Dirichlet series
| L(s) = 1 | − 2.44e4·2-s − 1.10e8·4-s + 1.48e10·5-s − 2.84e12·8-s + 1.08e15·9-s − 3.63e14·10-s + 7.34e16·13-s − 3.40e16·16-s − 4.41e18·17-s − 2.64e19·18-s − 1.64e18·20-s − 5.13e21·25-s − 1.79e21·26-s − 5.18e21·29-s + 1.51e22·32-s + 1.08e23·34-s − 1.19e23·36-s − 6.27e23·37-s − 4.23e22·40-s + 1.49e24·41-s + 1.60e25·45-s + 1.76e26·49-s + 1.25e26·50-s − 8.14e24·52-s − 1.62e26·53-s + 1.26e26·58-s − 7.81e26·61-s + ⋯ |
| L(s) = 1 | − 0.746·2-s − 0.103·4-s + 0.487·5-s − 0.0809·8-s + 5.24·9-s − 0.363·10-s + 1.43·13-s − 0.0295·16-s − 1.54·17-s − 3.91·18-s − 0.0502·20-s − 5.50·25-s − 1.07·26-s − 0.600·29-s + 0.400·32-s + 1.15·34-s − 0.541·36-s − 1.88·37-s − 0.0394·40-s + 0.960·41-s + 2.55·45-s + 7.82·49-s + 4.11·50-s − 0.148·52-s − 2.21·53-s + 0.448·58-s − 1.29·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(31-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s+15)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{28}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.02945\times 10^{19}\) |
| Root analytic conductor: | \(4.77553\) |
| Motivic weight: | \(30\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{28} ,\ ( \ : [15]^{14} ),\ 1 )\) |
Particular Values
| \(L(\frac{31}{2})\) | \(\approx\) | \(13.71495873\) |
| \(L(\frac12)\) | \(\approx\) | \(13.71495873\) |
| \(L(16)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 6119 p^{2} T + 11092669 p^{6} T^{2} + 5599704747 p^{12} T^{3} + 709401085677 p^{20} T^{4} + 7909248799671 p^{30} T^{5} + 118616959511173 p^{42} T^{6} + 228254589677531 p^{56} T^{7} + 118616959511173 p^{72} T^{8} + 7909248799671 p^{90} T^{9} + 709401085677 p^{110} T^{10} + 5599704747 p^{132} T^{11} + 11092669 p^{156} T^{12} + 6119 p^{182} T^{13} + p^{210} T^{14} \) |
| good | 3 | \( 1 - 360048997946698 p T^{2} + \)\(80\!\cdots\!59\)\( p^{6} T^{4} - \)\(39\!\cdots\!44\)\( p^{12} T^{6} + \)\(48\!\cdots\!83\)\( p^{19} T^{8} - \)\(20\!\cdots\!54\)\( p^{31} T^{10} + \)\(23\!\cdots\!03\)\( p^{42} T^{12} - \)\(80\!\cdots\!08\)\( p^{52} T^{14} + \)\(23\!\cdots\!03\)\( p^{102} T^{16} - \)\(20\!\cdots\!54\)\( p^{151} T^{18} + \)\(48\!\cdots\!83\)\( p^{199} T^{20} - \)\(39\!\cdots\!44\)\( p^{252} T^{22} + \)\(80\!\cdots\!59\)\( p^{306} T^{24} - 360048997946698 p^{361} T^{26} + p^{420} T^{28} \) |
| 5 | \( ( 1 - 1486479854 p T + \)\(52\!\cdots\!91\)\( p T^{2} - \)\(25\!\cdots\!08\)\( p^{4} T^{3} + \)\(85\!\cdots\!89\)\( p^{8} T^{4} + \)\(74\!\cdots\!18\)\( p^{11} T^{5} + \)\(10\!\cdots\!39\)\( p^{15} T^{6} + \)\(24\!\cdots\!52\)\( p^{20} T^{7} + \)\(10\!\cdots\!39\)\( p^{45} T^{8} + \)\(74\!\cdots\!18\)\( p^{71} T^{9} + \)\(85\!\cdots\!89\)\( p^{98} T^{10} - \)\(25\!\cdots\!08\)\( p^{124} T^{11} + \)\(52\!\cdots\!91\)\( p^{151} T^{12} - 1486479854 p^{181} T^{13} + p^{210} T^{14} )^{2} \) | |
