| L(s) = 1 | + 0.549·2-s + 2.96·3-s − 1.69·4-s − 0.0524·5-s + 1.63·6-s − 0.375·7-s − 2.03·8-s + 5.81·9-s − 0.0288·10-s + 4.73·11-s − 5.03·12-s + 2.79·13-s − 0.206·14-s − 0.155·15-s + 2.27·16-s − 17-s + 3.19·18-s − 1.64·19-s + 0.0889·20-s − 1.11·21-s + 2.60·22-s + 7.26·23-s − 6.03·24-s − 4.99·25-s + 1.53·26-s + 8.35·27-s + 0.637·28-s + ⋯ |
| L(s) = 1 | + 0.388·2-s + 1.71·3-s − 0.848·4-s − 0.0234·5-s + 0.666·6-s − 0.141·7-s − 0.718·8-s + 1.93·9-s − 0.00911·10-s + 1.42·11-s − 1.45·12-s + 0.774·13-s − 0.0551·14-s − 0.0401·15-s + 0.569·16-s − 0.242·17-s + 0.753·18-s − 0.376·19-s + 0.0198·20-s − 0.243·21-s + 0.555·22-s + 1.51·23-s − 1.23·24-s − 0.999·25-s + 0.301·26-s + 1.60·27-s + 0.120·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.812790065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.812790065\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 0.549T + 2T^{2} \) |
| 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 + 0.0524T + 5T^{2} \) |
| 7 | \( 1 + 0.375T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 23 | \( 1 - 7.26T + 23T^{2} \) |
| 29 | \( 1 + 6.97T + 29T^{2} \) |
| 31 | \( 1 - 3.09T + 31T^{2} \) |
| 37 | \( 1 - 0.721T + 37T^{2} \) |
| 41 | \( 1 + 2.68T + 41T^{2} \) |
| 47 | \( 1 - 1.05T + 47T^{2} \) |
| 53 | \( 1 - 7.14T + 53T^{2} \) |
| 59 | \( 1 + 6.60T + 59T^{2} \) |
| 61 | \( 1 + 3.97T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 3.01T + 73T^{2} \) |
| 79 | \( 1 - 9.25T + 79T^{2} \) |
| 83 | \( 1 + 6.02T + 83T^{2} \) |
| 89 | \( 1 + 8.44T + 89T^{2} \) |
| 97 | \( 1 + 8.77T + 97T^{2} \) |
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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
−9.994598223514282137248295010958, −9.120209140014621106163024783775, −8.954538652001351728884590404951, −8.058918972235197166076328460939, −7.02721812914839329072519962722, −5.95094872844151307973891382314, −4.47358061984529258036645254029, −3.80421407147670316810060702412, −3.05780326377421791460049135501, −1.52226764479173028323555084275,
1.52226764479173028323555084275, 3.05780326377421791460049135501, 3.80421407147670316810060702412, 4.47358061984529258036645254029, 5.95094872844151307973891382314, 7.02721812914839329072519962722, 8.058918972235197166076328460939, 8.954538652001351728884590404951, 9.120209140014621106163024783775, 9.994598223514282137248295010958
