Dirichlet series
| L(s) = 1 | − 2.73e4·2-s − 9.64e9·4-s − 2.13e10·5-s + 1.03e15·8-s + 9.95e16·9-s + 5.85e14·10-s − 5.23e18·13-s + 1.49e20·16-s − 6.03e20·17-s − 2.72e21·18-s + 2.06e20·20-s − 4.04e24·25-s + 1.43e23·26-s − 4.08e24·29-s − 8.06e24·32-s + 1.65e25·34-s − 9.59e26·36-s + 9.50e26·37-s − 2.20e25·40-s + 4.69e26·41-s − 2.12e27·45-s + 4.30e29·49-s + 1.10e29·50-s + 5.04e28·52-s − 2.87e29·53-s + 1.11e29·58-s − 7.36e30·61-s + ⋯ |
| L(s) = 1 | − 0.208·2-s − 0.561·4-s − 0.0280·5-s + 0.458·8-s + 5.96·9-s + 0.00585·10-s − 0.605·13-s + 0.506·16-s − 0.729·17-s − 1.24·18-s + 0.0157·20-s − 6.94·25-s + 0.126·26-s − 0.563·29-s − 0.208·32-s + 0.152·34-s − 3.34·36-s + 2.08·37-s − 0.0128·40-s + 0.179·41-s − 0.167·45-s + 7.94·49-s + 1.45·50-s + 0.339·52-s − 1.39·53-s + 0.117·58-s − 3.28·61-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(35-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+17)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{32}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.93475\times 10^{23}\) |
| Root analytic conductor: | \(5.41204\) |
| Motivic weight: | \(34\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{32} ,\ ( \ : [17]^{16} ),\ 1 )\) |
Particular Values
| \(L(\frac{35}{2})\) | \(\approx\) | \(0.03294773202\) |
| \(L(\frac12)\) | \(\approx\) | \(0.03294773202\) |
| \(L(18)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 6843 p^{2} T + 162360169 p^{6} T^{2} - 118131047133 p^{12} T^{3} - 86496181830827 p^{20} T^{4} - 12944117609387468 p^{30} T^{5} - 80080044589748272 p^{42} T^{6} + 11585454353502851072 p^{56} T^{7} + 1256064703040490240 p^{72} T^{8} + 11585454353502851072 p^{90} T^{9} - 80080044589748272 p^{110} T^{10} - 12944117609387468 p^{132} T^{11} - 86496181830827 p^{156} T^{12} - 118131047133 p^{182} T^{13} + 162360169 p^{210} T^{14} + 6843 p^{240} T^{15} + p^{272} T^{16} \) |
| good | 3 | \( 1 - 11055908974961168 p^{2} T^{2} + \)\(21\!\cdots\!48\)\( p^{5} T^{4} - \)\(34\!\cdots\!84\)\( p^{10} T^{6} + \)\(55\!\cdots\!72\)\( p^{19} T^{8} - \)\(83\!\cdots\!60\)\( p^{30} T^{10} + \)\(35\!\cdots\!88\)\( p^{42} T^{12} - \)\(15\!\cdots\!12\)\( p^{56} T^{14} + \)\(18\!\cdots\!94\)\( p^{71} T^{16} - \)\(15\!\cdots\!12\)\( p^{124} T^{18} + \)\(35\!\cdots\!88\)\( p^{178} T^{20} - \)\(83\!\cdots\!60\)\( p^{234} T^{22} + \)\(55\!\cdots\!72\)\( p^{291} T^{24} - \)\(34\!\cdots\!84\)\( p^{350} T^{26} + \)\(21\!\cdots\!48\)\( p^{413} T^{28} - 11055908974961168 p^{478} T^{30} + p^{544} T^{32} \) |
