| L(s) = 1 | − 0.0835·3-s + 3.81·5-s − 2.99·9-s − 1.01·11-s − 0.289·13-s − 0.319·15-s − 1.76·17-s − 6.29·19-s + 2.13·23-s + 9.58·25-s + 0.500·27-s − 7.26·29-s − 6.47·31-s + 0.0852·33-s + 7.89·37-s + 0.0242·39-s + 2.58·41-s − 7.62·43-s − 11.4·45-s + 1.07·47-s + 0.147·51-s + 3.62·53-s − 3.89·55-s + 0.526·57-s − 12.6·59-s + 7.39·61-s − 1.10·65-s + ⋯ |
| L(s) = 1 | − 0.0482·3-s + 1.70·5-s − 0.997·9-s − 0.307·11-s − 0.0803·13-s − 0.0823·15-s − 0.428·17-s − 1.44·19-s + 0.444·23-s + 1.91·25-s + 0.0963·27-s − 1.34·29-s − 1.16·31-s + 0.0148·33-s + 1.29·37-s + 0.00387·39-s + 0.404·41-s − 1.16·43-s − 1.70·45-s + 0.156·47-s + 0.0206·51-s + 0.497·53-s − 0.525·55-s + 0.0697·57-s − 1.65·59-s + 0.946·61-s − 0.137·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.0835T + 3T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 13 | \( 1 + 0.289T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 + 6.29T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 7.89T + 37T^{2} \) |
| 41 | \( 1 - 2.58T + 41T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 - 3.62T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 - 7.39T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 3.55T + 71T^{2} \) |
| 73 | \( 1 + 8.24T + 73T^{2} \) |
| 79 | \( 1 + 3.44T + 79T^{2} \) |
| 83 | \( 1 - 6.90T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.84566839205547623178237487329, −6.91926311775123337366427242426, −6.18468923521391563454619472763, −5.73686774040857075080971144396, −5.12441153501837222116217135693, −4.18334382281971033706687385869, −2.98265417414409412071370132414, −2.31746012927135557005198975478, −1.59492933845692987836043049442, 0,
1.59492933845692987836043049442, 2.31746012927135557005198975478, 2.98265417414409412071370132414, 4.18334382281971033706687385869, 5.12441153501837222116217135693, 5.73686774040857075080971144396, 6.18468923521391563454619472763, 6.91926311775123337366427242426, 7.84566839205547623178237487329
