| L(s) = 1 | + 8·2-s + 79.8·3-s + 64·4-s − 330.·5-s + 638.·6-s + 1.29e3·7-s + 512·8-s + 4.18e3·9-s − 2.64e3·10-s + 3.40e3·11-s + 5.10e3·12-s − 1.39e4·13-s + 1.03e4·14-s − 2.63e4·15-s + 4.09e3·16-s + 1.87e4·17-s + 3.34e4·18-s + 6.85e3·19-s − 2.11e4·20-s + 1.03e5·21-s + 2.72e4·22-s − 5.95e4·23-s + 4.08e4·24-s + 3.11e4·25-s − 1.11e5·26-s + 1.59e5·27-s + 8.28e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.70·3-s + 0.5·4-s − 1.18·5-s + 1.20·6-s + 1.42·7-s + 0.353·8-s + 1.91·9-s − 0.836·10-s + 0.770·11-s + 0.853·12-s − 1.76·13-s + 1.00·14-s − 2.01·15-s + 0.250·16-s + 0.924·17-s + 1.35·18-s + 0.229·19-s − 0.591·20-s + 2.43·21-s + 0.545·22-s − 1.02·23-s + 0.603·24-s + 0.398·25-s − 1.24·26-s + 1.55·27-s + 0.713·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.253651385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.253651385\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 19 | \( 1 - 6.85e3T \) |
| good | 3 | \( 1 - 79.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 330.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.29e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.40e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.39e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.87e4T + 4.10e8T^{2} \) |
| 23 | \( 1 + 5.95e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.97e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 6.58e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.36e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.83e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.90e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.24e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.97e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.78e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.26e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.67e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.37e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.24e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.46e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.31e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.72e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.75e5T + 8.07e13T^{2} \) |
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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
−14.69536864199050023781317807934, −14.08415586047052343419791972957, −12.44108127007635575292388540869, −11.47955933530601596831744550394, −9.575313490721812167918691502299, −7.918909106248510156017119889199, −7.58415274276687506447620182506, −4.66574625484201104346814702666, −3.55231470337234971894709657575, −1.93786134775705376739493591476,
1.93786134775705376739493591476, 3.55231470337234971894709657575, 4.66574625484201104346814702666, 7.58415274276687506447620182506, 7.918909106248510156017119889199, 9.575313490721812167918691502299, 11.47955933530601596831744550394, 12.44108127007635575292388540869, 14.08415586047052343419791972957, 14.69536864199050023781317807934
