| L(s) = 1 | − 2-s + 1.13·3-s + 4-s + 0.191·5-s − 1.13·6-s + 1.30·7-s − 8-s − 1.72·9-s − 0.191·10-s − 3.27·11-s + 1.13·12-s + 2.94·13-s − 1.30·14-s + 0.216·15-s + 16-s + 5.53·17-s + 1.72·18-s − 5.79·19-s + 0.191·20-s + 1.47·21-s + 3.27·22-s − 0.738·23-s − 1.13·24-s − 4.96·25-s − 2.94·26-s − 5.33·27-s + 1.30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.652·3-s + 0.5·4-s + 0.0855·5-s − 0.461·6-s + 0.493·7-s − 0.353·8-s − 0.573·9-s − 0.0604·10-s − 0.987·11-s + 0.326·12-s + 0.816·13-s − 0.348·14-s + 0.0558·15-s + 0.250·16-s + 1.34·17-s + 0.405·18-s − 1.32·19-s + 0.0427·20-s + 0.322·21-s + 0.698·22-s − 0.154·23-s − 0.230·24-s − 0.992·25-s − 0.577·26-s − 1.02·27-s + 0.246·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 1.13T + 3T^{2} \) |
| 5 | \( 1 - 0.191T + 5T^{2} \) |
| 7 | \( 1 - 1.30T + 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 13 | \( 1 - 2.94T + 13T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 + 5.79T + 19T^{2} \) |
| 23 | \( 1 + 0.738T + 23T^{2} \) |
| 29 | \( 1 + 2.47T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 + 5.10T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 + 4.72T + 47T^{2} \) |
| 53 | \( 1 - 8.97T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 7.94T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 + 7.74T + 73T^{2} \) |
| 79 | \( 1 + 6.50T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 - 6.26T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−8.038761705149894837977610569914, −7.84373560860863976303709336540, −6.73649022708710786284778465815, −5.83763856854124995642572361040, −5.30875583104826192275519945768, −4.04845818476922529274425418818, −3.21815375112523033252648602371, −2.36200951109602833486488302152, −1.51808681072530472900990844512, 0,
1.51808681072530472900990844512, 2.36200951109602833486488302152, 3.21815375112523033252648602371, 4.04845818476922529274425418818, 5.30875583104826192275519945768, 5.83763856854124995642572361040, 6.73649022708710786284778465815, 7.84373560860863976303709336540, 8.038761705149894837977610569914
