| L(s) = 1 | − 2.36·2-s + 0.842·3-s + 3.57·4-s − 1.49·5-s − 1.98·6-s − 1.11·7-s − 3.71·8-s − 2.29·9-s + 3.52·10-s − 4.02·11-s + 3.01·12-s + 13-s + 2.62·14-s − 1.25·15-s + 1.62·16-s − 0.552·17-s + 5.40·18-s + 3.31·19-s − 5.33·20-s − 0.935·21-s + 9.51·22-s + 6.65·23-s − 3.12·24-s − 2.77·25-s − 2.36·26-s − 4.45·27-s − 3.96·28-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.486·3-s + 1.78·4-s − 0.667·5-s − 0.811·6-s − 0.419·7-s − 1.31·8-s − 0.763·9-s + 1.11·10-s − 1.21·11-s + 0.868·12-s + 0.277·13-s + 0.700·14-s − 0.324·15-s + 0.406·16-s − 0.133·17-s + 1.27·18-s + 0.761·19-s − 1.19·20-s − 0.204·21-s + 2.02·22-s + 1.38·23-s − 0.638·24-s − 0.554·25-s − 0.463·26-s − 0.857·27-s − 0.750·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 617 | \( 1 - T \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 3 | \( 1 - 0.842T + 3T^{2} \) |
| 5 | \( 1 + 1.49T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 17 | \( 1 + 0.552T + 17T^{2} \) |
| 19 | \( 1 - 3.31T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 + 4.21T + 29T^{2} \) |
| 31 | \( 1 - 8.36T + 31T^{2} \) |
| 37 | \( 1 + 7.94T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 1.90T + 43T^{2} \) |
| 47 | \( 1 - 6.14T + 47T^{2} \) |
| 53 | \( 1 - 6.22T + 53T^{2} \) |
| 59 | \( 1 + 1.22T + 59T^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 - 2.00T + 67T^{2} \) |
| 71 | \( 1 - 0.614T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 0.421T + 79T^{2} \) |
| 83 | \( 1 - 4.75T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 5.73T + 97T^{2} \) |
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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
−7.50428128840363275201579749364, −7.42166772483250328279018960315, −6.35934883673098357353647425353, −5.60334511997378446300261789492, −4.68527911128645961544043187377, −3.48334508602096798403265024316, −2.87679977798896660277682202995, −2.17092486392147675308323227893, −0.904098657899079828660123474588, 0,
0.904098657899079828660123474588, 2.17092486392147675308323227893, 2.87679977798896660277682202995, 3.48334508602096798403265024316, 4.68527911128645961544043187377, 5.60334511997378446300261789492, 6.35934883673098357353647425353, 7.42166772483250328279018960315, 7.50428128840363275201579749364