| 5 | \( ( 1 + 2137225584 p T + \)\(40\!\cdots\!12\)\( p T^{2} - \)\(98\!\cdots\!72\)\( p^{3} T^{3} + \)\(29\!\cdots\!84\)\( p^{7} T^{4} - \)\(22\!\cdots\!64\)\( p^{10} T^{5} + \)\(30\!\cdots\!92\)\( p^{14} T^{6} - \)\(11\!\cdots\!72\)\( p^{19} T^{7} + \)\(20\!\cdots\!06\)\( p^{24} T^{8} - \)\(11\!\cdots\!72\)\( p^{53} T^{9} + \)\(30\!\cdots\!92\)\( p^{82} T^{10} - \)\(22\!\cdots\!64\)\( p^{112} T^{11} + \)\(29\!\cdots\!84\)\( p^{143} T^{12} - \)\(98\!\cdots\!72\)\( p^{173} T^{13} + \)\(40\!\cdots\!12\)\( p^{205} T^{14} + 2137225584 p^{239} T^{15} + p^{272} T^{16} )^{2} \) | |
| 7 | \( 1 - \)\(61\!\cdots\!16\)\( p T^{2} + \)\(28\!\cdots\!68\)\( p^{3} T^{4} - \)\(13\!\cdots\!64\)\( p^{6} T^{6} + \)\(47\!\cdots\!52\)\( p^{9} T^{8} - \)\(13\!\cdots\!40\)\( p^{12} T^{10} + \)\(46\!\cdots\!72\)\( p^{16} T^{12} - \)\(27\!\cdots\!88\)\( p^{22} T^{14} + \)\(27\!\cdots\!22\)\( p^{30} T^{16} - \)\(27\!\cdots\!88\)\( p^{90} T^{18} + \)\(46\!\cdots\!72\)\( p^{152} T^{20} - \)\(13\!\cdots\!40\)\( p^{216} T^{22} + \)\(47\!\cdots\!52\)\( p^{281} T^{24} - \)\(13\!\cdots\!64\)\( p^{346} T^{26} + \)\(28\!\cdots\!68\)\( p^{411} T^{28} - \)\(61\!\cdots\!16\)\( p^{477} T^{30} + p^{544} T^{32} \) | |
| 11 | \( 1 - \)\(17\!\cdots\!96\)\( p T^{2} + \)\(14\!\cdots\!00\)\( p^{3} T^{4} - \)\(82\!\cdots\!20\)\( p^{5} T^{6} + \)\(33\!\cdots\!00\)\( p^{8} T^{8} - \)\(10\!\cdots\!88\)\( p^{11} T^{10} + \)\(30\!\cdots\!08\)\( p^{14} T^{12} - \)\(71\!\cdots\!00\)\( p^{17} T^{14} + \)\(14\!\cdots\!90\)\( p^{20} T^{16} - \)\(71\!\cdots\!00\)\( p^{85} T^{18} + \)\(30\!\cdots\!08\)\( p^{150} T^{20} - \)\(10\!\cdots\!88\)\( p^{215} T^{22} + \)\(33\!\cdots\!00\)\( p^{280} T^{24} - \)\(82\!\cdots\!20\)\( p^{345} T^{26} + \)\(14\!\cdots\!00\)\( p^{411} T^{28} - \)\(17\!\cdots\!96\)\( p^{477} T^{30} + p^{544} T^{32} \) | |
| 13 | \( ( 1 + 201288732169679216 p T + \)\(29\!\cdots\!16\)\( T^{2} + \)\(14\!\cdots\!96\)\( p T^{3} + \)\(26\!\cdots\!32\)\( p^{2} T^{4} + \)\(17\!\cdots\!08\)\( p^{3} T^{5} + \)\(15\!\cdots\!88\)\( p^{5} T^{6} + \)\(61\!\cdots\!68\)\( p^{7} T^{7} + \)\(48\!\cdots\!54\)\( p^{9} T^{8} + \)\(61\!\cdots\!68\)\( p^{41} T^{9} + \)\(15\!\cdots\!88\)\( p^{73} T^{10} + \)\(17\!\cdots\!08\)\( p^{105} T^{11} + \)\(26\!\cdots\!32\)\( p^{138} T^{12} + \)\(14\!\cdots\!96\)\( p^{171} T^{13} + \)\(29\!\cdots\!16\)\( p^{204} T^{14} + 201288732169679216 p^{239} T^{15} + p^{272} T^{16} )^{2} \) | |
