| L(s) = 1 | + 2.55·2-s − 3-s + 4.51·4-s − 2.02·5-s − 2.55·6-s − 3.46·7-s + 6.42·8-s + 9-s − 5.17·10-s + 2.24·11-s − 4.51·12-s + 2.20·13-s − 8.85·14-s + 2.02·15-s + 7.36·16-s − 17-s + 2.55·18-s + 1.23·19-s − 9.15·20-s + 3.46·21-s + 5.72·22-s − 6.14·23-s − 6.42·24-s − 0.894·25-s + 5.63·26-s − 27-s − 15.6·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 0.577·3-s + 2.25·4-s − 0.906·5-s − 1.04·6-s − 1.31·7-s + 2.27·8-s + 0.333·9-s − 1.63·10-s + 0.675·11-s − 1.30·12-s + 0.612·13-s − 2.36·14-s + 0.523·15-s + 1.84·16-s − 0.242·17-s + 0.601·18-s + 0.283·19-s − 2.04·20-s + 0.757·21-s + 1.22·22-s − 1.28·23-s − 1.31·24-s − 0.178·25-s + 1.10·26-s − 0.192·27-s − 2.96·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + 6.14T + 23T^{2} \) |
| 29 | \( 1 - 6.11T + 29T^{2} \) |
| 31 | \( 1 - 7.08T + 31T^{2} \) |
| 37 | \( 1 + 8.56T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 0.242T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 + 2.50T + 53T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 + 4.75T + 67T^{2} \) |
| 71 | \( 1 + 7.79T + 71T^{2} \) |
| 73 | \( 1 - 9.40T + 73T^{2} \) |
| 79 | \( 1 + 2.68T + 79T^{2} \) |
| 83 | \( 1 + 7.70T + 83T^{2} \) |
| 89 | \( 1 - 1.43T + 89T^{2} \) |
| 97 | \( 1 + 8.34T + 97T^{2} \) |
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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
−6.93602169953862187711323298443, −6.60698321321913221789370228354, −6.09235794301796072008152876714, −5.38493802309528658082363600305, −4.49527624537505560475341887218, −3.94807672876137643782069113264, −3.45078436816593227449014971160, −2.71793666862493182303251751469, −1.47412444659380026612186357065, 0,
1.47412444659380026612186357065, 2.71793666862493182303251751469, 3.45078436816593227449014971160, 3.94807672876137643782069113264, 4.49527624537505560475341887218, 5.38493802309528658082363600305, 6.09235794301796072008152876714, 6.60698321321913221789370228354, 6.93602169953862187711323298443