| 7 | \( 1 - \)\(17\!\cdots\!94\)\( T^{2} + \)\(28\!\cdots\!39\)\( p^{2} T^{4} - \)\(40\!\cdots\!72\)\( p^{5} T^{6} + \)\(50\!\cdots\!23\)\( p^{9} T^{8} - \)\(55\!\cdots\!62\)\( p^{14} T^{10} + \)\(28\!\cdots\!89\)\( p^{19} T^{12} - \)\(51\!\cdots\!28\)\( p^{24} T^{14} + \)\(28\!\cdots\!89\)\( p^{79} T^{16} - \)\(55\!\cdots\!62\)\( p^{134} T^{18} + \)\(50\!\cdots\!23\)\( p^{189} T^{20} - \)\(40\!\cdots\!72\)\( p^{245} T^{22} + \)\(28\!\cdots\!39\)\( p^{302} T^{24} - \)\(17\!\cdots\!94\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 11 | \( 1 - \)\(11\!\cdots\!14\)\( T^{2} + \)\(68\!\cdots\!91\)\( T^{4} - \)\(20\!\cdots\!84\)\( p^{2} T^{6} + \)\(40\!\cdots\!61\)\( p^{4} T^{8} - \)\(53\!\cdots\!62\)\( p^{7} T^{10} + \)\(46\!\cdots\!43\)\( p^{12} T^{12} - \)\(51\!\cdots\!32\)\( p^{15} T^{14} + \)\(46\!\cdots\!43\)\( p^{72} T^{16} - \)\(53\!\cdots\!62\)\( p^{127} T^{18} + \)\(40\!\cdots\!61\)\( p^{184} T^{20} - \)\(20\!\cdots\!84\)\( p^{242} T^{22} + \)\(68\!\cdots\!91\)\( p^{300} T^{24} - \)\(11\!\cdots\!14\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 13 | \( ( 1 - 36736947062467766 T + \)\(74\!\cdots\!87\)\( p T^{2} - \)\(27\!\cdots\!88\)\( p^{2} T^{3} + \)\(24\!\cdots\!41\)\( p^{3} T^{4} - \)\(67\!\cdots\!78\)\( p^{5} T^{5} + \)\(31\!\cdots\!51\)\( p^{7} T^{6} - \)\(78\!\cdots\!92\)\( p^{9} T^{7} + \)\(31\!\cdots\!51\)\( p^{37} T^{8} - \)\(67\!\cdots\!78\)\( p^{65} T^{9} + \)\(24\!\cdots\!41\)\( p^{93} T^{10} - \)\(27\!\cdots\!88\)\( p^{122} T^{11} + \)\(74\!\cdots\!87\)\( p^{151} T^{12} - 36736947062467766 p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
| 17 | \( ( 1 + 2208600866705597906 T + \)\(30\!\cdots\!71\)\( T^{2} + \)\(42\!\cdots\!72\)\( T^{3} + \)\(27\!\cdots\!01\)\( p T^{4} + \)\(17\!\cdots\!26\)\( p^{2} T^{5} + \)\(10\!\cdots\!59\)\( p^{3} T^{6} + \)\(61\!\cdots\!96\)\( p^{4} T^{7} + \)\(10\!\cdots\!59\)\( p^{33} T^{8} + \)\(17\!\cdots\!26\)\( p^{62} T^{9} + \)\(27\!\cdots\!01\)\( p^{91} T^{10} + \)\(42\!\cdots\!72\)\( p^{120} T^{11} + \)\(30\!\cdots\!71\)\( p^{150} T^{12} + 2208600866705597906 p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
| 19 | \( 1 - \)\(10\!\cdots\!14\)\( T^{2} + \)\(71\!\cdots\!91\)\( T^{4} - \)\(18\!\cdots\!56\)\( p T^{6} + \)\(20\!\cdots\!39\)\( p^{3} T^{8} - \)\(18\!\cdots\!98\)\( p^{5} T^{10} + \)\(76\!\cdots\!83\)\( p^{8} T^{12} - \)\(99\!\cdots\!08\)\( p^{9} T^{14} + \)\(76\!\cdots\!83\)\( p^{68} T^{16} - \)\(18\!\cdots\!98\)\( p^{125} T^{18} + \)\(20\!\cdots\!39\)\( p^{183} T^{20} - \)\(18\!\cdots\!56\)\( p^{241} T^{22} + \)\(71\!