| 17 | \( ( 1 + 17755026644685586416 p T + \)\(16\!\cdots\!28\)\( p T^{2} + \)\(16\!\cdots\!28\)\( p^{2} T^{3} + \)\(15\!\cdots\!92\)\( p^{2} T^{4} + \)\(12\!\cdots\!08\)\( p^{3} T^{5} + \)\(97\!\cdots\!08\)\( p^{3} T^{6} + \)\(65\!\cdots\!44\)\( p^{4} T^{7} + \)\(45\!\cdots\!42\)\( p^{4} T^{8} + \)\(65\!\cdots\!44\)\( p^{38} T^{9} + \)\(97\!\cdots\!08\)\( p^{71} T^{10} + \)\(12\!\cdots\!08\)\( p^{105} T^{11} + \)\(15\!\cdots\!92\)\( p^{138} T^{12} + \)\(16\!\cdots\!28\)\( p^{172} T^{13} + \)\(16\!\cdots\!28\)\( p^{205} T^{14} + 17755026644685586416 p^{239} T^{15} + p^{272} T^{16} )^{2} \) | |
| 19 | \( 1 - \)\(21\!\cdots\!16\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{4} - \)\(23\!\cdots\!00\)\( T^{6} + \)\(81\!\cdots\!80\)\( p T^{8} - \)\(12\!\cdots\!72\)\( p^{3} T^{10} + \)\(15\!\cdots\!72\)\( p^{5} T^{12} - \)\(16\!\cdots\!60\)\( p^{7} T^{14} + \)\(14\!\cdots\!50\)\( p^{9} T^{16} - \)\(16\!\cdots\!60\)\( p^{75} T^{18} + \)\(15\!\cdots\!72\)\( p^{141} T^{20} - \)\(12\!\cdots\!72\)\( p^{207} T^{22} + \)\(81\!\cdots\!80\)\( p^{273} T^{24} - \)\(23\!\cdots\!00\)\( p^{340} T^{26} + \)\(26\!\cdots\!20\)\( p^{408} T^{28} - \)\(21\!\cdots\!16\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 23 | \( 1 - \)\(72\!\cdots\!32\)\( T^{2} + \)\(14\!\cdots\!08\)\( p T^{4} - \)\(22\!\cdots\!44\)\( p^{2} T^{6} + \)\(30\!\cdots\!72\)\( p^{3} T^{8} - \)\(15\!\cdots\!80\)\( p^{5} T^{10} + \)\(72\!\cdots\!96\)\( p^{7} T^{12} - \)\(31\!\cdots\!24\)\( p^{9} T^{14} + \)\(12\!\cdots\!74\)\( p^{11} T^{16} - \)\(31\!\cdots\!24\)\( p^{77} T^{18} + \)\(72\!\cdots\!96\)\( p^{143} T^{20} - \)\(15\!\cdots\!80\)\( p^{209} T^{22} + \)\(30\!\cdots\!72\)\( p^{275} T^{24} - \)\(22\!\cdots\!44\)\( p^{342} T^{26} + \)\(14\!\cdots\!08\)\( p^{409} T^{28} - \)\(72\!\cdots\!32\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 29 | \( ( 1 + \)\(20\!\cdots\!28\)\( T + \)\(74\!\cdots\!96\)\( p T^{2} + \)\(43\!\cdots\!84\)\( p^{2} T^{3} + \)\(32\!\cdots\!44\)\( p^{4} T^{4} - \)\(23\!\cdots\!60\)\( p^{4} T^{5} + \)\(88\!\cdots\!48\)\( p^{5} T^{6} - \)\(17\!\cdots\!32\)\( p^{6} T^{7} + \)\(65\!\cdots\!82\)\( p^{7} T^{8} - \)\(17\!\cdots\!32\)\( p^{40} T^{9} + \)\(88\!\cdots\!48\)\( p^{73} T^{10} - \)\(23\!\cdots\!60\)\( p^{106} T^{11} + \)\(32\!\cdots\!44\)\( p^{140} T^{12} + \)\(43\!\cdots\!84\)\( p^{172} T^{13} + \)\(74\!\cdots\!96\)\( p^{205} T^{14} + \)\(20\!\cdots\!28\)\( p^{238} T^{15} + p^{272} T^{16} )^{2} \) | |
