L(s) = 1 | + 2-s + 0.927·3-s + 4-s + 5-s + 0.927·6-s − 5.00·7-s + 8-s − 2.14·9-s + 10-s + 4.31·11-s + 0.927·12-s + 13-s − 5.00·14-s + 0.927·15-s + 16-s − 5.44·17-s − 2.14·18-s − 3.63·19-s + 20-s − 4.63·21-s + 4.31·22-s + 0.339·23-s + 0.927·24-s + 25-s + 26-s − 4.76·27-s − 5.00·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.535·3-s + 0.5·4-s + 0.447·5-s + 0.378·6-s − 1.89·7-s + 0.353·8-s − 0.713·9-s + 0.316·10-s + 1.30·11-s + 0.267·12-s + 0.277·13-s − 1.33·14-s + 0.239·15-s + 0.250·16-s − 1.32·17-s − 0.504·18-s − 0.832·19-s + 0.223·20-s − 1.01·21-s + 0.920·22-s + 0.0708·23-s + 0.189·24-s + 0.200·25-s + 0.196·26-s − 0.917·27-s − 0.945·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 - 0.927T + 3T^{2} \) |
| 7 | \( 1 + 5.00T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 17 | \( 1 + 5.44T + 17T^{2} \) |
| 19 | \( 1 + 3.63T + 19T^{2} \) |
| 23 | \( 1 - 0.339T + 23T^{2} \) |
| 29 | \( 1 + 1.23T + 29T^{2} \) |
| 37 | \( 1 + 2.03T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 5.54T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 - 5.37T + 61T^{2} \) |
| 67 | \( 1 + 0.421T + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 1.89T + 73T^{2} \) |
| 79 | \( 1 - 0.390T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 0.304T + 89T^{2} \) |
| 97 | \( 1 - 5.95T + 97T^{2} \) |
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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
−8.225652264839620552826893421988, −6.85104368228080008807317435022, −6.54663668413933512925061254551, −6.11903702490812197037885542037, −5.08156474082781353666280895943, −3.96977203221822252032054884615, −3.45510873869116673100520091864, −2.70872242142060449493998993937, −1.78796552991875759033397939702, 0,
1.78796552991875759033397939702, 2.70872242142060449493998993937, 3.45510873869116673100520091864, 3.96977203221822252032054884615, 5.08156474082781353666280895943, 6.11903702490812197037885542037, 6.54663668413933512925061254551, 6.85104368228080008807317435022, 8.225652264839620552826893421988