| L(s) = 1 | − 2-s − 0.381·3-s + 4-s + 0.381·6-s − 3·7-s − 8-s − 2.85·9-s + 4.23·11-s − 0.381·12-s + 13-s + 3·14-s + 16-s + 1.14·17-s + 2.85·18-s + 5.85·19-s + 1.14·21-s − 4.23·22-s − 1.76·23-s + 0.381·24-s − 26-s + 2.23·27-s − 3·28-s − 9.47·29-s − 0.236·31-s − 32-s − 1.61·33-s − 1.14·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.220·3-s + 0.5·4-s + 0.155·6-s − 1.13·7-s − 0.353·8-s − 0.951·9-s + 1.27·11-s − 0.110·12-s + 0.277·13-s + 0.801·14-s + 0.250·16-s + 0.277·17-s + 0.672·18-s + 1.34·19-s + 0.250·21-s − 0.903·22-s − 0.367·23-s + 0.0779·24-s − 0.196·26-s + 0.430·27-s − 0.566·28-s − 1.75·29-s − 0.0423·31-s − 0.176·32-s − 0.281·33-s − 0.196·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 1.14T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 + 9.47T + 29T^{2} \) |
| 31 | \( 1 + 0.236T + 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 6.23T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 + 8.85T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 + 4.52T + 83T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 - 9.56T + 97T^{2} \) |
| 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
−9.308624594590718829355598842595, −8.717094732779529229402441367208, −7.66104099912829551977679885248, −6.79290467285443458940181043493, −6.11455789499729706113166584997, −5.33553569681958815450739778221, −3.69978163618871470655948913228, −3.08926829191259052804792617530, −1.53229775322296022596641864282, 0,
1.53229775322296022596641864282, 3.08926829191259052804792617530, 3.69978163618871470655948913228, 5.33553569681958815450739778221, 6.11455789499729706113166584997, 6.79290467285443458940181043493, 7.66104099912829551977679885248, 8.717094732779529229402441367208, 9.308624594590718829355598842595
