L(s) = 1 | − 2-s − 2.64·3-s + 4-s + 3.69·5-s + 2.64·6-s − 4.07·7-s − 8-s + 3.99·9-s − 3.69·10-s − 5.71·11-s − 2.64·12-s + 4.07·14-s − 9.76·15-s + 16-s + 3.36·17-s − 3.99·18-s − 19-s + 3.69·20-s + 10.7·21-s + 5.71·22-s + 1.44·23-s + 2.64·24-s + 8.63·25-s − 2.63·27-s − 4.07·28-s + 3.22·29-s + 9.76·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.52·3-s + 0.5·4-s + 1.65·5-s + 1.07·6-s − 1.53·7-s − 0.353·8-s + 1.33·9-s − 1.16·10-s − 1.72·11-s − 0.763·12-s + 1.08·14-s − 2.52·15-s + 0.250·16-s + 0.816·17-s − 0.942·18-s − 0.229·19-s + 0.825·20-s + 2.35·21-s + 1.21·22-s + 0.301·23-s + 0.539·24-s + 1.72·25-s − 0.508·27-s − 0.769·28-s + 0.599·29-s + 1.78·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4498076716\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4498076716\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 23 | \( 1 - 1.44T + 23T^{2} \) |
| 29 | \( 1 - 3.22T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 - 1.05T + 43T^{2} \) |
| 47 | \( 1 - 3.95T + 47T^{2} \) |
| 53 | \( 1 + 7.94T + 53T^{2} \) |
| 59 | \( 1 - 1.81T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 + 1.81T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 7.57T + 83T^{2} \) |
| 89 | \( 1 - 4.91T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−7.936641963122060252350203147590, −6.97587076535837561930287361068, −6.60031573067667712445947324301, −5.81144891860919186042798076688, −5.54654030188770693806939798211, −4.90556483005522048600924125759, −3.32795378637141851612742728439, −2.60027437209605746233416439488, −1.59757209672983397690166097632, −0.41289113158626736094046493470,
0.41289113158626736094046493470, 1.59757209672983397690166097632, 2.60027437209605746233416439488, 3.32795378637141851612742728439, 4.90556483005522048600924125759, 5.54654030188770693806939798211, 5.81144891860919186042798076688, 6.60031573067667712445947324301, 6.97587076535837561930287361068, 7.936641963122060252350203147590