\cdots\!91\)\( p^{300} T^{24} - \)\(10\!\cdots\!14\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 23 | \( 1 - \)\(54\!\cdots\!94\)\( T^{2} + \)\(64\!\cdots\!57\)\( p T^{4} - \)\(21\!\cdots\!12\)\( p^{3} T^{6} + \)\(56\!\cdots\!27\)\( p^{5} T^{8} - \)\(11\!\cdots\!54\)\( p^{7} T^{10} + \)\(20\!\cdots\!29\)\( p^{9} T^{12} - \)\(29\!\cdots\!64\)\( p^{11} T^{14} + \)\(20\!\cdots\!29\)\( p^{69} T^{16} - \)\(11\!\cdots\!54\)\( p^{127} T^{18} + \)\(56\!\cdots\!27\)\( p^{185} T^{20} - \)\(21\!\cdots\!12\)\( p^{243} T^{22} + \)\(64\!\cdots\!57\)\( p^{301} T^{24} - \)\(54\!\cdots\!94\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 29 | \( ( 1 + 89364252316431140914 p T + \)\(32\!\cdots\!71\)\( p^{2} T^{2} + \)\(11\!\cdots\!64\)\( p^{3} T^{3} + \)\(53\!\cdots\!81\)\( p^{4} T^{4} + \)\(77\!\cdots\!38\)\( p^{5} T^{5} + \)\(61\!\cdots\!87\)\( p^{6} T^{6} + \)\(56\!\cdots\!08\)\( p^{7} T^{7} + \)\(61\!\cdots\!87\)\( p^{36} T^{8} + \)\(77\!\cdots\!38\)\( p^{65} T^{9} + \)\(53\!\cdots\!81\)\( p^{94} T^{10} + \)\(11\!\cdots\!64\)\( p^{123} T^{11} + \)\(32\!\cdots\!71\)\( p^{152} T^{12} + 89364252316431140914 p^{181} T^{13} + p^{210} T^{14} )^{2} \) | |
| 31 | \( 1 - \)\(44\!\cdots\!14\)\( T^{2} + \)\(10\!\cdots\!91\)\( T^{4} - \)\(16\!\cdots\!64\)\( T^{6} + \)\(19\!\cdots\!01\)\( T^{8} - \)\(18\!\cdots\!02\)\( T^{10} + \)\(13\!\cdots\!03\)\( T^{12} - \)\(84\!\cdots\!32\)\( T^{14} + \)\(13\!\cdots\!03\)\( p^{60} T^{16} - \)\(18\!\cdots\!02\)\( p^{120} T^{18} + \)\(19\!\cdots\!01\)\( p^{180} T^{20} - \)\(16\!\cdots\!64\)\( p^{240} T^{22} + \)\(10\!\cdots\!91\)\( p^{300} T^{24} - \)\(44\!\cdots\!14\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 37 | \( ( 1 + \)\(31\!\cdots\!46\)\( T + \)\(48\!\cdots\!11\)\( T^{2} + \)\(11\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!97\)\( T^{4} + \)\(20\!\cdots\!54\)\( T^{5} + \)\(14\!\cdots\!87\)\( T^{6} + \)\(24\!\cdots\!76\)\( T^{7} + \)\(14\!\cdots\!87\)\( p^{30} T^{8} + \)\(20\!\cdots\!54\)\( p^{60} T^{9} + \)\(10\!\cdots\!97\)\( p^{90} T^{10} + \)\(11\!\cdots\!12\)\( p^{120} T^{11} + \)\(48\!\cdots\!11\)\( p^{150} T^{12} + \)\(31\!\cdots\!46\)\( p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
| 41 | \( ( 1 - \)\(74\!\cdots\!14\)\( T + \)\(12\!\cdots\!91\)\( T^{2} - \)\(95\!\cdots\!64\)\( T^{3} + \)\(76\!\cdots\!01\)\( T^{4} - \)\(52\!\cdots\!02\)\( T^{5} + \)\(28\!\cdots\!03\)\( T^{6} - \)\(16\!\cdots\!32\)\( T^{7} + \)\(28\!\cdots\!03\)\( p^{30} T^{8} - \)\(52\!\cdots\!02\)\( p^{60} T^{9} + \)\(76\!\cdots\!01\)\( p^{90} T^{10} - \)\(95\!\cdots\!64\)\( p^{120} T^{11} + \)\(12\!\cdots\!91\)\( p^{150} T^{12} - \)\(74\!\cdots\!14\)\( p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
