| L(s) = 1 | − 2-s + 2.71·3-s + 4-s − 1.72·5-s − 2.71·6-s − 7-s − 8-s + 4.38·9-s + 1.72·10-s + 4.56·11-s + 2.71·12-s + 2.07·13-s + 14-s − 4.68·15-s + 16-s − 5.66·17-s − 4.38·18-s + 3.13·19-s − 1.72·20-s − 2.71·21-s − 4.56·22-s − 5.87·23-s − 2.71·24-s − 2.03·25-s − 2.07·26-s + 3.76·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.56·3-s + 0.5·4-s − 0.770·5-s − 1.10·6-s − 0.377·7-s − 0.353·8-s + 1.46·9-s + 0.544·10-s + 1.37·11-s + 0.784·12-s + 0.574·13-s + 0.267·14-s − 1.20·15-s + 0.250·16-s − 1.37·17-s − 1.03·18-s + 0.718·19-s − 0.385·20-s − 0.593·21-s − 0.973·22-s − 1.22·23-s − 0.554·24-s − 0.406·25-s − 0.406·26-s + 0.724·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.71T + 3T^{2} \) |
| 5 | \( 1 + 1.72T + 5T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 - 2.07T + 13T^{2} \) |
| 17 | \( 1 + 5.66T + 17T^{2} \) |
| 19 | \( 1 - 3.13T + 19T^{2} \) |
| 23 | \( 1 + 5.87T + 23T^{2} \) |
| 29 | \( 1 + 8.13T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 - 3.63T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 0.816T + 43T^{2} \) |
| 47 | \( 1 + 8.64T + 47T^{2} \) |
| 53 | \( 1 + 4.99T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 5.98T + 67T^{2} \) |
| 71 | \( 1 + 2.62T + 71T^{2} \) |
| 73 | \( 1 + 5.19T + 73T^{2} \) |
| 79 | \( 1 + 2.19T + 79T^{2} \) |
| 83 | \( 1 + 6.25T + 83T^{2} \) |
| 89 | \( 1 - 2.18T + 89T^{2} \) |
| 97 | \( 1 - 4.79T + 97T^{2} \) |
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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
−7.893100237977250210175667161238, −7.23720865656604928289991928027, −6.67371005833238484671522534614, −5.81818489394852783889027512921, −4.40749268402550845232246202661, −3.60311033225202110105589628579, −3.47215571081670901635547001315, −2.14748648563625158695081920569, −1.58252763474799108594726141430, 0,
1.58252763474799108594726141430, 2.14748648563625158695081920569, 3.47215571081670901635547001315, 3.60311033225202110105589628579, 4.40749268402550845232246202661, 5.81818489394852783889027512921, 6.67371005833238484671522534614, 7.23720865656604928289991928027, 7.893100237977250210175667161238
