| L(s) = 1 | + 16·2-s + 110.·3-s + 256·4-s − 1.29e3·5-s + 1.76e3·6-s + 6.62e3·7-s + 4.09e3·8-s − 7.47e3·9-s − 2.07e4·10-s + 7.95e4·11-s + 2.82e4·12-s + 1.68e5·13-s + 1.06e5·14-s − 1.43e5·15-s + 6.55e4·16-s − 1.53e5·17-s − 1.19e5·18-s − 1.30e5·19-s − 3.32e5·20-s + 7.32e5·21-s + 1.27e6·22-s + 1.01e6·23-s + 4.52e5·24-s − 2.67e5·25-s + 2.70e6·26-s − 3.00e6·27-s + 1.69e6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.787·3-s + 0.5·4-s − 0.928·5-s + 0.556·6-s + 1.04·7-s + 0.353·8-s − 0.379·9-s − 0.656·10-s + 1.63·11-s + 0.393·12-s + 1.64·13-s + 0.737·14-s − 0.731·15-s + 0.250·16-s − 0.445·17-s − 0.268·18-s − 0.229·19-s − 0.464·20-s + 0.821·21-s + 1.15·22-s + 0.754·23-s + 0.278·24-s − 0.137·25-s + 1.15·26-s − 1.08·27-s + 0.521·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.889812100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.889812100\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 3 | \( 1 - 110.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.29e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.62e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.95e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.68e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.53e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 1.01e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 5.40e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 3.60e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.20e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.81e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 6.47e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.68e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.21e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.66e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.59e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 9.61e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.86e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.45e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.54e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.15e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.66e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.14e8T + 7.60e17T^{2} \) |
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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
−14.42806627447853833882714219277, −13.37763228184279516855109111360, −11.69010756528583774105887740444, −11.19770577246732631396013944983, −8.878791392755431473201944960245, −8.020882403097535426032690016507, −6.37902282996600690107669156367, −4.40350550170063210126729924545, −3.37114172157925745113457582998, −1.44169155849399242590803488281,
1.44169155849399242590803488281, 3.37114172157925745113457582998, 4.40350550170063210126729924545, 6.37902282996600690107669156367, 8.020882403097535426032690016507, 8.878791392755431473201944960245, 11.19770577246732631396013944983, 11.69010756528583774105887740444, 13.37763228184279516855109111360, 14.42806627447853833882714219277
