| L(s) = 1 | + 2-s − 3.11·3-s + 4-s + 2.99·5-s − 3.11·6-s + 2.46·7-s + 8-s + 6.67·9-s + 2.99·10-s − 1.64·11-s − 3.11·12-s + 3.05·13-s + 2.46·14-s − 9.32·15-s + 16-s − 7.30·17-s + 6.67·18-s + 5.24·19-s + 2.99·20-s − 7.66·21-s − 1.64·22-s + 3.96·23-s − 3.11·24-s + 3.99·25-s + 3.05·26-s − 11.4·27-s + 2.46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.79·3-s + 0.5·4-s + 1.34·5-s − 1.26·6-s + 0.931·7-s + 0.353·8-s + 2.22·9-s + 0.948·10-s − 0.496·11-s − 0.897·12-s + 0.847·13-s + 0.658·14-s − 2.40·15-s + 0.250·16-s − 1.77·17-s + 1.57·18-s + 1.20·19-s + 0.670·20-s − 1.67·21-s − 0.350·22-s + 0.825·23-s − 0.634·24-s + 0.799·25-s + 0.599·26-s − 2.19·27-s + 0.465·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.728614867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.728614867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 - 3.05T + 13T^{2} \) |
| 17 | \( 1 + 7.30T + 17T^{2} \) |
| 19 | \( 1 - 5.24T + 19T^{2} \) |
| 23 | \( 1 - 3.96T + 23T^{2} \) |
| 29 | \( 1 + 7.42T + 29T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 43 | \( 1 - 6.77T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 0.0444T + 53T^{2} \) |
| 59 | \( 1 - 2.31T + 59T^{2} \) |
| 61 | \( 1 + 7.95T + 61T^{2} \) |
| 67 | \( 1 - 7.01T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 3.73T + 79T^{2} \) |
| 83 | \( 1 + 5.47T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
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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
−7.68385132043476262971380640133, −7.12374739959547256734624974773, −6.22195304553692866036514998504, −5.88660612693644863296372258733, −5.33774495205157558677523242891, −4.70824289832772568727211393708, −4.09896174088386049300066101558, −2.60651957975954934338023560753, −1.73250867625891787481755735595, −0.913365088216689142018798672529,
0.913365088216689142018798672529, 1.73250867625891787481755735595, 2.60651957975954934338023560753, 4.09896174088386049300066101558, 4.70824289832772568727211393708, 5.33774495205157558677523242891, 5.88660612693644863296372258733, 6.22195304553692866036514998504, 7.12374739959547256734624974773, 7.68385132043476262971380640133
