| L(s) = 1 | + 0.539·2-s + 2.48·3-s − 1.70·4-s + 2.73·5-s + 1.33·6-s + 1.35·7-s − 2.00·8-s + 3.15·9-s + 1.47·10-s − 4.30·11-s − 4.23·12-s + 5.54·13-s + 0.731·14-s + 6.77·15-s + 2.33·16-s − 5.42·17-s + 1.70·18-s − 3.09·19-s − 4.66·20-s + 3.36·21-s − 2.32·22-s + 1.64·23-s − 4.96·24-s + 2.46·25-s + 2.99·26-s + 0.384·27-s − 2.31·28-s + ⋯ |
| L(s) = 1 | + 0.381·2-s + 1.43·3-s − 0.854·4-s + 1.22·5-s + 0.546·6-s + 0.511·7-s − 0.707·8-s + 1.05·9-s + 0.466·10-s − 1.29·11-s − 1.22·12-s + 1.53·13-s + 0.195·14-s + 1.74·15-s + 0.583·16-s − 1.31·17-s + 0.401·18-s − 0.710·19-s − 1.04·20-s + 0.733·21-s − 0.495·22-s + 0.343·23-s − 1.01·24-s + 0.492·25-s + 0.587·26-s + 0.0739·27-s − 0.437·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.449593541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.449593541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 383 | \( 1 - T \) |
| good | 2 | \( 1 - 0.539T + 2T^{2} \) |
| 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 + 4.30T + 11T^{2} \) |
| 13 | \( 1 - 5.54T + 13T^{2} \) |
| 17 | \( 1 + 5.42T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 8.10T + 41T^{2} \) |
| 43 | \( 1 + 4.50T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 + 3.35T + 53T^{2} \) |
| 59 | \( 1 - 7.79T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 1.79T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 - 5.41T + 79T^{2} \) |
| 83 | \( 1 - 2.79T + 83T^{2} \) |
| 89 | \( 1 + 3.32T + 89T^{2} \) |
| 97 | \( 1 - 6.40T + 97T^{2} \) |
| 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
−11.17084228079869564646258016299, −10.20638039579161506479290398935, −9.310040670450115390300738083369, −8.568099180461786915796849894403, −8.103889767809695101913166576691, −6.44521380780238297035135894535, −5.37571828966961323549967565296, −4.29379723834300177476871531074, −3.05122512669388530863877212551, −1.93524168195246958189899113371,
1.93524168195246958189899113371, 3.05122512669388530863877212551, 4.29379723834300177476871531074, 5.37571828966961323549967565296, 6.44521380780238297035135894535, 8.103889767809695101913166576691, 8.568099180461786915796849894403, 9.310040670450115390300738083369, 10.20638039579161506479290398935, 11.17084228079869564646258016299