| 43 | \( 1 - \)\(92\!\cdots\!94\)\( T^{2} + \)\(42\!\cdots\!11\)\( T^{4} - \)\(12\!\cdots\!04\)\( T^{6} + \)\(28\!\cdots\!61\)\( T^{8} - \)\(11\!\cdots\!66\)\( p T^{10} + \)\(67\!\cdots\!27\)\( T^{12} - \)\(74\!\cdots\!28\)\( T^{14} + \)\(67\!\cdots\!27\)\( p^{60} T^{16} - \)\(11\!\cdots\!66\)\( p^{121} T^{18} + \)\(28\!\cdots\!61\)\( p^{180} T^{20} - \)\(12\!\cdots\!04\)\( p^{240} T^{22} + \)\(42\!\cdots\!11\)\( p^{300} T^{24} - \)\(92\!\cdots\!94\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 47 | \( 1 - \)\(10\!\cdots\!94\)\( T^{2} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{6} + \)\(49\!\cdots\!61\)\( T^{8} - \)\(10\!\cdots\!38\)\( T^{10} + \)\(20\!\cdots\!27\)\( T^{12} - \)\(32\!\cdots\!28\)\( T^{14} + \)\(20\!\cdots\!27\)\( p^{60} T^{16} - \)\(10\!\cdots\!38\)\( p^{120} T^{18} + \)\(49\!\cdots\!61\)\( p^{180} T^{20} - \)\(18\!\cdots\!04\)\( p^{240} T^{22} + \)\(51\!\cdots\!11\)\( p^{300} T^{24} - \)\(10\!\cdots\!94\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 53 | \( ( 1 + \)\(81\!\cdots\!34\)\( T + \)\(16\!\cdots\!31\)\( T^{2} + \)\(10\!\cdots\!68\)\( T^{3} + \)\(13\!\cdots\!37\)\( T^{4} + \)\(78\!\cdots\!66\)\( T^{5} + \)\(83\!\cdots\!07\)\( T^{6} + \)\(41\!\cdots\!64\)\( T^{7} + \)\(83\!\cdots\!07\)\( p^{30} T^{8} + \)\(78\!\cdots\!66\)\( p^{60} T^{9} + \)\(13\!\cdots\!37\)\( p^{90} T^{10} + \)\(10\!\cdots\!68\)\( p^{120} T^{11} + \)\(16\!\cdots\!31\)\( p^{150} T^{12} + \)\(81\!\cdots\!34\)\( p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
| 59 | \( 1 - \)\(53\!\cdots\!14\)\( T^{2} + \)\(17\!\cdots\!91\)\( T^{4} - \)\(37\!\cdots\!64\)\( T^{6} + \)\(60\!\cdots\!01\)\( T^{8} - \)\(74\!\cdots\!02\)\( T^{10} + \)\(73\!\cdots\!03\)\( T^{12} - \)\(82\!\cdots\!32\)\( T^{14} + \)\(73\!\cdots\!03\)\( p^{60} T^{16} - \)\(74\!\cdots\!02\)\( p^{120} T^{18} + \)\(60\!\cdots\!01\)\( p^{180} T^{20} - \)\(37\!\cdots\!64\)\( p^{240} T^{22} + \)\(17\!\cdots\!91\)\( p^{300} T^{24} - \)\(53\!\cdots\!14\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 61 | \( ( 1 + \)\(39\!\cdots\!86\)\( T + \)\(18\!\cdots\!91\)\( T^{2} + \)\(55\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!01\)\( T^{4} + \)\(39\!\cdots\!98\)\( T^{5} + \)\(84\!\cdots\!03\)\( T^{6} + \)\(17\!\cdots\!68\)\( T^{7} + \)\(84\!\cdots\!03\)\( p^{30} T^{8} + \)\(39\!\cdots\!98\)\( p^{60} T^{9} + \)\(15\!\cdots\!01\)\( p^{90} T^{10} + \)\(55\!\cdots\!36\)\( p^{120} T^{11} + \)\(18\!\cdots\!91\)\( p^{150} T^{12} + \)\(39\!\cdots\!86\)\( p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
