| L(s) = 1 | − 4·2-s + 16.2·3-s + 16·4-s + 25·5-s − 65.1·6-s − 85.5·7-s − 64·8-s + 22.6·9-s − 100·10-s − 403.·11-s + 260.·12-s + 841.·13-s + 342.·14-s + 407.·15-s + 256·16-s − 2.29e3·17-s − 90.5·18-s + 2.49e3·19-s + 400·20-s − 1.39e3·21-s + 1.61e3·22-s − 795.·23-s − 1.04e3·24-s + 625·25-s − 3.36e3·26-s − 3.59e3·27-s − 1.36e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.04·3-s + 0.5·4-s + 0.447·5-s − 0.739·6-s − 0.659·7-s − 0.353·8-s + 0.0931·9-s − 0.316·10-s − 1.00·11-s + 0.522·12-s + 1.38·13-s + 0.466·14-s + 0.467·15-s + 0.250·16-s − 1.92·17-s − 0.0658·18-s + 1.58·19-s + 0.223·20-s − 0.689·21-s + 0.710·22-s − 0.313·23-s − 0.369·24-s + 0.200·25-s − 0.976·26-s − 0.948·27-s − 0.329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.051867877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051867877\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 79 | \( 1 + 6.24e3T \) |
| good | 3 | \( 1 - 16.2T + 243T^{2} \) |
| 7 | \( 1 + 85.5T + 1.68e4T^{2} \) |
| 11 | \( 1 + 403.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 841.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.29e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 795.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.53e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.92e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.04e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.38e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.84e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.56e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 885.T + 2.07e9T^{2} \) |
| 83 | \( 1 - 1.10e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.50e3T + 8.58e9T^{2} \) |
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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
−9.407737760772623627260197599965, −8.591034060395507334466841887438, −8.239213460669405100381083458870, −7.03051234441437244228734059074, −6.29295838021807371238366215259, −5.19020140647803655111252113051, −3.66409107645101611116539312493, −2.82902958915374277968066262630, −2.03111699242524661144245315940, −0.67220054482339968490003643646,
0.67220054482339968490003643646, 2.03111699242524661144245315940, 2.82902958915374277968066262630, 3.66409107645101611116539312493, 5.19020140647803655111252113051, 6.29295838021807371238366215259, 7.03051234441437244228734059074, 8.239213460669405100381083458870, 8.591034060395507334466841887438, 9.407737760772623627260197599965
