L(s) = 1 | − 1.53e15·2-s + 1.09e23·3-s + 1.73e30·4-s − 1.42e34·5-s − 1.68e38·6-s − 4.25e41·7-s − 1.69e45·8-s − 1.59e47·9-s + 2.19e49·10-s − 5.04e51·11-s + 1.90e53·12-s − 1.33e55·13-s + 6.54e56·14-s − 1.55e57·15-s + 1.51e60·16-s + 8.54e60·17-s + 2.45e62·18-s − 3.57e63·19-s − 2.47e64·20-s − 4.65e64·21-s + 7.76e66·22-s − 1.32e67·23-s − 1.85e68·24-s − 1.37e69·25-s + 2.05e70·26-s − 3.63e70·27-s − 7.37e71·28-s + ⋯ |
L(s) = 1 | − 1.93·2-s + 0.264·3-s + 2.73·4-s − 0.358·5-s − 0.510·6-s − 0.625·7-s − 3.36·8-s − 0.930·9-s + 0.693·10-s − 1.42·11-s + 0.723·12-s − 0.965·13-s + 1.20·14-s − 0.0947·15-s + 3.76·16-s + 1.05·17-s + 1.79·18-s − 1.80·19-s − 0.981·20-s − 0.165·21-s + 2.75·22-s − 0.521·23-s − 0.887·24-s − 0.871·25-s + 1.86·26-s − 0.509·27-s − 1.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(100-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+99/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(50)\) |
\(\approx\) |
\(0.003122643029\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.003122643029\) |
\(L(\frac{101}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
good | 2 | \( 1 + 1.53e15T + 6.33e29T^{2} \) |
| 3 | \( 1 - 1.09e23T + 1.71e47T^{2} \) |
| 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
−15.01146275052369743077562627650, −12.32112540394735054472525046695, −10.79369323364500928403001998356, −9.726817868569553135269251489866, −8.356708014895255088732055096616, −7.51695230002812897172164015992, −5.92501523376097362495176638710, −3.02403823756730519695357521273, −2.07466531707269433382459856444, −0.03599348156476027294440974586,
0.03599348156476027294440974586, 2.07466531707269433382459856444, 3.02403823756730519695357521273, 5.92501523376097362495176638710, 7.51695230002812897172164015992, 8.356708014895255088732055096616, 9.726817868569553135269251489866, 10.79369323364500928403001998356, 12.32112540394735054472525046695, 15.01146275052369743077562627650