| L(s) = 1 | + 2-s − 3-s + 4-s − 1.49·5-s − 6-s + 7-s + 8-s + 9-s − 1.49·10-s + 5.89·11-s − 12-s + 4.21·13-s + 14-s + 1.49·15-s + 16-s + 2.39·17-s + 18-s + 1.94·19-s − 1.49·20-s − 21-s + 5.89·22-s − 0.170·23-s − 24-s − 2.76·25-s + 4.21·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.668·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.472·10-s + 1.77·11-s − 0.288·12-s + 1.16·13-s + 0.267·14-s + 0.385·15-s + 0.250·16-s + 0.579·17-s + 0.235·18-s + 0.446·19-s − 0.334·20-s − 0.218·21-s + 1.25·22-s − 0.0355·23-s − 0.204·24-s − 0.553·25-s + 0.826·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.390702967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.390702967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 + 1.49T + 5T^{2} \) |
| 11 | \( 1 - 5.89T + 11T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 17 | \( 1 - 2.39T + 17T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 + 0.170T + 23T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 + 9.44T + 53T^{2} \) |
| 59 | \( 1 + 7.51T + 59T^{2} \) |
| 61 | \( 1 - 3.20T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 17.2T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 + 8.77T + 89T^{2} \) |
| 97 | \( 1 - 5.49T + 97T^{2} \) |
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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
−7.68332663231961684892223892369, −7.00197614678808011761943737825, −6.18920354483469840095618212320, −5.98831984268124500550469151073, −4.86885574657365076965846337426, −4.27290097768558013915174079417, −3.73939666334087683052261710974, −2.96194933017793474901764523387, −1.53877891433945387161578938209, −0.961033605185082854866244639685,
0.961033605185082854866244639685, 1.53877891433945387161578938209, 2.96194933017793474901764523387, 3.73939666334087683052261710974, 4.27290097768558013915174079417, 4.86885574657365076965846337426, 5.98831984268124500550469151073, 6.18920354483469840095618212320, 7.00197614678808011761943737825, 7.68332663231961684892223892369
