| L(s) = 1 | + 1.53e15·2-s − 3.78e23·3-s + 1.70e30·4-s − 5.47e34·5-s − 5.78e38·6-s − 5.03e41·7-s + 1.64e45·8-s − 2.88e46·9-s − 8.38e49·10-s + 3.39e50·11-s − 6.45e53·12-s + 1.35e55·13-s − 7.71e56·14-s + 2.07e58·15-s + 1.43e60·16-s + 1.35e61·17-s − 4.42e61·18-s + 1.25e63·19-s − 9.35e64·20-s + 1.90e65·21-s + 5.18e65·22-s + 5.45e66·23-s − 6.21e68·24-s + 1.42e69·25-s + 2.08e70·26-s + 7.58e70·27-s − 8.60e71·28-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 0.912·3-s + 2.69·4-s − 1.37·5-s − 1.75·6-s − 0.741·7-s + 3.25·8-s − 0.168·9-s − 2.65·10-s + 0.0957·11-s − 2.45·12-s + 0.984·13-s − 1.42·14-s + 1.25·15-s + 3.57·16-s + 1.67·17-s − 0.323·18-s + 0.632·19-s − 3.71·20-s + 0.675·21-s + 0.184·22-s + 0.214·23-s − 2.97·24-s + 0.901·25-s + 1.89·26-s + 1.06·27-s − 1.99·28-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
| \(L(50)\) |
\(\approx\) |
\(4.278205927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.278205927\) |
| \(L(\frac{101}{2})\) |
|
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 | \( 1 + 3.78e23T + 1.71e47T^{2} \) |
| 5 | \( 1 + 5.47e34T + 1.57e69T^{2} \) |
| 7 | \( 1 + 5.03e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 3.39e50T + 1.25e103T^{2} \) |
| 13 | \( 1 - 1.35e55T + 1.90e110T^{2} \) |
| 17 | \( 1 - 1.35e61T + 6.52e121T^{2} \) |
| 19 | \( 1 - 1.25e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 5.45e66T + 6.47e134T^{2} \) |
| 29 | \( 1 - 1.00e72T + 5.98e144T^{2} \) |
| 31 | \( 1 + 5.43e73T + 4.41e147T^{2} \) |
| 37 | \( 1 + 2.04e77T + 1.78e155T^{2} \) |
| 41 | \( 1 - 5.05e79T + 4.63e159T^{2} \) |
| 43 | \( 1 - 9.21e80T + 5.16e161T^{2} \) |
| 47 | \( 1 - 4.56e82T + 3.44e165T^{2} \) |
| 53 | \( 1 + 2.37e85T + 5.05e170T^{2} \) |
| 59 | \( 1 - 1.52e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 1.15e88T + 5.59e176T^{2} \) |
| 67 | \( 1 - 1.58e90T + 6.04e180T^{2} \) |
| 71 | \( 1 + 7.68e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 2.31e92T + 2.94e184T^{2} \) |
| 79 | \( 1 - 1.21e94T + 7.32e187T^{2} \) |
| 83 | \( 1 - 6.01e94T + 9.74e189T^{2} \) |
| 89 | \( 1 - 1.16e96T + 9.76e192T^{2} \) |
| 97 | \( 1 + 1.98e98T + 4.90e196T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.27844326460858480694026348011, −12.60673720544253022339517973684, −11.81615653107932955654799785206, −10.85780058728852992241177836864, −7.58436674272474694121875037373, −6.26584012015693567905993073762, −5.27087040441203420687234062522, −3.85412196400214236288439201822, −3.12561784033871415900743439535, −0.890337043804568232452066276259,
0.890337043804568232452066276259, 3.12561784033871415900743439535, 3.85412196400214236288439201822, 5.27087040441203420687234062522, 6.26584012015693567905993073762, 7.58436674272474694121875037373, 10.85780058728852992241177836864, 11.81615653107932955654799785206, 12.60673720544253022339517973684, 14.27844326460858480694026348011
