| L(s) = 1 | − 2.60·2-s − 2.49·3-s + 4.79·4-s − 1.82·5-s + 6.49·6-s + 1.00·7-s − 7.27·8-s + 3.20·9-s + 4.75·10-s + 0.511·11-s − 11.9·12-s − 1.90·13-s − 2.62·14-s + 4.54·15-s + 9.36·16-s + 2.82·17-s − 8.34·18-s − 6.99·19-s − 8.73·20-s − 2.51·21-s − 1.33·22-s + 0.587·23-s + 18.1·24-s − 1.67·25-s + 4.95·26-s − 0.508·27-s + 4.83·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s − 1.43·3-s + 2.39·4-s − 0.815·5-s + 2.64·6-s + 0.381·7-s − 2.57·8-s + 1.06·9-s + 1.50·10-s + 0.154·11-s − 3.44·12-s − 0.527·13-s − 0.702·14-s + 1.17·15-s + 2.34·16-s + 0.685·17-s − 1.96·18-s − 1.60·19-s − 1.95·20-s − 0.548·21-s − 0.284·22-s + 0.122·23-s + 3.69·24-s − 0.335·25-s + 0.972·26-s − 0.0978·27-s + 0.913·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5077 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1308924730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1308924730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5077 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 7 | \( 1 - 1.00T + 7T^{2} \) |
| 11 | \( 1 - 0.511T + 11T^{2} \) |
| 13 | \( 1 + 1.90T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 6.99T + 19T^{2} \) |
| 23 | \( 1 - 0.587T + 23T^{2} \) |
| 29 | \( 1 + 2.47T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 - 6.23T + 41T^{2} \) |
| 43 | \( 1 - 6.10T + 43T^{2} \) |
| 47 | \( 1 + 7.83T + 47T^{2} \) |
| 53 | \( 1 - 8.97T + 53T^{2} \) |
| 59 | \( 1 - 8.10T + 59T^{2} \) |
| 61 | \( 1 + 7.44T + 61T^{2} \) |
| 67 | \( 1 - 3.14T + 67T^{2} \) |
| 71 | \( 1 + 9.63T + 71T^{2} \) |
| 73 | \( 1 + 1.31T + 73T^{2} \) |
| 79 | \( 1 + 3.25T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 - 9.47T + 97T^{2} \) |
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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
−8.238532792726384783766581139619, −7.51172605667540320055820939474, −7.07764263607966092359333323986, −6.27139288356758088736899360216, −5.64615284115830672655122133163, −4.67960084157639627077336406091, −3.69794171022615418254210770893, −2.34492369380409546234602949088, −1.38901004200175079590498247331, −0.29681675434096977682239350870,
0.29681675434096977682239350870, 1.38901004200175079590498247331, 2.34492369380409546234602949088, 3.69794171022615418254210770893, 4.67960084157639627077336406091, 5.64615284115830672655122133163, 6.27139288356758088736899360216, 7.07764263607966092359333323986, 7.51172605667540320055820939474, 8.238532792726384783766581139619
