| L(s) = 1 | + 1.10e15·2-s + 5.08e23·3-s + 5.82e29·4-s + 3.48e34·5-s + 5.61e38·6-s + 4.01e41·7-s − 5.63e43·8-s + 8.70e46·9-s + 3.83e49·10-s + 4.15e51·11-s + 2.96e53·12-s − 4.59e54·13-s + 4.42e56·14-s + 1.77e58·15-s − 4.31e59·16-s + 1.61e61·17-s + 9.60e61·18-s − 2.54e63·19-s + 2.02e64·20-s + 2.04e65·21-s + 4.57e66·22-s + 2.64e67·23-s − 2.86e67·24-s − 3.66e68·25-s − 5.06e69·26-s − 4.30e70·27-s + 2.33e71·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 1.22·3-s + 0.919·4-s + 0.876·5-s + 1.70·6-s + 0.590·7-s − 0.111·8-s + 0.506·9-s + 1.21·10-s + 1.17·11-s + 1.12·12-s − 0.332·13-s + 0.818·14-s + 1.07·15-s − 1.07·16-s + 1.99·17-s + 0.702·18-s − 1.28·19-s + 0.805·20-s + 0.725·21-s + 1.62·22-s + 1.04·23-s − 0.137·24-s − 0.232·25-s − 0.460·26-s − 0.605·27-s + 0.543·28-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
| \(L(50)\) |
\(\approx\) |
\(8.840834701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.840834701\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.10e15T + 6.33e29T^{2} \) |
| 3 | \( 1 - 5.08e23T + 1.71e47T^{2} \) |
| 5 | \( 1 - 3.48e34T + 1.57e69T^{2} \) |
| 7 | \( 1 - 4.01e41T + 4.62e83T^{2} \) |
| 11 | \( 1 - 4.15e51T + 1.25e103T^{2} \) |
| 13 | \( 1 + 4.59e54T + 1.90e110T^{2} \) |
| 17 | \( 1 - 1.61e61T + 6.52e121T^{2} \) |
| 19 | \( 1 + 2.54e63T + 3.95e126T^{2} \) |
| 23 | \( 1 - 2.64e67T + 6.47e134T^{2} \) |
| 29 | \( 1 - 1.04e72T + 5.98e144T^{2} \) |
| 31 | \( 1 - 1.04e74T + 4.41e147T^{2} \) |
| 37 | \( 1 - 1.95e76T + 1.78e155T^{2} \) |
| 41 | \( 1 + 7.32e79T + 4.63e159T^{2} \) |
| 43 | \( 1 + 3.18e80T + 5.16e161T^{2} \) |
| 47 | \( 1 + 8.27e82T + 3.44e165T^{2} \) |
| 53 | \( 1 - 8.37e84T + 5.05e170T^{2} \) |
| 59 | \( 1 - 4.40e87T + 2.06e175T^{2} \) |
| 61 | \( 1 - 2.09e87T + 5.59e176T^{2} \) |
| 67 | \( 1 - 2.84e90T + 6.04e180T^{2} \) |
| 71 | \( 1 + 5.10e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 1.71e92T + 2.94e184T^{2} \) |
| 79 | \( 1 + 7.83e93T + 7.32e187T^{2} \) |
| 83 | \( 1 - 1.37e95T + 9.74e189T^{2} \) |
| 89 | \( 1 - 4.51e95T + 9.76e192T^{2} \) |
| 97 | \( 1 - 1.05e98T + 4.90e196T^{2} \) |
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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
−14.38454602461702219168005900650, −13.34879988757407612438060324428, −11.90319924269441399352162601825, −9.719156204172572300524671265154, −8.359736075876195732734654975172, −6.47400641507921899293382303297, −5.09375347066378560007054935099, −3.73300991387566089608525041043, −2.68094627333850606771841645426, −1.49818656562129496114105808424,
1.49818656562129496114105808424, 2.68094627333850606771841645426, 3.73300991387566089608525041043, 5.09375347066378560007054935099, 6.47400641507921899293382303297, 8.359736075876195732734654975172, 9.719156204172572300524671265154, 11.90319924269441399352162601825, 13.34879988757407612438060324428, 14.38454602461702219168005900650
