| L(s) = 1 | + 1.73·2-s + 0.999·4-s − 3.46·5-s − 4·7-s − 1.73·8-s − 5.99·10-s + 5.19·11-s − 4·13-s − 6.92·14-s − 5·16-s − 3.46·17-s + 19-s − 3.46·20-s + 9·22-s + 3.46·23-s + 6.99·25-s − 6.92·26-s − 3.99·28-s − 1.73·29-s − 7·31-s − 5.19·32-s − 5.99·34-s + 13.8·35-s − 10·37-s + 1.73·38-s + 6.00·40-s + 1.73·41-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s − 1.54·5-s − 1.51·7-s − 0.612·8-s − 1.89·10-s + 1.56·11-s − 1.10·13-s − 1.85·14-s − 1.25·16-s − 0.840·17-s + 0.229·19-s − 0.774·20-s + 1.91·22-s + 0.722·23-s + 1.39·25-s − 1.35·26-s − 0.755·28-s − 0.321·29-s − 1.25·31-s − 0.918·32-s − 1.02·34-s + 2.34·35-s − 1.64·37-s + 0.280·38-s + 0.948·40-s + 0.270·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 8.66T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 5.19T + 83T^{2} \) |
| 89 | \( 1 + 1.73T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.76412089653393983705330587551, −9.300077086928726772190853497920, −8.931725993212289344768679941233, −7.20430045745813027210848812373, −6.84995266092123194518914291863, −5.65347996771669268067169811178, −4.32479491171357245746657987967, −3.81016492236720231822055285811, −2.91436988970677574286495778349, 0,
2.91436988970677574286495778349, 3.81016492236720231822055285811, 4.32479491171357245746657987967, 5.65347996771669268067169811178, 6.84995266092123194518914291863, 7.20430045745813027210848812373, 8.931725993212289344768679941233, 9.300077086928726772190853497920, 10.76412089653393983705330587551
