| L(s) = 1 | − 2-s + 2.11·3-s + 4-s − 5-s − 2.11·6-s − 3.58·7-s − 8-s + 1.47·9-s + 10-s − 11-s + 2.11·12-s + 5.70·13-s + 3.58·14-s − 2.11·15-s + 16-s − 1.35·17-s − 1.47·18-s − 0.243·19-s − 20-s − 7.58·21-s + 22-s + 6.22·23-s − 2.11·24-s + 25-s − 5.70·26-s − 3.22·27-s − 3.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.22·3-s + 0.5·4-s − 0.447·5-s − 0.863·6-s − 1.35·7-s − 0.353·8-s + 0.490·9-s + 0.316·10-s − 0.301·11-s + 0.610·12-s + 1.58·13-s + 0.958·14-s − 0.546·15-s + 0.250·16-s − 0.329·17-s − 0.347·18-s − 0.0557·19-s − 0.223·20-s − 1.65·21-s + 0.213·22-s + 1.29·23-s − 0.431·24-s + 0.200·25-s − 1.11·26-s − 0.621·27-s − 0.678·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 3 | \( 1 - 2.11T + 3T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 + 0.243T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 + 9.10T + 29T^{2} \) |
| 31 | \( 1 - 5.58T + 31T^{2} \) |
| 37 | \( 1 + 5.58T + 37T^{2} \) |
| 41 | \( 1 - 3.05T + 41T^{2} \) |
| 43 | \( 1 - 1.60T + 43T^{2} \) |
| 47 | \( 1 - 3.60T + 47T^{2} \) |
| 53 | \( 1 - 0.0737T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 2.22T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.52894998124220174089741300222, −7.09293067572053562514158832581, −6.24671709116291276527045072238, −5.68270281930637856168444003569, −4.35069113155604112144780290700, −3.45025494663409000223677597295, −3.20725645543578603701505639300, −2.34950108898838854547891654278, −1.24536178388036227556971348788, 0,
1.24536178388036227556971348788, 2.34950108898838854547891654278, 3.20725645543578603701505639300, 3.45025494663409000223677597295, 4.35069113155604112144780290700, 5.68270281930637856168444003569, 6.24671709116291276527045072238, 7.09293067572053562514158832581, 7.52894998124220174089741300222
