L(s) = 1 | + 2-s − 0.523·3-s + 4-s + 1.37·5-s − 0.523·6-s − 3.92·7-s + 8-s − 2.72·9-s + 1.37·10-s + 1.18·11-s − 0.523·12-s + 5.13·13-s − 3.92·14-s − 0.720·15-s + 16-s − 3.19·17-s − 2.72·18-s − 6.02·19-s + 1.37·20-s + 2.05·21-s + 1.18·22-s + 8.56·23-s − 0.523·24-s − 3.10·25-s + 5.13·26-s + 2.99·27-s − 3.92·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.302·3-s + 0.5·4-s + 0.615·5-s − 0.213·6-s − 1.48·7-s + 0.353·8-s − 0.908·9-s + 0.435·10-s + 0.356·11-s − 0.151·12-s + 1.42·13-s − 1.04·14-s − 0.186·15-s + 0.250·16-s − 0.775·17-s − 0.642·18-s − 1.38·19-s + 0.307·20-s + 0.448·21-s + 0.251·22-s + 1.78·23-s − 0.106·24-s − 0.621·25-s + 1.00·26-s + 0.577·27-s − 0.741·28-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 | \( 1 + 0.523T + 3T^{2} \) |
| 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 + 3.92T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 + 6.02T + 19T^{2} \) |
| 23 | \( 1 - 8.56T + 23T^{2} \) |
| 29 | \( 1 + 4.87T + 29T^{2} \) |
| 31 | \( 1 - 5.99T + 31T^{2} \) |
| 37 | \( 1 + 5.74T + 37T^{2} \) |
| 41 | \( 1 + 0.0599T + 41T^{2} \) |
| 43 | \( 1 + 9.15T + 43T^{2} \) |
| 47 | \( 1 - 5.61T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 4.93T + 59T^{2} \) |
| 61 | \( 1 + 7.66T + 61T^{2} \) |
| 67 | \( 1 + 6.06T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 6.40T + 83T^{2} \) |
| 89 | \( 1 - 6.10T + 89T^{2} \) |
| 97 | \( 1 - 7.36T + 97T^{2} \) |
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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
−8.141932470440259757955416571662, −6.81715933618855654125137248506, −6.41270617630214290827575547771, −6.04423513861634288015853888670, −5.21599345069883503458597926622, −4.20734869098385417997886196912, −3.35564433722888934555804290895, −2.75545202963653204719999362642, −1.56994652115675373683589604220, 0,
1.56994652115675373683589604220, 2.75545202963653204719999362642, 3.35564433722888934555804290895, 4.20734869098385417997886196912, 5.21599345069883503458597926622, 6.04423513861634288015853888670, 6.41270617630214290827575547771, 6.81715933618855654125137248506, 8.141932470440259757955416571662