| L(s) = 1 | + 2.27·2-s + 0.241·3-s + 3.19·4-s − 5-s + 0.551·6-s + 7-s + 2.71·8-s − 2.94·9-s − 2.27·10-s − 0.945·11-s + 0.772·12-s + 1.27·13-s + 2.27·14-s − 0.241·15-s − 0.197·16-s − 4.16·17-s − 6.70·18-s − 1.73·19-s − 3.19·20-s + 0.241·21-s − 2.15·22-s + 5.41·23-s + 0.656·24-s + 25-s + 2.91·26-s − 1.43·27-s + 3.19·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 0.139·3-s + 1.59·4-s − 0.447·5-s + 0.225·6-s + 0.377·7-s + 0.959·8-s − 0.980·9-s − 0.720·10-s − 0.284·11-s + 0.222·12-s + 0.354·13-s + 0.608·14-s − 0.0624·15-s − 0.0494·16-s − 1.01·17-s − 1.57·18-s − 0.397·19-s − 0.713·20-s + 0.0528·21-s − 0.459·22-s + 1.12·23-s + 0.134·24-s + 0.200·25-s + 0.570·26-s − 0.276·27-s + 0.603·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 3 | \( 1 - 0.241T + 3T^{2} \) |
| 11 | \( 1 + 0.945T + 11T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 17 | \( 1 + 4.16T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 - 0.304T + 29T^{2} \) |
| 31 | \( 1 - 7.25T + 31T^{2} \) |
| 37 | \( 1 + 6.80T + 37T^{2} \) |
| 41 | \( 1 + 4.74T + 41T^{2} \) |
| 43 | \( 1 + 0.410T + 43T^{2} \) |
| 47 | \( 1 + 2.39T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 7.75T + 59T^{2} \) |
| 61 | \( 1 + 8.72T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 - 2.75T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 + 0.807T + 83T^{2} \) |
| 89 | \( 1 - 2.47T + 89T^{2} \) |
| 97 | \( 1 - 3.64T + 97T^{2} \) |
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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
−7.27312847029767899302508872837, −6.48143210615780509765915489631, −6.09484740164418010754953010211, −5.05212714667663479279976993680, −4.81725648021221466439272595357, −3.97829840217431181284719918858, −3.14264165487855694408249087876, −2.70956867856226032002976229462, −1.66011781552368404456114291648, 0,
1.66011781552368404456114291648, 2.70956867856226032002976229462, 3.14264165487855694408249087876, 3.97829840217431181284719918858, 4.81725648021221466439272595357, 5.05212714667663479279976993680, 6.09484740164418010754953010211, 6.48143210615780509765915489631, 7.27312847029767899302508872837
