| L(s) = 1 | − 2-s − 3.37·3-s + 4-s + 3.37·5-s + 3.37·6-s − 8-s + 8.37·9-s − 3.37·10-s + 1.37·11-s − 3.37·12-s − 0.372·13-s − 11.3·15-s + 16-s + 4.74·17-s − 8.37·18-s + 5.37·19-s + 3.37·20-s − 1.37·22-s − 8.74·23-s + 3.37·24-s + 6.37·25-s + 0.372·26-s − 18.1·27-s − 7.11·29-s + 11.3·30-s − 2.74·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.94·3-s + 0.5·4-s + 1.50·5-s + 1.37·6-s − 0.353·8-s + 2.79·9-s − 1.06·10-s + 0.413·11-s − 0.973·12-s − 0.103·13-s − 2.93·15-s + 0.250·16-s + 1.15·17-s − 1.97·18-s + 1.23·19-s + 0.754·20-s − 0.292·22-s − 1.82·23-s + 0.688·24-s + 1.27·25-s + 0.0730·26-s − 3.48·27-s − 1.32·29-s + 2.07·30-s − 0.492·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 3.37T + 3T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 + 0.372T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 + 8.74T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 + 2.74T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 43 | \( 1 + 6.37T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 0.372T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 - 0.372T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.84550287854155103548981645388, −7.21762784210767678889474145165, −6.42059711437723404237460200608, −5.73180706934821513024584011981, −5.61757763343157302413636243051, −4.62284262770275185999294553372, −3.39894159064024539565350543420, −1.75867203670746081516867686957, −1.40249507977420917152770130666, 0,
1.40249507977420917152770130666, 1.75867203670746081516867686957, 3.39894159064024539565350543420, 4.62284262770275185999294553372, 5.61757763343157302413636243051, 5.73180706934821513024584011981, 6.42059711437723404237460200608, 7.21762784210767678889474145165, 7.84550287854155103548981645388
