| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s − 4.71·13-s + 14-s + 16-s − 7.70·17-s − 6.98·19-s − 20-s − 22-s − 0.493·23-s + 25-s + 4.71·26-s − 28-s + 3.78·29-s − 32-s + 7.70·34-s + 35-s − 6.27·37-s + 6.98·38-s + 40-s + 8.76·41-s − 9.20·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 1.30·13-s + 0.267·14-s + 0.250·16-s − 1.86·17-s − 1.60·19-s − 0.223·20-s − 0.213·22-s − 0.102·23-s + 0.200·25-s + 0.924·26-s − 0.188·28-s + 0.701·29-s − 0.176·32-s + 1.32·34-s + 0.169·35-s − 1.03·37-s + 1.13·38-s + 0.158·40-s + 1.36·41-s − 1.40·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4085114031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4085114031\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 + 4.71T + 13T^{2} \) |
| 17 | \( 1 + 7.70T + 17T^{2} \) |
| 19 | \( 1 + 6.98T + 19T^{2} \) |
| 23 | \( 1 + 0.493T + 23T^{2} \) |
| 29 | \( 1 - 3.78T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6.27T + 37T^{2} \) |
| 41 | \( 1 - 8.76T + 41T^{2} \) |
| 43 | \( 1 + 9.20T + 43T^{2} \) |
| 47 | \( 1 + 5.20T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 + 8.98T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 3.48T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 1.01T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 6.76T + 83T^{2} \) |
| 89 | \( 1 - 8.71T + 89T^{2} \) |
| 97 | \( 1 - 2.98T + 97T^{2} \) |
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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
−8.024089178250409736866488397997, −7.28743473588143417106184422799, −6.53782732696074954719453325293, −6.31990785717964031964667311411, −4.92056834318588182982208530536, −4.46559147265563429700960516072, −3.49395911864807577618368211753, −2.49184690431812544153227145049, −1.89571310225842180010762899493, −0.34010912813952876871794806200,
0.34010912813952876871794806200, 1.89571310225842180010762899493, 2.49184690431812544153227145049, 3.49395911864807577618368211753, 4.46559147265563429700960516072, 4.92056834318588182982208530536, 6.31990785717964031964667311411, 6.53782732696074954719453325293, 7.28743473588143417106184422799, 8.024089178250409736866488397997
