| L(s) = 1 | + 17.1·2-s + 27·3-s + 166.·4-s + 69.3·5-s + 463.·6-s + 343·7-s + 663.·8-s + 729·9-s + 1.19e3·10-s − 910.·11-s + 4.49e3·12-s − 8.35e3·13-s + 5.88e3·14-s + 1.87e3·15-s − 9.94e3·16-s + 2.04e4·17-s + 1.25e4·18-s − 4.01e3·19-s + 1.15e4·20-s + 9.26e3·21-s − 1.56e4·22-s − 8.48e4·23-s + 1.79e4·24-s − 7.33e4·25-s − 1.43e5·26-s + 1.96e4·27-s + 5.71e4·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 0.577·3-s + 1.30·4-s + 0.248·5-s + 0.875·6-s + 0.377·7-s + 0.458·8-s + 0.333·9-s + 0.376·10-s − 0.206·11-s + 0.751·12-s − 1.05·13-s + 0.573·14-s + 0.143·15-s − 0.606·16-s + 1.01·17-s + 0.505·18-s − 0.134·19-s + 0.323·20-s + 0.218·21-s − 0.313·22-s − 1.45·23-s + 0.264·24-s − 0.938·25-s − 1.60·26-s + 0.192·27-s + 0.492·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.002466616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.002466616\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 - 17.1T + 128T^{2} \) |
| 5 | \( 1 - 69.3T + 7.81e4T^{2} \) |
| 11 | \( 1 + 910.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.35e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.04e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.01e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.48e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.42e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.14e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.32e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.30e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.13e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.38e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.67e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.26e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.92e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.76e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.32e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.31e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.45e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 4.35e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.82e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.54e6T + 8.07e13T^{2} \) |
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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
−16.04889461414684691918806916828, −14.75153507942405915388675418582, −14.09877546473603991349555709370, −12.87094098339307939618146129844, −11.74138685894363142520133168536, −9.806187473939465656697441510307, −7.72276876112654559353729139660, −5.79806591580752762914344915807, −4.23131677541820408548172923622, −2.46838957608683795260208025584,
2.46838957608683795260208025584, 4.23131677541820408548172923622, 5.79806591580752762914344915807, 7.72276876112654559353729139660, 9.806187473939465656697441510307, 11.74138685894363142520133168536, 12.87094098339307939618146129844, 14.09877546473603991349555709370, 14.75153507942405915388675418582, 16.04889461414684691918806916828
