| L(s) = 1 | + 2-s − 0.434·3-s + 4-s + 5-s − 0.434·6-s + 3.24·7-s + 8-s − 2.81·9-s + 10-s − 11-s − 0.434·12-s + 0.968·13-s + 3.24·14-s − 0.434·15-s + 16-s − 7.25·17-s − 2.81·18-s − 4.37·19-s + 20-s − 1.40·21-s − 22-s − 7.37·23-s − 0.434·24-s + 25-s + 0.968·26-s + 2.52·27-s + 3.24·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.250·3-s + 0.5·4-s + 0.447·5-s − 0.177·6-s + 1.22·7-s + 0.353·8-s − 0.937·9-s + 0.316·10-s − 0.301·11-s − 0.125·12-s + 0.268·13-s + 0.867·14-s − 0.112·15-s + 0.250·16-s − 1.76·17-s − 0.662·18-s − 1.00·19-s + 0.223·20-s − 0.307·21-s − 0.213·22-s − 1.53·23-s − 0.0886·24-s + 0.200·25-s + 0.189·26-s + 0.485·27-s + 0.613·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 + 0.434T + 3T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 13 | \( 1 - 0.968T + 13T^{2} \) |
| 17 | \( 1 + 7.25T + 17T^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 + 7.37T + 23T^{2} \) |
| 29 | \( 1 + 1.71T + 29T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 + 0.800T + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 0.633T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 5.27T + 73T^{2} \) |
| 79 | \( 1 - 8.30T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 0.849T + 89T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−7.940421102437893978714298232356, −7.08925377081114365250509507397, −6.18738096449432161809628035861, −5.79233466070307124499283108731, −4.89905592961351820806281425430, −4.41447750363982924827960223609, −3.42715732421380913352780263510, −2.18301760939844646324842446250, −1.87212869955207462422956252338, 0,
1.87212869955207462422956252338, 2.18301760939844646324842446250, 3.42715732421380913352780263510, 4.41447750363982924827960223609, 4.89905592961351820806281425430, 5.79233466070307124499283108731, 6.18738096449432161809628035861, 7.08925377081114365250509507397, 7.940421102437893978714298232356
