L(s) = 1 | − 2-s − 2.56·3-s + 4-s − 5-s + 2.56·6-s + 3.60·7-s − 8-s + 3.57·9-s + 10-s − 4.57·11-s − 2.56·12-s + 5.17·13-s − 3.60·14-s + 2.56·15-s + 16-s + 3.02·17-s − 3.57·18-s − 1.57·19-s − 20-s − 9.25·21-s + 4.57·22-s + 0.529·23-s + 2.56·24-s + 25-s − 5.17·26-s − 1.47·27-s + 3.60·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.48·3-s + 0.5·4-s − 0.447·5-s + 1.04·6-s + 1.36·7-s − 0.353·8-s + 1.19·9-s + 0.316·10-s − 1.38·11-s − 0.740·12-s + 1.43·13-s − 0.964·14-s + 0.662·15-s + 0.250·16-s + 0.732·17-s − 0.842·18-s − 0.360·19-s − 0.223·20-s − 2.01·21-s + 0.975·22-s + 0.110·23-s + 0.523·24-s + 0.200·25-s − 1.01·26-s − 0.283·27-s + 0.681·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 - 5.17T + 13T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 + 1.57T + 19T^{2} \) |
| 23 | \( 1 - 0.529T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 + 0.541T + 31T^{2} \) |
| 37 | \( 1 + 1.81T + 37T^{2} \) |
| 41 | \( 1 - 8.81T + 41T^{2} \) |
| 43 | \( 1 + 5.60T + 43T^{2} \) |
| 47 | \( 1 + 8.56T + 47T^{2} \) |
| 53 | \( 1 + 7.29T + 53T^{2} \) |
| 59 | \( 1 - 4.84T + 59T^{2} \) |
| 61 | \( 1 - 9.54T + 61T^{2} \) |
| 67 | \( 1 + 3.59T + 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 1.32T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 9.51T + 89T^{2} \) |
| 97 | \( 1 - 9.40T + 97T^{2} \) |
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
−7.80375874542898255073395134313, −7.15035723432197972174281714193, −6.19975590000284085480693317817, −5.59836709732700837857957426770, −5.07516841891464041226069616405, −4.26734428079638650512300316382, −3.21186807356829236184699160064, −1.88293545178097197448067683312, −1.07720731421713702176680721450, 0,
1.07720731421713702176680721450, 1.88293545178097197448067683312, 3.21186807356829236184699160064, 4.26734428079638650512300316382, 5.07516841891464041226069616405, 5.59836709732700837857957426770, 6.19975590000284085480693317817, 7.15035723432197972174281714193, 7.80375874542898255073395134313