| L(s) = 1 | − 1.73·2-s − 1.68·3-s + 1.00·4-s + 2.92·6-s + 1.46·7-s + 1.72·8-s − 0.149·9-s + 1.36·11-s − 1.69·12-s − 3.63·13-s − 2.54·14-s − 4.99·16-s − 2.07·17-s + 0.258·18-s + 5.36·19-s − 2.47·21-s − 2.36·22-s + 1.31·23-s − 2.91·24-s + 6.29·26-s + 5.31·27-s + 1.47·28-s − 0.362·29-s + 3.80·31-s + 5.22·32-s − 2.29·33-s + 3.60·34-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.974·3-s + 0.502·4-s + 1.19·6-s + 0.554·7-s + 0.609·8-s − 0.0497·9-s + 0.410·11-s − 0.490·12-s − 1.00·13-s − 0.680·14-s − 1.24·16-s − 0.504·17-s + 0.0609·18-s + 1.23·19-s − 0.540·21-s − 0.503·22-s + 0.273·23-s − 0.594·24-s + 1.23·26-s + 1.02·27-s + 0.278·28-s − 0.0672·29-s + 0.682·31-s + 0.922·32-s − 0.400·33-s + 0.618·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 1.68T + 3T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 3.63T + 13T^{2} \) |
| 17 | \( 1 + 2.07T + 17T^{2} \) |
| 19 | \( 1 - 5.36T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 0.362T + 29T^{2} \) |
| 31 | \( 1 - 3.80T + 31T^{2} \) |
| 37 | \( 1 + 7.99T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 8.35T + 43T^{2} \) |
| 47 | \( 1 + 0.363T + 47T^{2} \) |
| 53 | \( 1 + 8.25T + 53T^{2} \) |
| 59 | \( 1 + 1.86T + 59T^{2} \) |
| 61 | \( 1 - 6.13T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 9.81T + 71T^{2} \) |
| 73 | \( 1 + 0.508T + 73T^{2} \) |
| 79 | \( 1 - 6.08T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 + 2.88T + 89T^{2} \) |
| 97 | \( 1 + 19.0T + 97T^{2} \) |
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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
−7.969095585527524765054293657033, −6.90278980486959145993730259972, −6.72355290182176102572376375255, −5.43215942664604615892071343573, −5.04661957488782473982634216328, −4.28301021520800056142761036751, −3.01875174366890726627100800774, −1.90207873010499735647976434170, −0.986380497613381038098870504301, 0,
0.986380497613381038098870504301, 1.90207873010499735647976434170, 3.01875174366890726627100800774, 4.28301021520800056142761036751, 5.04661957488782473982634216328, 5.43215942664604615892071343573, 6.72355290182176102572376375255, 6.90278980486959145993730259972, 7.969095585527524765054293657033
