| L(s) = 1 | + 1.53e15·2-s + 1.73e30·4-s + 1.42e34·5-s − 4.25e41·7-s + 1.69e45·8-s + 2.19e49·10-s + 5.04e51·11-s − 1.33e55·13-s − 6.54e56·14-s + 1.51e60·16-s − 8.54e60·17-s − 3.57e63·19-s + 2.47e64·20-s + 7.76e66·22-s + 1.32e67·23-s − 1.37e69·25-s − 2.05e70·26-s − 7.37e71·28-s + 1.57e71·29-s − 3.89e73·31-s + 1.25e75·32-s − 1.31e76·34-s − 6.05e75·35-s − 1.80e77·37-s − 5.50e78·38-s + 2.41e79·40-s − 2.13e79·41-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 2.73·4-s + 0.358·5-s − 0.625·7-s + 3.36·8-s + 0.693·10-s + 1.42·11-s − 0.965·13-s − 1.20·14-s + 3.76·16-s − 1.05·17-s − 1.80·19-s + 0.981·20-s + 2.75·22-s + 0.521·23-s − 0.871·25-s − 1.86·26-s − 1.71·28-s + 0.0644·29-s − 0.586·31-s + 3.90·32-s − 2.04·34-s − 0.224·35-s − 0.426·37-s − 3.48·38-s + 1.20·40-s − 0.313·41-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 | \( 1 - 1.53e15T + 6.33e29T^{2} \) |
| 5 | \( 1 - 1.42e34T + 1.57e69T^{2} \) |
| 7 | \( 1 + 4.25e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 5.04e51T + 1.25e103T^{2} \) |
| 13 | \( 1 + 1.33e55T + 1.90e110T^{2} \) |
| 17 | \( 1 + 8.54e60T + 6.52e121T^{2} \) |
| 19 | \( 1 + 3.57e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 1.32e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 1.57e71T + 5.98e144T^{2} \) |
| 31 | \( 1 + 3.89e73T + 4.41e147T^{2} \) |
| 37 | \( 1 + 1.80e77T + 1.78e155T^{2} \) |
| 41 | \( 1 + 2.13e79T + 4.63e159T^{2} \) |
| 43 | \( 1 - 8.43e79T + 5.16e161T^{2} \) |
| 47 | \( 1 + 2.19e82T + 3.44e165T^{2} \) |
| 53 | \( 1 - 2.60e85T + 5.05e170T^{2} \) |
| 59 | \( 1 + 3.91e87T + 2.06e175T^{2} \) |
| 61 | \( 1 + 3.67e88T + 5.59e176T^{2} \) |
| 67 | \( 1 + 3.53e90T + 6.04e180T^{2} \) |
| 71 | \( 1 + 1.43e91T + 1.88e183T^{2} \) |
| 73 | \( 1 + 5.96e91T + 2.94e184T^{2} \) |
| 79 | \( 1 + 1.81e93T + 7.32e187T^{2} \) |
| 83 | \( 1 - 6.69e93T + 9.74e189T^{2} \) |
| 89 | \( 1 - 5.60e95T + 9.76e192T^{2} \) |
| 97 | \( 1 + 2.13e98T + 4.90e196T^{2} \) |
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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
−8.979739549251866188513315864863, −7.28300063729663601052662441947, −6.49995632043947868670912924311, −5.99667239557934529974417545290, −4.72708719165945584584420905655, −4.14054077809464803700411285295, −3.22372787963539950314157526171, −2.23005032908389101065810638461, −1.62943717670595377240499103177, 0,
1.62943717670595377240499103177, 2.23005032908389101065810638461, 3.22372787963539950314157526171, 4.14054077809464803700411285295, 4.72708719165945584584420905655, 5.99667239557934529974417545290, 6.49995632043947868670912924311, 7.28300063729663601052662441947, 8.979739549251866188513315864863
