\)\( T^{11} + \)\(20\!\cdots\!43\)\( T^{12} - \)\(31\!\cdots\!88\)\( T^{13} + \)\(48\!\cdots\!58\)\( T^{14} - \)\(31\!\cdots\!88\)\( p^{17} T^{15} + \)\(20\!\cdots\!43\)\( p^{34} T^{16} - \)\(12\!\cdots\!97\)\( p^{51} T^{17} + \)\(74\!\cdots\!32\)\( p^{68} T^{18} - \)\(41\!\cdots\!16\)\( p^{85} T^{19} + \)\(23\!\cdots\!55\)\( p^{102} T^{20} - \)\(11\!\cdots\!86\)\( p^{119} T^{21} + \)\(59\!\cdots\!85\)\( p^{136} T^{22} - \)\(26\!\cdots\!02\)\( p^{153} T^{23} + \)\(11\!\cdots\!17\)\( p^{170} T^{24} - \)\(41\!\cdots\!18\)\( p^{187} T^{25} + \)\(15\!\cdots\!40\)\( p^{204} T^{26} - 3615823349440805 p^{221} T^{27} + p^{238} T^{28} \) | |
| 67 | \( 1 - 5025950285916071 T + \)\(12\!\cdots\!64\)\( T^{2} - \)\(52\!\cdots\!52\)\( T^{3} + \)\(70\!\cdots\!97\)\( T^{4} - \)\(26\!\cdots\!96\)\( T^{5} + \)\(25\!\cdots\!93\)\( T^{6} - \)\(87\!\cdots\!16\)\( T^{7} + \)\(68\!\cdots\!11\)\( T^{8} - \)\(20\!\cdots\!98\)\( T^{9} + \)\(13\!\cdots\!36\)\( T^{10} - \)\(36\!\cdots\!53\)\( T^{11} + \)\(31\!\cdots\!65\)\( p T^{12} - \)\(51\!\cdots\!92\)\( T^{13} + \)\(26\!\cdots\!54\)\( T^{14} - \)\(51\!\cdots\!92\)\( p^{17} T^{15} + \)\(31\!\cdots\!65\)\( p^{35} T^{16} - \)\(36\!\cdots\!53\)\( p^{51} T^{17} + \)\(13\!\cdots\!36\)\( p^{68} T^{18} - \)\(20\!\cdots\!98\)\( p^{85} T^{19} + \)\(68\!\cdots\!11\)\( p^{102} T^{20} - \)\(87\!\cdots\!16\)\( p^{119} T^{21} + \)\(25\!\cdots\!93\)\( p^{136} T^{22} - \)\(26\!\cdots\!96\)\( p^{153} T^{23} + \)\(70\!\cdots\!97\)\( p^{170} T^{24} - \)\(52\!\cdots\!52\)\( p^{187} T^{25} + \)\(12\!\cdots\!64\)\( p^{204} T^{26} - 5025950285916071 p^{221} T^{27} + p^{238} T^{28} \) | |
| 71 | \( 1 + 3134424844113936 T + \)\(19\!\cdots\!71\)\( T^{2} + \)\(43\!\cdots\!40\)\( T^{3} + \)\(18\!\cdots\!34\)\( T^{4} + \)\(36\!\cdots\!32\)\( T^{5} + \)\(13\!\cdots\!75\)\( T^{6} + \)\(24\!\cdots\!36\)\( T^{7} + \)\(70\!\cdots\!72\)\( T^{8} + \)\(13\!\cdots\!88\)\( T^{9} + \)\(30\!\cdots\!61\)\( T^{10} + \)\(57\!\cdots\!28\)\( T^{11} + \)\(11\!\cdots\!85\)\( T^{12} + \)\(20\!\cdots\!00\)\( T^{13} + \)\(36\!\cdots\!70\)\( T^{14} + \)\(20\!\cdots\!00\)\( p^{17} T^{15} + \)\(11\!\cdots\!85\)\( p^{34} T^{16} + \)\(57\!\cdots\!28\)\( p^{51} T^{17} + \)\(30\!\cdots\!61\)\( p^{68} T^{18} + \)\(13\!\cdots\!88\)\( p^{85} T^{19} + \)\(70\!\cdots\!72\)\( p^{102} T^{20} + \)\(24\!\cdots\!36\)\( p^{119} T^{21} + \)\(13\!\cdots\!75\)\( p^{136} T^{22} + \)\(36\!\cdots\!32\)\( p^{153} T^{23} + \)\(18\!\cdots\!34\)\( p^{170} T^{24} + \)\(43\!\cdots\!40\)\( p^{187} T^{25} + \)\(19\!\cdots\!71\)\( p^{204} T^{26} + 3134424844113936 p^{221} T^{27} + p^{238} T^{28} \) | |
| 73 | \( 1 - 14476967538956495 T + \)\(42\!\cdots\!11\)\( T^{2} - \)\(48\!\cdots\!91\)\( T^{3} + \)\(86\!\cdots\!32\)\( T^{4} - \)\(83\!\cdots\!43\)\( T^{5} + \)\(11\!\cdots\!20\)\( T^{6} - \)\(99\!\cdots\!92\)\( T^{7} + \)\(11\!\cdots\!13\)\( T^{8} - \)\(89\!\cdots\!88\)\( T^{9} + \)\(89\!\cdots\!74\)\( T^{10} - \)\(64\!\cdots\!25\)\( T^{11} + \)\(57\!\cdots\!09\)\( T^{12} - \)\(37\!\cdots\!18\)\( T^{13} + \)\(29\!\cdots\!80\)\( T^{14} - \)\(37\!\cdots\!18\)\( p^{17} T^{15} + \)\(57\!\cdots\!09\)\( p^{34} T^{16} - \)\(64\!\cdots\!25\)\( p^{51} T^{17} + \)\(89\!\cdots\!74\)\( p^{68} T^{18} - \)\(89\!\cdots\!88\)\( p^{85} T^{19} + \)\(11\!\cdots\!13\)\( p^{102} T^{20} - \)\(99\!\cdots\!92\)\( p^{119} T^{21} + \)\(11\!\cdots\!20\)\( p^{136} T^{22} - \)\(83\!\cdots\!43\)\( p^{153} T^{23} + \)\(86\!\cdots\!32\)\( p^{170} T^{24} - \)\(48\!\cdots\!91\)\( p^{187} T^{25} + \)\(42\!\cdots\!11\)\( p^{204} T^{26} - 14476967538956495 p^{221} T^{27} + p^{238} T^{28} \) | |
