| L(s) = 1 | − 2.36·2-s + 3.59·4-s − 5-s + 3.86·7-s − 3.76·8-s + 2.36·10-s + 1.19·11-s + 13-s − 9.12·14-s + 1.72·16-s + 0.670·17-s + 3.86·19-s − 3.59·20-s − 2.82·22-s + 2.76·23-s + 25-s − 2.36·26-s + 13.8·28-s + 6.30·29-s + 2.29·31-s + 3.45·32-s − 1.58·34-s − 3.86·35-s − 10.5·37-s − 9.14·38-s + 3.76·40-s + 4.66·41-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.79·4-s − 0.447·5-s + 1.45·7-s − 1.33·8-s + 0.747·10-s + 0.360·11-s + 0.277·13-s − 2.44·14-s + 0.431·16-s + 0.162·17-s + 0.886·19-s − 0.803·20-s − 0.602·22-s + 0.575·23-s + 0.200·25-s − 0.463·26-s + 2.62·28-s + 1.17·29-s + 0.411·31-s + 0.610·32-s − 0.272·34-s − 0.652·35-s − 1.72·37-s − 1.48·38-s + 0.596·40-s + 0.729·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9250596687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9250596687\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.36T + 2T^{2} \) |
| 7 | \( 1 - 3.86T + 7T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 17 | \( 1 - 0.670T + 17T^{2} \) |
| 19 | \( 1 - 3.86T + 19T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 + 1.57T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 3.75T + 59T^{2} \) |
| 61 | \( 1 - 5.60T + 61T^{2} \) |
| 67 | \( 1 + 1.82T + 67T^{2} \) |
| 71 | \( 1 - 0.529T + 71T^{2} \) |
| 73 | \( 1 + 7.01T + 73T^{2} \) |
| 79 | \( 1 - 5.49T + 79T^{2} \) |
| 83 | \( 1 + 7.05T + 83T^{2} \) |
| 89 | \( 1 + 9.58T + 89T^{2} \) |
| 97 | \( 1 + 9.33T + 97T^{2} \) |
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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
−9.084561344573882403187334216156, −8.542396785212489255538892452676, −7.904090328816766927806320577452, −7.33244919970821224067879387234, −6.50394937347304843895176794757, −5.26429235349008600671868139618, −4.37422355415499292331351902008, −3.00775291181640423376633603628, −1.72979036511403248873269611966, −0.919603423848574778308012591706,
0.919603423848574778308012591706, 1.72979036511403248873269611966, 3.00775291181640423376633603628, 4.37422355415499292331351902008, 5.26429235349008600671868139618, 6.50394937347304843895176794757, 7.33244919970821224067879387234, 7.904090328816766927806320577452, 8.542396785212489255538892452676, 9.084561344573882403187334216156
