| L(s) = 1 | − 0.659·2-s − 1.56·4-s − 1.36·5-s − 4.12·7-s + 2.35·8-s + 0.901·10-s + 11-s + 5.19·13-s + 2.71·14-s + 1.58·16-s + 3.22·17-s + 19-s + 2.14·20-s − 0.659·22-s − 8.38·23-s − 3.12·25-s − 3.42·26-s + 6.45·28-s + 8.06·29-s + 2.96·31-s − 5.74·32-s − 2.12·34-s + 5.63·35-s + 4.26·37-s − 0.659·38-s − 3.21·40-s − 5.15·41-s + ⋯ |
| L(s) = 1 | − 0.466·2-s − 0.782·4-s − 0.611·5-s − 1.55·7-s + 0.831·8-s + 0.285·10-s + 0.301·11-s + 1.44·13-s + 0.726·14-s + 0.395·16-s + 0.781·17-s + 0.229·19-s + 0.478·20-s − 0.140·22-s − 1.74·23-s − 0.625·25-s − 0.671·26-s + 1.21·28-s + 1.49·29-s + 0.532·31-s − 1.01·32-s − 0.364·34-s + 0.952·35-s + 0.701·37-s − 0.106·38-s − 0.508·40-s − 0.805·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 0.659T + 2T^{2} \) |
| 5 | \( 1 + 1.36T + 5T^{2} \) |
| 7 | \( 1 + 4.12T + 7T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 23 | \( 1 + 8.38T + 23T^{2} \) |
| 29 | \( 1 - 8.06T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 - 4.26T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 0.149T + 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 9.58T + 61T^{2} \) |
| 67 | \( 1 + 6.34T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 3.01T + 73T^{2} \) |
| 79 | \( 1 - 3.15T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.875012304351055577582776307706, −8.122050899981571035128086868535, −7.52273350650954286359075385661, −6.25972627105720736992773593151, −5.94831165281827373716970345736, −4.45918411025618372088073500529, −3.79915234127507938695078431538, −3.07218553585831349978409655560, −1.22865812950014865736965954955, 0,
1.22865812950014865736965954955, 3.07218553585831349978409655560, 3.79915234127507938695078431538, 4.45918411025618372088073500529, 5.94831165281827373716970345736, 6.25972627105720736992773593151, 7.52273350650954286359075385661, 8.122050899981571035128086868535, 8.875012304351055577582776307706
