| L(s) = 1 | − 2-s + 4-s − 1.84·5-s + 2.73·7-s − 8-s + 1.84·10-s + 3.84·11-s − 1.70·13-s − 2.73·14-s + 16-s − 1.33·17-s + 4.04·19-s − 1.84·20-s − 3.84·22-s + 3.24·23-s − 1.60·25-s + 1.70·26-s + 2.73·28-s − 1.77·29-s − 3.60·31-s − 32-s + 1.33·34-s − 5.02·35-s − 4.85·37-s − 4.04·38-s + 1.84·40-s − 11.5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.823·5-s + 1.03·7-s − 0.353·8-s + 0.582·10-s + 1.16·11-s − 0.472·13-s − 0.729·14-s + 0.250·16-s − 0.324·17-s + 0.928·19-s − 0.411·20-s − 0.820·22-s + 0.676·23-s − 0.321·25-s + 0.333·26-s + 0.515·28-s − 0.329·29-s − 0.647·31-s − 0.176·32-s + 0.229·34-s − 0.850·35-s − 0.798·37-s − 0.656·38-s + 0.291·40-s − 1.80·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 + 1.84T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 4.04T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 7.48T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 9.23T + 59T^{2} \) |
| 61 | \( 1 - 7.09T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 7.94T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 8.02T + 79T^{2} \) |
| 83 | \( 1 + 6.08T + 83T^{2} \) |
| 89 | \( 1 - 8.43T + 89T^{2} \) |
| 97 | \( 1 - 7.75T + 97T^{2} \) |
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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
−7.63722491145883129557512420468, −7.00773451940075095439246261875, −6.34007760355807356309623959675, −5.28676723218206348085204810132, −4.71139865572955622036481519530, −3.81006611659117081074401407884, −3.15144455281082406046291369563, −1.90733084906180289057549703386, −1.27101028633967225389002667361, 0,
1.27101028633967225389002667361, 1.90733084906180289057549703386, 3.15144455281082406046291369563, 3.81006611659117081074401407884, 4.71139865572955622036481519530, 5.28676723218206348085204810132, 6.34007760355807356309623959675, 7.00773451940075095439246261875, 7.63722491145883129557512420468