| 79 | \( 1 - 34318750984472891 T + \)\(17\!\cdots\!16\)\( T^{2} - \)\(47\!\cdots\!88\)\( T^{3} + \)\(14\!\cdots\!25\)\( T^{4} - \)\(33\!\cdots\!04\)\( T^{5} + \)\(79\!\cdots\!05\)\( T^{6} - \)\(15\!\cdots\!72\)\( T^{7} + \)\(30\!\cdots\!83\)\( T^{8} - \)\(53\!\cdots\!74\)\( T^{9} + \)\(92\!\cdots\!88\)\( T^{10} - \)\(14\!\cdots\!81\)\( T^{11} + \)\(22\!\cdots\!55\)\( T^{12} - \)\(31\!\cdots\!76\)\( T^{13} + \)\(45\!\cdots\!86\)\( T^{14} - \)\(31\!\cdots\!76\)\( p^{17} T^{15} + \)\(22\!\cdots\!55\)\( p^{34} T^{16} - \)\(14\!\cdots\!81\)\( p^{51} T^{17} + \)\(92\!\cdots\!88\)\( p^{68} T^{18} - \)\(53\!\cdots\!74\)\( p^{85} T^{19} + \)\(30\!\cdots\!83\)\( p^{102} T^{20} - \)\(15\!\cdots\!72\)\( p^{119} T^{21} + \)\(79\!\cdots\!05\)\( p^{136} T^{22} - \)\(33\!\cdots\!04\)\( p^{153} T^{23} + \)\(14\!\cdots\!25\)\( p^{170} T^{24} - \)\(47\!\cdots\!88\)\( p^{187} T^{25} + \)\(17\!\cdots\!16\)\( p^{204} T^{26} - 34318750984472891 p^{221} T^{27} + p^{238} T^{28} \) | |
| 83 | \( 1 - 4695300158441348 T + \)\(29\!\cdots\!39\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(43\!\cdots\!62\)\( T^{4} - \)\(13\!\cdots\!08\)\( T^{5} + \)\(43\!\cdots\!27\)\( T^{6} - \)\(12\!\cdots\!20\)\( T^{7} + \)\(33\!\cdots\!00\)\( T^{8} - \)\(10\!\cdots\!00\)\( T^{9} + \)\(21\!\cdots\!17\)\( T^{10} - \)\(78\!\cdots\!48\)\( p T^{11} + \)\(11\!\cdots\!09\)\( T^{12} - \)\(33\!\cdots\!68\)\( T^{13} + \)\(51\!\cdots\!98\)\( T^{14} - \)\(33\!\cdots\!68\)\( p^{17} T^{15} + \)\(11\!\cdots\!09\)\( p^{34} T^{16} - \)\(78\!\cdots\!48\)\( p^{52} T^{17} + \)\(21\!\cdots\!17\)\( p^{68} T^{18} - \)\(10\!\cdots\!00\)\( p^{85} T^{19} + \)\(33\!\cdots\!00\)\( p^{102} T^{20} - \)\(12\!\cdots\!20\)\( p^{119} T^{21} + \)\(43\!\cdots\!27\)\( p^{136} T^{22} - \)\(13\!\cdots\!08\)\( p^{153} T^{23} + \)\(43\!\cdots\!62\)\( p^{170} T^{24} - \)\(10\!\cdots\!60\)\( p^{187} T^{25} + \)\(29\!\cdots\!39\)\( p^{204} T^{26} - 4695300158441348 p^{221} T^{27} + p^{238} T^{28} \) | |
| 89 | \( 1 - 53855337394859876 T + \)\(10\!\cdots\!62\)\( T^{2} - \)\(54\!\cdots\!32\)\( T^{3} + \)\(58\!\cdots\!47\)\( T^{4} - \)\(27\!\cdots\!20\)\( T^{5} + \)\(20\!\cdots\!20\)\( T^{6} - \)\(89\!\cdots\!68\)\( T^{7} + \)\(52\!\cdots\!25\)\( T^{8} - \)\(21\!\cdots\!40\)\( T^{9} + \)\(10\!\cdots\!86\)\( T^{10} - \)\(42\!\cdots\!08\)\( T^{11} + \)\(18\!\cdots\!47\)\( T^{12} - \)\(67\!\cdots\!92\)\( T^{13} + \)\(27\!\cdots\!04\)\( T^{14} - \)\(67\!\cdots\!92\)\( p^{17} T^{15} + \)\(18\!\cdots\!47\)\( p^{34} T^{16} - \)\(42\!\cdots\!08\)\( p^{51} T^{17} + \)\(10\!\cdots\!86\)\( p^{68} T^{18} - \)\(21\!\cdots\!40\)\( p^{85} T^{19} + \)\(52\!\cdots\!25\)\( p^{102} T^{20} - \)\(89\!\cdots\!68\)\( p^{119} T^{21} + \)\(20\!\cdots\!20\)\( p^{136} T^{22} - \)\(27\!\cdots\!20\)\( p^{153} T^{23} + \)\(58\!\cdots\!47\)\( p^{170} T^{24} - \)\(54\!\cdots\!32\)\( p^{187} T^{25} + \)\(10\!\cdots\!62\)\( p^{204} T^{26} - 53855337394859876 p^{221} T^{27} + p^{238} T^{28} \) | |