| 67 | \( 1 - \)\(41\!\cdots\!94\)\( T^{2} + \)\(90\!\cdots\!11\)\( T^{4} - \)\(13\!\cdots\!04\)\( T^{6} + \)\(15\!\cdots\!61\)\( T^{8} - \)\(14\!\cdots\!38\)\( T^{10} + \)\(11\!\cdots\!27\)\( T^{12} - \)\(76\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!27\)\( p^{60} T^{16} - \)\(14\!\cdots\!38\)\( p^{120} T^{18} + \)\(15\!\cdots\!61\)\( p^{180} T^{20} - \)\(13\!\cdots\!04\)\( p^{240} T^{22} + \)\(90\!\cdots\!11\)\( p^{300} T^{24} - \)\(41\!\cdots\!94\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 71 | \( 1 - \)\(29\!\cdots\!14\)\( T^{2} + \)\(45\!\cdots\!91\)\( T^{4} - \)\(45\!\cdots\!64\)\( T^{6} + \)\(33\!\cdots\!01\)\( T^{8} - \)\(19\!\cdots\!02\)\( T^{10} + \)\(89\!\cdots\!03\)\( T^{12} - \)\(34\!\cdots\!32\)\( T^{14} + \)\(89\!\cdots\!03\)\( p^{60} T^{16} - \)\(19\!\cdots\!02\)\( p^{120} T^{18} + \)\(33\!\cdots\!01\)\( p^{180} T^{20} - \)\(45\!\cdots\!64\)\( p^{240} T^{22} + \)\(45\!\cdots\!91\)\( p^{300} T^{24} - \)\(29\!\cdots\!14\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 73 | \( ( 1 + \)\(69\!\cdots\!54\)\( T + \)\(27\!\cdots\!11\)\( T^{2} + \)\(13\!\cdots\!68\)\( T^{3} + \)\(34\!\cdots\!17\)\( T^{4} + \)\(80\!\cdots\!26\)\( T^{5} + \)\(28\!\cdots\!27\)\( T^{6} + \)\(31\!\cdots\!04\)\( T^{7} + \)\(28\!\cdots\!27\)\( p^{30} T^{8} + \)\(80\!\cdots\!26\)\( p^{60} T^{9} + \)\(34\!\cdots\!17\)\( p^{90} T^{10} + \)\(13\!\cdots\!68\)\( p^{120} T^{11} + \)\(27\!\cdots\!11\)\( p^{150} T^{12} + \)\(69\!\cdots\!54\)\( p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
| 79 | \( 1 - \)\(76\!\cdots\!14\)\( T^{2} + \)\(28\!\cdots\!91\)\( T^{4} - \)\(70\!\cdots\!64\)\( T^{6} + \)\(12\!\cdots\!01\)\( T^{8} - \)\(17\!\cdots\!02\)\( T^{10} + \)\(20\!\cdots\!03\)\( T^{12} - \)\(19\!\cdots\!32\)\( T^{14} + \)\(20\!\cdots\!03\)\( p^{60} T^{16} - \)\(17\!\cdots\!02\)\( p^{120} T^{18} + \)\(12\!\cdots\!01\)\( p^{180} T^{20} - \)\(70\!\cdots\!64\)\( p^{240} T^{22} + \)\(28\!\cdots\!91\)\( p^{300} T^{24} - \)\(76\!\cdots\!14\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 83 | \( 1 - \)\(31\!\cdots\!94\)\( T^{2} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(55\!\cdots\!04\)\( T^{6} + \)\(44\!\cdots\!61\)\( T^{8} - \)\(28\!\cdots\!38\)\( T^{10} + \)\(14\!\cdots\!27\)\( T^{12} - \)\(59\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!27\)\( p^{60} T^{16} - \)\(28\!\cdots\!38\)\( p^{120} T^{18} + \)\(44\!\cdots\!61\)\( p^{180} T^{20} - \)\(55\!\cdots\!04\)\( p^{240} T^{22} + \)\(51\!\cdots\!11\)\( p^{300} T^{24} - \)\(31\!\cdots\!94\)\( p^{360} T^{26} + p^{420} T^{28} \) | |
