| L(s) = 1 | − 2.61·2-s + 0.381·3-s + 4.85·4-s + 2.61·5-s − 6-s − 7.47·8-s − 2.85·9-s − 6.85·10-s − 0.381·11-s + 1.85·12-s + 2.85·13-s + 15-s + 9.85·16-s − 6.70·17-s + 7.47·18-s − 19-s + 12.7·20-s + 22-s − 8.61·23-s − 2.85·24-s + 1.85·25-s − 7.47·26-s − 2.23·27-s + 7.47·29-s − 2.61·30-s + 5.70·31-s − 10.8·32-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.220·3-s + 2.42·4-s + 1.17·5-s − 0.408·6-s − 2.64·8-s − 0.951·9-s − 2.16·10-s − 0.115·11-s + 0.535·12-s + 0.791·13-s + 0.258·15-s + 2.46·16-s − 1.62·17-s + 1.76·18-s − 0.229·19-s + 2.84·20-s + 0.213·22-s − 1.79·23-s − 0.582·24-s + 0.370·25-s − 1.46·26-s − 0.430·27-s + 1.38·29-s − 0.477·30-s + 1.02·31-s − 1.91·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 2.61T + 5T^{2} \) |
| 11 | \( 1 + 0.381T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 8.61T + 23T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 + 5.94T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 0.145T + 61T^{2} \) |
| 67 | \( 1 - 1.29T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 1.85T + 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 - 1.14T + 97T^{2} \) |
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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
−8.404400381101505217154325989098, −7.86902043861493517741709871863, −6.62967756733259943050558467264, −6.31040254105551322217552084567, −5.66221526160577523161323753021, −4.24178651834661282836987608572, −2.73726936502845604339447106631, −2.31500758003264200610322529250, −1.36085409798688991219681225282, 0,
1.36085409798688991219681225282, 2.31500758003264200610322529250, 2.73726936502845604339447106631, 4.24178651834661282836987608572, 5.66221526160577523161323753021, 6.31040254105551322217552084567, 6.62967756733259943050558467264, 7.86902043861493517741709871863, 8.404400381101505217154325989098
