| L(s) = 1 | − 1.39e15·2-s + 1.31e30·4-s − 5.19e33·5-s + 3.00e41·7-s − 9.58e44·8-s + 7.25e48·10-s + 5.65e51·11-s − 2.39e55·13-s − 4.20e56·14-s + 5.03e59·16-s + 6.59e60·17-s + 5.96e62·19-s − 6.85e63·20-s − 7.90e66·22-s + 1.57e67·23-s − 1.55e69·25-s + 3.34e70·26-s + 3.96e71·28-s + 4.99e71·29-s − 7.23e73·31-s − 9.58e73·32-s − 9.21e75·34-s − 1.56e75·35-s − 9.44e76·37-s − 8.34e77·38-s + 4.97e78·40-s + 1.72e78·41-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 2.08·4-s − 0.130·5-s + 0.442·7-s − 1.89·8-s + 0.229·10-s + 1.59·11-s − 1.73·13-s − 0.776·14-s + 1.25·16-s + 0.816·17-s + 0.300·19-s − 0.272·20-s − 2.80·22-s + 0.618·23-s − 0.982·25-s + 3.04·26-s + 0.921·28-s + 0.204·29-s − 1.08·31-s − 0.299·32-s − 1.43·34-s − 0.0578·35-s − 0.223·37-s − 0.527·38-s + 0.248·40-s + 0.0253·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(100-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+99/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(50)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 1.39e15T + 6.33e29T^{2} \) |
| 5 | \( 1 + 5.19e33T + 1.57e69T^{2} \) |
| 7 | \( 1 - 3.00e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 5.65e51T + 1.25e103T^{2} \) |
| 13 | \( 1 + 2.39e55T + 1.90e110T^{2} \) |
| 17 | \( 1 - 6.59e60T + 6.52e121T^{2} \) |
| 19 | \( 1 - 5.96e62T + 3.95e126T^{2} \) |
| 23 | \( 1 - 1.57e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 4.99e71T + 5.98e144T^{2} \) |
| 31 | \( 1 + 7.23e73T + 4.41e147T^{2} \) |
| 37 | \( 1 + 9.44e76T + 1.78e155T^{2} \) |
| 41 | \( 1 - 1.72e78T + 4.63e159T^{2} \) |
| 43 | \( 1 - 7.49e80T + 5.16e161T^{2} \) |
| 47 | \( 1 - 3.93e82T + 3.44e165T^{2} \) |
| 53 | \( 1 + 5.71e84T + 5.05e170T^{2} \) |
| 59 | \( 1 + 8.13e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 3.51e88T + 5.59e176T^{2} \) |
| 67 | \( 1 + 4.30e88T + 6.04e180T^{2} \) |
| 71 | \( 1 + 2.22e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 3.08e92T + 2.94e184T^{2} \) |
| 79 | \( 1 - 9.66e93T + 7.32e187T^{2} \) |
| 83 | \( 1 - 5.89e94T + 9.74e189T^{2} \) |
| 89 | \( 1 - 3.01e96T + 9.76e192T^{2} \) |
| 97 | \( 1 - 3.23e98T + 4.90e196T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.216318317983913334307038991175, −7.915493519948340231990936504141, −7.35145983237200475767317921933, −6.43293917139040479826108072861, −5.10305627197044406814152655422, −3.77117677765498069051441178871, −2.51335941359418646901621290653, −1.64882108508636180427005410540, −0.918691423343271442486864658452, 0,
0.918691423343271442486864658452, 1.64882108508636180427005410540, 2.51335941359418646901621290653, 3.77117677765498069051441178871, 5.10305627197044406814152655422, 6.43293917139040479826108072861, 7.35145983237200475767317921933, 7.915493519948340231990936504141, 9.216318317983913334307038991175
