L(s) = 1 | − 2-s − 2.42·3-s + 4-s − 5-s + 2.42·6-s − 2.05·7-s − 8-s + 2.86·9-s + 10-s + 5.02·11-s − 2.42·12-s − 13-s + 2.05·14-s + 2.42·15-s + 16-s + 2.89·17-s − 2.86·18-s + 5.54·19-s − 20-s + 4.98·21-s − 5.02·22-s − 7.87·23-s + 2.42·24-s + 25-s + 26-s + 0.320·27-s − 2.05·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.447·5-s + 0.988·6-s − 0.777·7-s − 0.353·8-s + 0.955·9-s + 0.316·10-s + 1.51·11-s − 0.699·12-s − 0.277·13-s + 0.549·14-s + 0.625·15-s + 0.250·16-s + 0.702·17-s − 0.675·18-s + 1.27·19-s − 0.223·20-s + 1.08·21-s − 1.07·22-s − 1.64·23-s + 0.494·24-s + 0.200·25-s + 0.196·26-s + 0.0616·27-s − 0.388·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6097830891\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6097830891\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 - 5.02T + 11T^{2} \) |
| 17 | \( 1 - 2.89T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 + 7.87T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 37 | \( 1 + 0.769T + 37T^{2} \) |
| 41 | \( 1 + 2.79T + 41T^{2} \) |
| 43 | \( 1 - 1.72T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 5.73T + 53T^{2} \) |
| 59 | \( 1 - 4.54T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 + 6.54T + 83T^{2} \) |
| 89 | \( 1 + 6.00T + 89T^{2} \) |
| 97 | \( 1 - 1.81T + 97T^{2} \) |
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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
−8.419108828991078985262637624537, −7.65496787249752713351907613944, −6.73035960426197039168329261782, −6.44457281028885519804735654476, −5.70132414356025834154565919042, −4.80094249440269524658495875518, −3.84529531061962480738953526456, −3.00292024607008142858887319644, −1.44823506623313743267469664374, −0.57836570931296115123040963606,
0.57836570931296115123040963606, 1.44823506623313743267469664374, 3.00292024607008142858887319644, 3.84529531061962480738953526456, 4.80094249440269524658495875518, 5.70132414356025834154565919042, 6.44457281028885519804735654476, 6.73035960426197039168329261782, 7.65496787249752713351907613944, 8.419108828991078985262637624537