| L(s) = 1 | + 9.15·2-s − 9·3-s + 51.7·4-s + 95.5·5-s − 82.3·6-s + 209.·7-s + 180.·8-s + 81·9-s + 874.·10-s − 465.·12-s + 335.·13-s + 1.91e3·14-s − 859.·15-s − 1.19·16-s + 799.·17-s + 741.·18-s + 658.·19-s + 4.94e3·20-s − 1.88e3·21-s − 4.11e3·23-s − 1.62e3·24-s + 5.99e3·25-s + 3.07e3·26-s − 729·27-s + 1.08e4·28-s − 559.·29-s − 7.86e3·30-s + ⋯ |
| L(s) = 1 | + 1.61·2-s − 0.577·3-s + 1.61·4-s + 1.70·5-s − 0.934·6-s + 1.61·7-s + 0.999·8-s + 0.333·9-s + 2.76·10-s − 0.933·12-s + 0.551·13-s + 2.61·14-s − 0.986·15-s − 0.00117·16-s + 0.670·17-s + 0.539·18-s + 0.418·19-s + 2.76·20-s − 0.933·21-s − 1.62·23-s − 0.576·24-s + 1.91·25-s + 0.891·26-s − 0.192·27-s + 2.61·28-s − 0.123·29-s − 1.59·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.618610529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.618610529\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 9.15T + 32T^{2} \) |
| 5 | \( 1 - 95.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 209.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 335.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 799.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 658.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 559.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.05e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.40e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.84e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.62e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.58e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.61e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.70e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.63e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.39e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.45e5T + 8.58e9T^{2} \) |
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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
−10.86752830821271002037401979463, −10.06338697120673918867709152928, −8.757435635734643461696033989687, −7.36698870037065148560342998601, −6.17349193589250423788786064266, −5.51332440625329750973043016669, −5.03267020794150003241371454619, −3.80349605868615583409602666246, −2.18014954881080629809643886662, −1.48076218823276650986331412550,
1.48076218823276650986331412550, 2.18014954881080629809643886662, 3.80349605868615583409602666246, 5.03267020794150003241371454619, 5.51332440625329750973043016669, 6.17349193589250423788786064266, 7.36698870037065148560342998601, 8.757435635734643461696033989687, 10.06338697120673918867709152928, 10.86752830821271002037401979463
