| L(s) = 1 | − 8.07e14·2-s − 6.25e23·3-s + 1.88e28·4-s − 5.34e34·5-s + 5.05e38·6-s + 1.04e42·7-s + 4.96e44·8-s + 2.18e47·9-s + 4.32e49·10-s + 2.09e51·11-s − 1.18e52·12-s − 2.27e55·13-s − 8.40e56·14-s + 3.34e58·15-s − 4.13e59·16-s + 7.56e60·17-s − 1.76e62·18-s + 5.81e62·19-s − 1.01e63·20-s − 6.50e65·21-s − 1.69e66·22-s − 2.12e67·23-s − 3.10e68·24-s + 1.28e69·25-s + 1.83e70·26-s − 2.94e70·27-s + 1.96e70·28-s + ⋯ |
| L(s) = 1 | − 1.01·2-s − 1.50·3-s + 0.0297·4-s − 1.34·5-s + 1.53·6-s + 1.53·7-s + 0.984·8-s + 1.27·9-s + 1.36·10-s + 0.592·11-s − 0.0449·12-s − 1.64·13-s − 1.55·14-s + 2.03·15-s − 1.02·16-s + 0.937·17-s − 1.29·18-s + 0.292·19-s − 0.0401·20-s − 2.30·21-s − 0.601·22-s − 0.836·23-s − 1.48·24-s + 0.812·25-s + 1.67·26-s − 0.414·27-s + 0.0456·28-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
| \(L(50)\) |
\(\approx\) |
\(0.2983520417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2983520417\) |
| \(L(\frac{101}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 8.07e14T + 6.33e29T^{2} \) |
| 3 | \( 1 + 6.25e23T + 1.71e47T^{2} \) |
| 5 | \( 1 + 5.34e34T + 1.57e69T^{2} \) |
| 7 | \( 1 - 1.04e42T + 4.62e83T^{2} \) |
| 11 | \( 1 - 2.09e51T + 1.25e103T^{2} \) |
| 13 | \( 1 + 2.27e55T + 1.90e110T^{2} \) |
| 17 | \( 1 - 7.56e60T + 6.52e121T^{2} \) |
| 19 | \( 1 - 5.81e62T + 3.95e126T^{2} \) |
| 23 | \( 1 + 2.12e67T + 6.47e134T^{2} \) |
| 29 | \( 1 + 1.52e72T + 5.98e144T^{2} \) |
| 31 | \( 1 - 2.72e73T + 4.41e147T^{2} \) |
| 37 | \( 1 - 4.28e77T + 1.78e155T^{2} \) |
| 41 | \( 1 + 5.02e79T + 4.63e159T^{2} \) |
| 43 | \( 1 + 1.05e81T + 5.16e161T^{2} \) |
| 47 | \( 1 + 3.56e82T + 3.44e165T^{2} \) |
| 53 | \( 1 - 1.04e85T + 5.05e170T^{2} \) |
| 59 | \( 1 - 3.60e87T + 2.06e175T^{2} \) |
| 61 | \( 1 + 9.18e87T + 5.59e176T^{2} \) |
| 67 | \( 1 + 1.79e90T + 6.04e180T^{2} \) |
| 71 | \( 1 + 3.40e91T + 1.88e183T^{2} \) |
| 73 | \( 1 - 1.68e92T + 2.94e184T^{2} \) |
| 79 | \( 1 + 1.45e94T + 7.32e187T^{2} \) |
| 83 | \( 1 + 9.55e94T + 9.74e189T^{2} \) |
| 89 | \( 1 - 4.74e96T + 9.76e192T^{2} \) |
| 97 | \( 1 + 1.76e98T + 4.90e196T^{2} \) |
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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
−14.70395037772116029659720562461, −11.97903058750936891264059489767, −11.44540301698086782045215542876, −10.06214482059453919623419117848, −8.117318923488563502662472537949, −7.28720383749918722831972576839, −5.13877859888061441671681646587, −4.31026581966752052291287874800, −1.47885692592129847625270994374, −0.40253771477770699112491134797,
0.40253771477770699112491134797, 1.47885692592129847625270994374, 4.31026581966752052291287874800, 5.13877859888061441671681646587, 7.28720383749918722831972576839, 8.117318923488563502662472537949, 10.06214482059453919623419117848, 11.44540301698086782045215542876, 11.97903058750936891264059489767, 14.70395037772116029659720562461