| 89 | \( ( 1 - \)\(49\!\cdots\!94\)\( T + \)\(10\!\cdots\!11\)\( T^{2} + \)\(32\!\cdots\!96\)\( T^{3} + \)\(52\!\cdots\!61\)\( T^{4} + \)\(43\!\cdots\!62\)\( T^{5} + \)\(19\!\cdots\!27\)\( T^{6} + \)\(18\!\cdots\!72\)\( T^{7} + \)\(19\!\cdots\!27\)\( p^{30} T^{8} + \)\(43\!\cdots\!62\)\( p^{60} T^{9} + \)\(52\!\cdots\!61\)\( p^{90} T^{10} + \)\(32\!\cdots\!96\)\( p^{120} T^{11} + \)\(10\!\cdots\!11\)\( p^{150} T^{12} - \)\(49\!\cdots\!94\)\( p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
| 97 | \( ( 1 + \)\(51\!\cdots\!66\)\( T + \)\(18\!\cdots\!31\)\( T^{2} + \)\(52\!\cdots\!12\)\( T^{3} + \)\(15\!\cdots\!17\)\( T^{4} + \)\(23\!\cdots\!54\)\( T^{5} + \)\(82\!\cdots\!07\)\( T^{6} + \)\(85\!\cdots\!36\)\( T^{7} + \)\(82\!\cdots\!07\)\( p^{30} T^{8} + \)\(23\!\cdots\!54\)\( p^{60} T^{9} + \)\(15\!\cdots\!17\)\( p^{90} T^{10} + \)\(52\!\cdots\!12\)\( p^{120} T^{11} + \)\(18\!\cdots\!31\)\( p^{150} T^{12} + \)\(51\!\cdots\!66\)\( p^{180} T^{13} + p^{210} T^{14} )^{2} \) | |
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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
−3.59406682790909835561701720957, −3.41687732261880330921837622351, −3.36874845314533794741329050892, −3.32507947119376303688436073393, −2.89696560593351002492719260762, −2.50583043757164586851803345580, −2.35905368187672395096709788212, −2.33427962320729293997423161588, −2.30023310661525568401436291027, −2.27455263409269599240721214980, −2.14155723968586459212124363812, −1.70135151736718443727783498521, −1.63435991863808669470619861024, −1.56156658322239310380327328511, −1.55002605541363024802057154338, −1.54167435546098276669630813328, −1.33201035150437085966084173361, −1.07814525830191649607035030700, −0.977215028155075648973096024427, −0.825266930033462763756761978483, −0.58440785191142448839047639949, −0.55069287761485674810682094466, −0.33555476370600057654406371424, −0.32308620357347413236557239655, −0.13945143904763071298612477702, 0.13945143904763071298612477702, 0.32308620357347413236557239655, 0.33555476370600057654406371424, 0.55069287761485674810682094466, 0.58440785191142448839047639949, 0.825266930033462763756761978483, 0.977215028155075648973096024427, 1.07814525830191649607035030700, 1.33201035150437085966084173361, 1.54167435546098276669630813328, 1.55002605541363024802057154338, 1.56156658322239310380327328511, 1.63435991863808669470619861024, 1.70135151736718443727783498521, 2.14155723968586459212124363812, 2.27455263409269599240721214980, 2.30023310661525568401436291027, 2.33427962320729293997423161588, 2.35905368187672395096709788212, 2.50583043757164586851803345580, 2.89696560593351002492719260762, 3.32507947119376303688436073393, 3.36874845314533794741329050892, 3.41687732261880330921837622351, 3.59406682790909835561701720957
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.