| L(s) = 1 | − 2.09·2-s − 3.33·3-s + 2.39·4-s − 5-s + 6.98·6-s − 7-s − 0.817·8-s + 8.10·9-s + 2.09·10-s − 7.96·12-s + 1.20·13-s + 2.09·14-s + 3.33·15-s − 3.06·16-s − 1.11·17-s − 16.9·18-s + 4.13·19-s − 2.39·20-s + 3.33·21-s + 3.71·23-s + 2.72·24-s + 25-s − 2.51·26-s − 17.0·27-s − 2.39·28-s − 5.97·29-s − 6.98·30-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 1.92·3-s + 1.19·4-s − 0.447·5-s + 2.85·6-s − 0.377·7-s − 0.289·8-s + 2.70·9-s + 0.662·10-s − 2.29·12-s + 0.333·13-s + 0.559·14-s + 0.860·15-s − 0.766·16-s − 0.270·17-s − 4.00·18-s + 0.947·19-s − 0.534·20-s + 0.727·21-s + 0.774·23-s + 0.556·24-s + 0.200·25-s − 0.494·26-s − 3.27·27-s − 0.451·28-s − 1.10·29-s − 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 3 | \( 1 + 3.33T + 3T^{2} \) |
| 13 | \( 1 - 1.20T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 23 | \( 1 - 3.71T + 23T^{2} \) |
| 29 | \( 1 + 5.97T + 29T^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 4.29T + 59T^{2} \) |
| 61 | \( 1 + 9.13T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 - 6.71T + 73T^{2} \) |
| 79 | \( 1 - 9.58T + 79T^{2} \) |
| 83 | \( 1 - 1.98T + 83T^{2} \) |
| 89 | \( 1 + 0.455T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−7.85256877687198813244259730333, −7.32433287766448657287830131006, −6.70901483121410571302422931275, −6.06718390807187541682954164144, −5.16556528603501163996388648020, −4.49803179484198121691099337910, −3.39674430411068734993148798300, −1.75761936980412429056130313773, −0.882162354307916074623768565899, 0,
0.882162354307916074623768565899, 1.75761936980412429056130313773, 3.39674430411068734993148798300, 4.49803179484198121691099337910, 5.16556528603501163996388648020, 6.06718390807187541682954164144, 6.70901483121410571302422931275, 7.32433287766448657287830131006, 7.85256877687198813244259730333