| 31 | \( 1 - \)\(10\!\cdots\!36\)\( p T^{2} + \)\(18\!\cdots\!20\)\( p^{3} T^{4} - \)\(22\!\cdots\!00\)\( p^{5} T^{6} + \)\(23\!\cdots\!20\)\( p^{7} T^{8} - \)\(19\!\cdots\!88\)\( p^{9} T^{10} + \)\(14\!\cdots\!88\)\( p^{11} T^{12} - \)\(89\!\cdots\!40\)\( p^{13} T^{14} + \)\(50\!\cdots\!50\)\( p^{15} T^{16} - \)\(89\!\cdots\!40\)\( p^{81} T^{18} + \)\(14\!\cdots\!88\)\( p^{147} T^{20} - \)\(19\!\cdots\!88\)\( p^{213} T^{22} + \)\(23\!\cdots\!20\)\( p^{279} T^{24} - \)\(22\!\cdots\!00\)\( p^{345} T^{26} + \)\(18\!\cdots\!20\)\( p^{411} T^{28} - \)\(10\!\cdots\!36\)\( p^{477} T^{30} + p^{544} T^{32} \) | |
| 37 | \( ( 1 - \)\(47\!\cdots\!28\)\( T + \)\(89\!\cdots\!96\)\( T^{2} - \)\(42\!\cdots\!48\)\( T^{3} + \)\(40\!\cdots\!48\)\( T^{4} - \)\(52\!\cdots\!28\)\( p T^{5} + \)\(12\!\cdots\!84\)\( T^{6} - \)\(59\!\cdots\!56\)\( T^{7} + \)\(30\!\cdots\!02\)\( T^{8} - \)\(59\!\cdots\!56\)\( p^{34} T^{9} + \)\(12\!\cdots\!84\)\( p^{68} T^{10} - \)\(52\!\cdots\!28\)\( p^{103} T^{11} + \)\(40\!\cdots\!48\)\( p^{136} T^{12} - \)\(42\!\cdots\!48\)\( p^{170} T^{13} + \)\(89\!\cdots\!96\)\( p^{204} T^{14} - \)\(47\!\cdots\!28\)\( p^{238} T^{15} + p^{272} T^{16} )^{2} \) | |
| 41 | \( ( 1 - \)\(23\!\cdots\!96\)\( T + \)\(37\!\cdots\!40\)\( T^{2} - \)\(29\!\cdots\!00\)\( T^{3} + \)\(67\!\cdots\!40\)\( T^{4} - \)\(19\!\cdots\!68\)\( T^{5} + \)\(77\!\cdots\!48\)\( T^{6} - \)\(12\!\cdots\!80\)\( T^{7} + \)\(62\!\cdots\!50\)\( T^{8} - \)\(12\!\cdots\!80\)\( p^{34} T^{9} + \)\(77\!\cdots\!48\)\( p^{68} T^{10} - \)\(19\!\cdots\!68\)\( p^{102} T^{11} + \)\(67\!\cdots\!40\)\( p^{136} T^{12} - \)\(29\!\cdots\!00\)\( p^{170} T^{13} + \)\(37\!\cdots\!40\)\( p^{204} T^{14} - \)\(23\!\cdots\!96\)\( p^{238} T^{15} + p^{272} T^{16} )^{2} \) | |
| 43 | \( 1 - \)\(42\!\cdots\!72\)\( T^{2} + \)\(86\!\cdots\!24\)\( T^{4} - \)\(11\!\cdots\!16\)\( T^{6} + \)\(10\!\cdots\!64\)\( T^{8} - \)\(76\!\cdots\!00\)\( T^{10} + \)\(43\!\cdots\!52\)\( T^{12} - \)\(20\!\cdots\!12\)\( T^{14} + \)\(76\!\cdots\!18\)\( T^{16} - \)\(20\!\cdots\!12\)\( p^{68} T^{18} + \)\(43\!\cdots\!52\)\( p^{136} T^{20} - \)\(76\!\cdots\!00\)\( p^{204} T^{22} + \)\(10\!\cdots\!64\)\( p^{272} T^{24} - \)\(11\!\cdots\!16\)\( p^{340} T^{26} + \)\(86\!\cdots\!24\)\( p^{408} T^{28} - \)\(42\!\cdots\!72\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 47 | \( 1 - \)\(48\!\cdots\!32\)\( T^{2} + \)\(13\!\cdots\!44\)\( T^{4} - \)\(25\!\cdots\!36\)\( T^{6} + \)\(37\!\cdots\!04\)\( T^{8} - \)\(46\!\cdots\!60\)\( T^{10} + \)\(48\!\cdots\!52\)\( T^{12} - \)\(42\!\cdots\!92\)\( T^{14} + \)\(32\!\cdots\!78\)\( T^{16} - \)\(42\!\cdots\!92\)\( p^{68} T^{18} + \)\(48\!\cdots\!52\)\( p^{136} T^{20} - \)\(46\!\cdots\!60\)\( p^{204} T^{22} + \)\(37\!\cdots\!04\)\( p^{272} T^{24} - \)\(25\!\cdots\!36\)\( p^{340} T^{26} + \)\(13\!\cdots\!44\)\( p^{408} T^{28} - \)\(48\!