| L(s) = 1 | + 2-s − 1.39·3-s + 4-s + 0.539·5-s − 1.39·6-s + 5.05·7-s + 8-s − 1.04·9-s + 0.539·10-s − 1.12·11-s − 1.39·12-s − 2.68·13-s + 5.05·14-s − 0.754·15-s + 16-s − 4.30·17-s − 1.04·18-s − 19-s + 0.539·20-s − 7.07·21-s − 1.12·22-s + 7.92·23-s − 1.39·24-s − 4.70·25-s − 2.68·26-s + 5.65·27-s + 5.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.807·3-s + 0.5·4-s + 0.241·5-s − 0.570·6-s + 1.91·7-s + 0.353·8-s − 0.348·9-s + 0.170·10-s − 0.339·11-s − 0.403·12-s − 0.745·13-s + 1.35·14-s − 0.194·15-s + 0.250·16-s − 1.04·17-s − 0.246·18-s − 0.229·19-s + 0.120·20-s − 1.54·21-s − 0.239·22-s + 1.65·23-s − 0.285·24-s − 0.941·25-s − 0.527·26-s + 1.08·27-s + 0.955·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.823964691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.823964691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 - 0.539T + 5T^{2} \) |
| 7 | \( 1 - 5.05T + 7T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 23 | \( 1 - 7.92T + 23T^{2} \) |
| 29 | \( 1 + 0.281T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 - 1.71T + 37T^{2} \) |
| 41 | \( 1 - 0.408T + 41T^{2} \) |
| 43 | \( 1 - 5.34T + 43T^{2} \) |
| 47 | \( 1 + 3.58T + 47T^{2} \) |
| 53 | \( 1 + 3.58T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 7.49T + 67T^{2} \) |
| 71 | \( 1 - 9.98T + 71T^{2} \) |
| 73 | \( 1 + 7.18T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 - 7.00T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
| 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
−7.65468525712461011933210344143, −7.09288022714840386305559300881, −6.29261642965882708706946518963, −5.45450789885037841515749746018, −5.07052497081745451418224641841, −4.65310426706034694113225316781, −3.72275668046139438395917820629, −2.44654422246039743139480355444, −1.99773434100919927613276242010, −0.77451743182509831186676039861,
0.77451743182509831186676039861, 1.99773434100919927613276242010, 2.44654422246039743139480355444, 3.72275668046139438395917820629, 4.65310426706034694113225316781, 5.07052497081745451418224641841, 5.45450789885037841515749746018, 6.29261642965882708706946518963, 7.09288022714840386305559300881, 7.65468525712461011933210344143
