| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s + 4·11-s + 6·13-s + 15-s − 2·17-s + 4·19-s − 4·21-s + 23-s + 25-s − 27-s − 2·29-s − 4·33-s − 4·35-s + 2·37-s − 6·39-s + 2·41-s − 45-s + 9·49-s + 2·51-s − 6·53-s − 4·55-s − 4·57-s − 8·59-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.872·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.696·33-s − 0.676·35-s + 0.328·37-s − 0.960·39-s + 0.312·41-s − 0.149·45-s + 9/7·49-s + 0.280·51-s − 0.824·53-s − 0.539·55-s − 0.529·57-s − 1.04·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.046598384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.046598384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.806480192823997895871946374503, −8.029337883098196498863583243235, −7.37293036306150686087320617270, −6.43762655051920177550179884992, −5.78994082717123639691181841685, −4.80808245166801612643367619630, −4.19876143808494066851113195373, −3.35232292796494675832983962415, −1.70710395963740807204908024553, −1.04289536099632999180787261257,
1.04289536099632999180787261257, 1.70710395963740807204908024553, 3.35232292796494675832983962415, 4.19876143808494066851113195373, 4.80808245166801612643367619630, 5.78994082717123639691181841685, 6.43762655051920177550179884992, 7.37293036306150686087320617270, 8.029337883098196498863583243235, 8.806480192823997895871946374503