\cdots\!32\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 53 | \( ( 1 + \)\(14\!\cdots\!68\)\( T + \)\(20\!\cdots\!36\)\( T^{2} + \)\(37\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!88\)\( T^{4} + \)\(45\!\cdots\!16\)\( T^{5} + \)\(12\!\cdots\!04\)\( T^{6} + \)\(30\!\cdots\!76\)\( T^{7} + \)\(59\!\cdots\!22\)\( T^{8} + \)\(30\!\cdots\!76\)\( p^{34} T^{9} + \)\(12\!\cdots\!04\)\( p^{68} T^{10} + \)\(45\!\cdots\!16\)\( p^{102} T^{11} + \)\(19\!\cdots\!88\)\( p^{136} T^{12} + \)\(37\!\cdots\!28\)\( p^{170} T^{13} + \)\(20\!\cdots\!36\)\( p^{204} T^{14} + \)\(14\!\cdots\!68\)\( p^{238} T^{15} + p^{272} T^{16} )^{2} \) | |
| 59 | \( 1 - \)\(11\!\cdots\!16\)\( T^{2} + \)\(65\!\cdots\!60\)\( T^{4} - \)\(46\!\cdots\!40\)\( p T^{6} + \)\(87\!\cdots\!60\)\( T^{8} - \)\(23\!\cdots\!08\)\( T^{10} + \)\(53\!\cdots\!68\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{14} + \)\(18\!\cdots\!70\)\( T^{16} - \)\(10\!\cdots\!20\)\( p^{68} T^{18} + \)\(53\!\cdots\!68\)\( p^{136} T^{20} - \)\(23\!\cdots\!08\)\( p^{204} T^{22} + \)\(87\!\cdots\!60\)\( p^{272} T^{24} - \)\(46\!\cdots\!40\)\( p^{341} T^{26} + \)\(65\!\cdots\!60\)\( p^{408} T^{28} - \)\(11\!\cdots\!16\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 61 | \( ( 1 + \)\(36\!\cdots\!64\)\( T + \)\(19\!\cdots\!40\)\( T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!40\)\( T^{4} + \)\(40\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} + \)\(25\!\cdots\!20\)\( T^{7} + \)\(72\!\cdots\!50\)\( T^{8} + \)\(25\!\cdots\!20\)\( p^{34} T^{9} + \)\(12\!\cdots\!68\)\( p^{68} T^{10} + \)\(40\!\cdots\!32\)\( p^{102} T^{11} + \)\(18\!\cdots\!40\)\( p^{136} T^{12} + \)\(49\!\cdots\!00\)\( p^{170} T^{13} + \)\(19\!\cdots\!40\)\( p^{204} T^{14} + \)\(36\!\cdots\!64\)\( p^{238} T^{15} + p^{272} T^{16} )^{2} \) | |
| 67 | \( 1 - \)\(37\!\cdots\!12\)\( T^{2} + \)\(91\!\cdots\!84\)\( T^{4} - \)\(19\!\cdots\!96\)\( T^{6} + \)\(35\!\cdots\!64\)\( T^{8} - \)\(56\!\cdots\!60\)\( T^{10} + \)\(82\!\cdots\!32\)\( T^{12} - \)\(11\!\cdots\!12\)\( T^{14} + \)\(14\!\cdots\!58\)\( T^{16} - \)\(11\!\cdots\!12\)\( p^{68} T^{18} + \)\(82\!\cdots\!32\)\( p^{136} T^{20} - \)\(56\!\cdots\!60\)\( p^{204} T^{22} + \)\(35\!\cdots\!64\)\( p^{272} T^{24} - \)\(19\!\cdots\!96\)\( p^{340} T^{26} + \)\(91\!\cdots\!84\)\( p^{408} T^{28} - \)\(37\!\cdots\!12\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 71 | \( 1 - \)\(42\!\cdots\!16\)\( T^{2} + \)\(95\!\cdots\!80\)\( T^{4} - \)\(16\!\cdots\!40\)\( T^{6} + \)\(24\!\cdots\!80\)\( T^{8} - \)\(31\!