| L(s) = 1 | − 2-s + 0.214·3-s + 4-s + 2.53·5-s − 0.214·6-s + 4.82·7-s − 8-s − 2.95·9-s − 2.53·10-s + 2.22·11-s + 0.214·12-s − 5.27·13-s − 4.82·14-s + 0.541·15-s + 16-s + 4.70·17-s + 2.95·18-s + 7.85·19-s + 2.53·20-s + 1.03·21-s − 2.22·22-s + 23-s − 0.214·24-s + 1.40·25-s + 5.27·26-s − 1.27·27-s + 4.82·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.123·3-s + 0.5·4-s + 1.13·5-s − 0.0874·6-s + 1.82·7-s − 0.353·8-s − 0.984·9-s − 0.800·10-s + 0.671·11-s + 0.0618·12-s − 1.46·13-s − 1.29·14-s + 0.139·15-s + 0.250·16-s + 1.14·17-s + 0.696·18-s + 1.80·19-s + 0.566·20-s + 0.225·21-s − 0.474·22-s + 0.208·23-s − 0.0437·24-s + 0.281·25-s + 1.03·26-s − 0.245·27-s + 0.912·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.453646842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.453646842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 0.214T + 3T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 - 4.70T + 17T^{2} \) |
| 19 | \( 1 - 7.85T + 19T^{2} \) |
| 29 | \( 1 - 6.80T + 29T^{2} \) |
| 31 | \( 1 + 2.30T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 0.658T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 5.91T + 53T^{2} \) |
| 59 | \( 1 - 7.72T + 59T^{2} \) |
| 61 | \( 1 - 6.31T + 61T^{2} \) |
| 67 | \( 1 + 3.82T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 1.63T + 73T^{2} \) |
| 79 | \( 1 - 2.73T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.180255676511874040464729092425, −7.42456236319783940264126729902, −6.96074566175238515618392215613, −5.63113153027980958936582391760, −5.45751603057740597659966528739, −4.74471022855151332683855766040, −3.36730280709093080722456795518, −2.49332345692901673023125997928, −1.74771606960980614260814021241, −0.978417268157910373911934503880,
0.978417268157910373911934503880, 1.74771606960980614260814021241, 2.49332345692901673023125997928, 3.36730280709093080722456795518, 4.74471022855151332683855766040, 5.45751603057740597659966528739, 5.63113153027980958936582391760, 6.96074566175238515618392215613, 7.42456236319783940264126729902, 8.180255676511874040464729092425
