| L(s) = 1 | − 1.31·3-s + 0.319·5-s + 1.93·7-s − 1.25·9-s + 3.25·11-s − 4.25·13-s − 0.421·15-s + 3·17-s + 19-s − 2.55·21-s − 5.61·23-s − 4.89·25-s + 5.61·27-s + 6.55·29-s − 6.93·31-s − 4.30·33-s + 0.619·35-s + 7.57·37-s + 5.61·39-s − 4.30·41-s − 7.13·43-s − 0.402·45-s − 6.31·47-s − 3.23·49-s − 3.95·51-s + 5.31·53-s + 1.04·55-s + ⋯ |
| L(s) = 1 | − 0.761·3-s + 0.142·5-s + 0.732·7-s − 0.419·9-s + 0.982·11-s − 1.18·13-s − 0.108·15-s + 0.727·17-s + 0.229·19-s − 0.558·21-s − 1.17·23-s − 0.979·25-s + 1.08·27-s + 1.21·29-s − 1.24·31-s − 0.748·33-s + 0.104·35-s + 1.24·37-s + 0.899·39-s − 0.671·41-s − 1.08·43-s − 0.0599·45-s − 0.921·47-s − 0.462·49-s − 0.554·51-s + 0.730·53-s + 0.140·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.31T + 3T^{2} \) |
| 5 | \( 1 - 0.319T + 5T^{2} \) |
| 7 | \( 1 - 1.93T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 4.25T + 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 - 6.55T + 29T^{2} \) |
| 31 | \( 1 + 6.93T + 31T^{2} \) |
| 37 | \( 1 - 7.57T + 37T^{2} \) |
| 41 | \( 1 + 4.30T + 41T^{2} \) |
| 43 | \( 1 + 7.13T + 43T^{2} \) |
| 47 | \( 1 + 6.31T + 47T^{2} \) |
| 53 | \( 1 - 5.31T + 53T^{2} \) |
| 59 | \( 1 - 1.74T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 8.68T + 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 + 6.93T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.983748983830288774658714832362, −7.11164500750352288311776071755, −6.37441112336655812885759214709, −5.68024623281198387399105203562, −5.06125943044118270544243376019, −4.33917770369584986442592602072, −3.37466311214651816734002112885, −2.26593588388207109014663463163, −1.31768749454975935584995074449, 0,
1.31768749454975935584995074449, 2.26593588388207109014663463163, 3.37466311214651816734002112885, 4.33917770369584986442592602072, 5.06125943044118270544243376019, 5.68024623281198387399105203562, 6.37441112336655812885759214709, 7.11164500750352288311776071755, 7.983748983830288774658714832362
