| L(s) = 1 | + 4·2-s − 17.9·3-s + 16·4-s − 25·5-s − 71.7·6-s + 215.·7-s + 64·8-s + 78.7·9-s − 100·10-s − 270.·11-s − 287.·12-s + 474.·13-s + 862.·14-s + 448.·15-s + 256·16-s − 426.·17-s + 315.·18-s − 577.·19-s − 400·20-s − 3.86e3·21-s − 1.08e3·22-s − 3.66e3·23-s − 1.14e3·24-s + 625·25-s + 1.89e3·26-s + 2.94e3·27-s + 3.45e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s − 0.447·5-s − 0.813·6-s + 1.66·7-s + 0.353·8-s + 0.324·9-s − 0.316·10-s − 0.673·11-s − 0.575·12-s + 0.778·13-s + 1.17·14-s + 0.514·15-s + 0.250·16-s − 0.357·17-s + 0.229·18-s − 0.367·19-s − 0.223·20-s − 1.91·21-s − 0.476·22-s − 1.44·23-s − 0.406·24-s + 0.200·25-s + 0.550·26-s + 0.777·27-s + 0.831·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 17.9T + 243T^{2} \) |
| 7 | \( 1 - 215.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 270.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 474.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 426.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 577.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.66e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.72e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.09e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.70e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.75e4T + 2.07e9T^{2} \) |
| 83 | \( 1 + 7.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.70e4T + 8.58e9T^{2} \) |
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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
−8.984632155892176707073357541699, −7.943936008493277712955536954860, −7.42330807994340752024748736794, −6.00083729197833893778633368948, −5.64261631795635684009515439487, −4.59902636505912813286232082525, −4.03713516245689931154208796239, −2.40187623916191700667058614627, −1.29182473476823037159045331196, 0,
1.29182473476823037159045331196, 2.40187623916191700667058614627, 4.03713516245689931154208796239, 4.59902636505912813286232082525, 5.64261631795635684009515439487, 6.00083729197833893778633368948, 7.42330807994340752024748736794, 7.943936008493277712955536954860, 8.984632155892176707073357541699
