| L(s) = 1 | + 2-s + 0.618·3-s + 4-s − 3.61·5-s + 0.618·6-s − 4.23·7-s + 8-s − 2.61·9-s − 3.61·10-s + 0.236·11-s + 0.618·12-s − 13-s − 4.23·14-s − 2.23·15-s + 16-s + 0.236·17-s − 2.61·18-s + 2·19-s − 3.61·20-s − 2.61·21-s + 0.236·22-s − 8.23·23-s + 0.618·24-s + 8.09·25-s − 26-s − 3.47·27-s − 4.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.356·3-s + 0.5·4-s − 1.61·5-s + 0.252·6-s − 1.60·7-s + 0.353·8-s − 0.872·9-s − 1.14·10-s + 0.0711·11-s + 0.178·12-s − 0.277·13-s − 1.13·14-s − 0.577·15-s + 0.250·16-s + 0.0572·17-s − 0.617·18-s + 0.458·19-s − 0.809·20-s − 0.571·21-s + 0.0503·22-s − 1.71·23-s + 0.126·24-s + 1.61·25-s − 0.196·26-s − 0.668·27-s − 0.800·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 269 | \( 1 - T \) |
| good | 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 0.236T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 - 0.854T + 29T^{2} \) |
| 31 | \( 1 - 7.47T + 31T^{2} \) |
| 37 | \( 1 + 8.70T + 37T^{2} \) |
| 41 | \( 1 - 7.70T + 41T^{2} \) |
| 43 | \( 1 - 2.23T + 43T^{2} \) |
| 47 | \( 1 + 6.85T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6.56T + 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 - 1.76T + 67T^{2} \) |
| 71 | \( 1 - 9.70T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 5.23T + 83T^{2} \) |
| 89 | \( 1 + 9.79T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−10.48423796127544971822527316674, −9.522969433472425107289092431849, −8.409896087057840088796039549985, −7.65757781263824684391871602608, −6.69189875945640086358163731787, −5.78771380616476145548109461474, −4.34816336712005722797211848473, −3.49314352731394953195257893184, −2.81533988877349973915721995346, 0,
2.81533988877349973915721995346, 3.49314352731394953195257893184, 4.34816336712005722797211848473, 5.78771380616476145548109461474, 6.69189875945640086358163731787, 7.65757781263824684391871602608, 8.409896087057840088796039549985, 9.522969433472425107289092431849, 10.48423796127544971822527316674