\cdots\!88\)\( T^{10} + \)\(35\!\cdots\!88\)\( T^{12} - \)\(36\!\cdots\!60\)\( T^{14} + \)\(33\!\cdots\!30\)\( T^{16} - \)\(36\!\cdots\!60\)\( p^{68} T^{18} + \)\(35\!\cdots\!88\)\( p^{136} T^{20} - \)\(31\!\cdots\!88\)\( p^{204} T^{22} + \)\(24\!\cdots\!80\)\( p^{272} T^{24} - \)\(16\!\cdots\!40\)\( p^{340} T^{26} + \)\(95\!\cdots\!80\)\( p^{408} T^{28} - \)\(42\!\cdots\!16\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 73 | \( ( 1 + \)\(10\!\cdots\!48\)\( T + \)\(11\!\cdots\!56\)\( T^{2} + \)\(40\!\cdots\!48\)\( T^{3} + \)\(61\!\cdots\!08\)\( T^{4} - \)\(59\!\cdots\!24\)\( T^{5} + \)\(20\!\cdots\!84\)\( T^{6} - \)\(32\!\cdots\!84\)\( T^{7} + \)\(50\!\cdots\!42\)\( T^{8} - \)\(32\!\cdots\!84\)\( p^{34} T^{9} + \)\(20\!\cdots\!84\)\( p^{68} T^{10} - \)\(59\!\cdots\!24\)\( p^{102} T^{11} + \)\(61\!\cdots\!08\)\( p^{136} T^{12} + \)\(40\!\cdots\!48\)\( p^{170} T^{13} + \)\(11\!\cdots\!56\)\( p^{204} T^{14} + \)\(10\!\cdots\!48\)\( p^{238} T^{15} + p^{272} T^{16} )^{2} \) | |
| 79 | \( 1 - \)\(29\!\cdots\!76\)\( T^{2} + \)\(43\!\cdots\!20\)\( T^{4} - \)\(43\!\cdots\!00\)\( T^{6} + \)\(31\!\cdots\!20\)\( T^{8} - \)\(18\!\cdots\!48\)\( T^{10} + \)\(93\!\cdots\!08\)\( T^{12} - \)\(38\!\cdots\!40\)\( T^{14} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(38\!\cdots\!40\)\( p^{68} T^{18} + \)\(93\!\cdots\!08\)\( p^{136} T^{20} - \)\(18\!\cdots\!48\)\( p^{204} T^{22} + \)\(31\!\cdots\!20\)\( p^{272} T^{24} - \)\(43\!\cdots\!00\)\( p^{340} T^{26} + \)\(43\!\cdots\!20\)\( p^{408} T^{28} - \)\(29\!\cdots\!76\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 83 | \( 1 - \)\(17\!\cdots\!92\)\( T^{2} + \)\(14\!\cdots\!24\)\( T^{4} - \)\(80\!\cdots\!16\)\( T^{6} + \)\(32\!\cdots\!04\)\( T^{8} - \)\(10\!\cdots\!40\)\( T^{10} + \)\(27\!\cdots\!72\)\( T^{12} - \)\(60\!\cdots\!32\)\( T^{14} + \)\(11\!\cdots\!18\)\( T^{16} - \)\(60\!\cdots\!32\)\( p^{68} T^{18} + \)\(27\!\cdots\!72\)\( p^{136} T^{20} - \)\(10\!\cdots\!40\)\( p^{204} T^{22} + \)\(32\!\cdots\!04\)\( p^{272} T^{24} - \)\(80\!\cdots\!16\)\( p^{340} T^{26} + \)\(14\!\cdots\!24\)\( p^{408} T^{28} - \)\(17\!\cdots\!92\)\( p^{476} T^{30} + p^{544} T^{32} \) | |
| 89 | \( ( 1 + \)\(12\!\cdots\!88\)\( T + \)\(96\!\cdots\!84\)\( T^{2} + \)\(72\!\cdots\!24\)\( T^{3} + \)\(41\!\cdots\!84\)\( T^{4} + \)\(21\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!92\)\( T^{6} + \)\(56\!\cdots\!08\)\( T^{7} + \)\(27\!\cdots\!98\)\( T^{8} + \)\(56\!\cdots\!08\)\( p^{34} T^{9} + \)\(12\!\cdots\!92\)\( p^{68} T^{10} + \)\(21\!\cdots\!00\)\( p^{102} T^{11} + \)\(41\!\cdots\!84\)\( p^{136} T^{12} + \)\(72\!\cdots\!24\)\( p^{170} T^{13} + \)\(96\!\cdots\!84\)\( p^{204} T^{14} + \)\(12\!\cdots\!88\)\( p^{238} T^{15} + p^{272} T^{16} )^{2} \) | |
