| L(s) = 1 | + 2-s + 4-s + 3.39·5-s + 0.912·7-s + 8-s + 3.39·10-s + 4.59·11-s − 0.963·13-s + 0.912·14-s + 16-s + 2.36·17-s − 6.97·19-s + 3.39·20-s + 4.59·22-s − 4.67·23-s + 6.53·25-s − 0.963·26-s + 0.912·28-s + 4.38·29-s − 8.13·31-s + 32-s + 2.36·34-s + 3.09·35-s + 2.94·37-s − 6.97·38-s + 3.39·40-s − 6.42·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.51·5-s + 0.344·7-s + 0.353·8-s + 1.07·10-s + 1.38·11-s − 0.267·13-s + 0.243·14-s + 0.250·16-s + 0.574·17-s − 1.60·19-s + 0.759·20-s + 0.980·22-s − 0.974·23-s + 1.30·25-s − 0.189·26-s + 0.172·28-s + 0.815·29-s − 1.46·31-s + 0.176·32-s + 0.406·34-s + 0.523·35-s + 0.483·37-s − 1.13·38-s + 0.536·40-s − 1.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.265119319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.265119319\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 3.39T + 5T^{2} \) |
| 7 | \( 1 - 0.912T + 7T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 + 0.963T + 13T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 + 4.67T + 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 + 8.13T + 31T^{2} \) |
| 37 | \( 1 - 2.94T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 - 9.90T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 + 7.33T + 61T^{2} \) |
| 67 | \( 1 - 0.214T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 - 8.97T + 73T^{2} \) |
| 79 | \( 1 - 0.227T + 79T^{2} \) |
| 83 | \( 1 - 8.95T + 83T^{2} \) |
| 89 | \( 1 - 0.839T + 89T^{2} \) |
| 97 | \( 1 - 4.12T + 97T^{2} \) |
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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
−7.70358511888401713911980893301, −6.80723917839444143980322570171, −6.34878884127728059165986692780, −5.75335291156955712878643027511, −5.17742914637802742145047151063, −4.21037285078287401232667609672, −3.73590891527888956620948358120, −2.41110955969021719329099232254, −2.03358186478262921822655523360, −1.08854577341274599497864217122,
1.08854577341274599497864217122, 2.03358186478262921822655523360, 2.41110955969021719329099232254, 3.73590891527888956620948358120, 4.21037285078287401232667609672, 5.17742914637802742145047151063, 5.75335291156955712878643027511, 6.34878884127728059165986692780, 6.80723917839444143980322570171, 7.70358511888401713911980893301
