L(s) = 1 | + 2-s + 1.84·3-s + 4-s + 5-s + 1.84·6-s − 7-s + 8-s + 0.392·9-s + 10-s + 1.84·12-s − 1.78·13-s − 14-s + 1.84·15-s + 16-s − 7.29·17-s + 0.392·18-s − 3.17·19-s + 20-s − 1.84·21-s − 4.72·23-s + 1.84·24-s + 25-s − 1.78·26-s − 4.80·27-s − 28-s + 4.51·29-s + 1.84·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.06·3-s + 0.5·4-s + 0.447·5-s + 0.751·6-s − 0.377·7-s + 0.353·8-s + 0.130·9-s + 0.316·10-s + 0.531·12-s − 0.494·13-s − 0.267·14-s + 0.475·15-s + 0.250·16-s − 1.76·17-s + 0.0924·18-s − 0.727·19-s + 0.223·20-s − 0.401·21-s − 0.985·23-s + 0.375·24-s + 0.200·25-s − 0.349·26-s − 0.924·27-s − 0.188·28-s + 0.838·29-s + 0.336·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 1.84T + 3T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 + 3.17T + 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 - 4.51T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 4.73T + 37T^{2} \) |
| 41 | \( 1 - 4.17T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 3.74T + 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 - 9.71T + 59T^{2} \) |
| 61 | \( 1 + 9.80T + 61T^{2} \) |
| 67 | \( 1 + 6.96T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 - 6.05T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 17.5T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 5.08T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.31370607870860565429060604888, −6.72221128602981941593829212078, −6.11724105190863556269895873068, −5.31833625435808471805295376077, −4.46545907570758899360160493656, −3.86725054454041746865647200102, −3.02493676580132302758944729485, −2.31462741553372275049346561968, −1.84851509265254020088176189606, 0,
1.84851509265254020088176189606, 2.31462741553372275049346561968, 3.02493676580132302758944729485, 3.86725054454041746865647200102, 4.46545907570758899360160493656, 5.31833625435808471805295376077, 6.11724105190863556269895873068, 6.72221128602981941593829212078, 7.31370607870860565429060604888