| 97 | \( ( 1 - \)\(35\!\cdots\!28\)\( T + \)\(13\!\cdots\!76\)\( T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + \)\(67\!\cdots\!48\)\( T^{4} + \)\(33\!\cdots\!84\)\( T^{5} + \)\(38\!\cdots\!84\)\( T^{6} + \)\(11\!\cdots\!84\)\( T^{7} + \)\(19\!\cdots\!62\)\( T^{8} + \)\(11\!\cdots\!84\)\( p^{34} T^{9} + \)\(38\!\cdots\!84\)\( p^{68} T^{10} + \)\(33\!\cdots\!84\)\( p^{102} T^{11} + \)\(67\!\cdots\!48\)\( p^{136} T^{12} + \)\(16\!\cdots\!72\)\( p^{170} T^{13} + \)\(13\!\cdots\!76\)\( p^{204} T^{14} - \)\(35\!\cdots\!28\)\( p^{238} T^{15} + p^{272} T^{16} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.72433726568946652421558102877, −2.69239635388059255134173640007, −2.52291032234559824930997825329, −2.41669861344605221151815117843, −2.29979147870366534863031816351, −2.08321204623520256439199838210, −2.00228834143390156668787101981, −1.95397172089531446702733261190, −1.92004680346816415903108190133, −1.84409657320076678139604685911, −1.73329452346760934239077206923, −1.72208499843946004891847878004, −1.34949165147225555236872818920, −1.26897052935649912368545642270, −1.10979193040354849194613693049, −1.09313244416236281959965221241, −1.03266288384757778830726289236, −0.995784800873868343617475605959, −0.941728456103529836444797527310, −0.804993658995093528904837607940, −0.43989850130560666229749287241, −0.43063992820816813056840680257, −0.22166409380604966581265957004, −0.04370443037541181685944836283, −0.03101637715391467973773578133, 0.03101637715391467973773578133, 0.04370443037541181685944836283, 0.22166409380604966581265957004, 0.43063992820816813056840680257, 0.43989850130560666229749287241, 0.804993658995093528904837607940, 0.941728456103529836444797527310, 0.995784800873868343617475605959, 1.03266288384757778830726289236, 1.09313244416236281959965221241, 1.10979193040354849194613693049, 1.26897052935649912368545642270, 1.34949165147225555236872818920, 1.72208499843946004891847878004, 1.73329452346760934239077206923, 1.84409657320076678139604685911, 1.92004680346816415903108190133, 1.95397172089531446702733261190, 2.00228834143390156668787101981, 2.08321204623520256439199838210, 2.29979147870366534863031816351, 2.41669861344605221151815117843, 2.52291032234559824930997825329, 2.69239635388059255134173640007, 2.72433726568946652421558102877
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.