| 97 | \( 1 - 231442923067496260 T + \)\(73\!\cdots\!42\)\( T^{2} - \)\(12\!\cdots\!84\)\( T^{3} + \)\(23\!\cdots\!87\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{5} + \)\(45\!\cdots\!44\)\( T^{6} - \)\(52\!\cdots\!96\)\( T^{7} + \)\(63\!\cdots\!65\)\( T^{8} - \)\(63\!\cdots\!16\)\( T^{9} + \)\(66\!\cdots\!54\)\( T^{10} - \)\(59\!\cdots\!88\)\( T^{11} + \)\(55\!\cdots\!23\)\( T^{12} - \)\(44\!\cdots\!20\)\( T^{13} + \)\(36\!\cdots\!40\)\( T^{14} - \)\(44\!\cdots\!20\)\( p^{17} T^{15} + \)\(55\!\cdots\!23\)\( p^{34} T^{16} - \)\(59\!\cdots\!88\)\( p^{51} T^{17} + \)\(66\!\cdots\!54\)\( p^{68} T^{18} - \)\(63\!\cdots\!16\)\( p^{85} T^{19} + \)\(63\!\cdots\!65\)\( p^{102} T^{20} - \)\(52\!\cdots\!96\)\( p^{119} T^{21} + \)\(45\!\cdots\!44\)\( p^{136} T^{22} - \)\(31\!\cdots\!80\)\( p^{153} T^{23} + \)\(23\!\cdots\!87\)\( p^{170} T^{24} - \)\(12\!\cdots\!84\)\( p^{187} T^{25} + \)\(73\!\cdots\!42\)\( p^{204} T^{26} - 231442923067496260 p^{221} T^{27} + p^{238} T^{28} \) | |
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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
−2.36317901544488271517002765279, −2.27424910398116803662434805822, −2.15508250772441323458803577655, −2.10702839456570882440127199369, −2.07631026802594383685695156911, −2.02023642594736412960411248228, −1.95513907945884176363683427227, −1.94394026339572724329900502886, −1.84985052350160407927604431602, −1.80216536752192099691555368769, −1.48207668712832074049815271016, −1.47877276989361526015857388184, −1.28418767158209911218666327232, −1.25606477469063119860036709239, −1.10228761998625233723555992002, −1.02308125402105658698463860922, −0.986732343504701190066167206965, −0.837131240486705564588919823641, −0.75543202279790887119395472201, −0.71005915580245530337515117086, −0.57213928062825308744711126938, −0.53028994125556576974857920892, −0.44192548525394943046510053934, −0.33650387721415465816734331528, −0.31252067221982046406673684268, 0.31252067221982046406673684268, 0.33650387721415465816734331528, 0.44192548525394943046510053934, 0.53028994125556576974857920892, 0.57213928062825308744711126938, 0.71005915580245530337515117086, 0.75543202279790887119395472201, 0.837131240486705564588919823641, 0.986732343504701190066167206965, 1.02308125402105658698463860922, 1.10228761998625233723555992002, 1.25606477469063119860036709239, 1.28418767158209911218666327232, 1.47877276989361526015857388184, 1.48207668712832074049815271016, 1.80216536752192099691555368769, 1.84985052350160407927604431602, 1.94394026339572724329900502886, 1.95513907945884176363683427227, 2.02023642594736412960411248228, 2.07631026802594383685695156911, 2.10702839456570882440127199369, 2.15508250772441323458803577655, 2.27424910398116803662434805822, 2.36317901544488271517002765279
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.