| L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.0120 + 1.73i)3-s + (−0.939 + 0.342i)4-s + (2.57 + 2.16i)5-s + (1.70 − 0.312i)6-s + (−0.939 − 0.342i)7-s + (0.5 + 0.866i)8-s + (−2.99 + 0.0416i)9-s + (1.68 − 2.91i)10-s + (3.50 − 2.94i)11-s + (−0.603 − 1.62i)12-s + (−0.288 + 1.63i)13-s + (−0.173 + 0.984i)14-s + (−3.71 + 4.49i)15-s + (0.766 − 0.642i)16-s + (−3.24 + 5.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (0.00693 + 0.999i)3-s + (−0.469 + 0.171i)4-s + (1.15 + 0.967i)5-s + (0.695 − 0.127i)6-s + (−0.355 − 0.129i)7-s + (0.176 + 0.306i)8-s + (−0.999 + 0.0138i)9-s + (0.532 − 0.921i)10-s + (1.05 − 0.887i)11-s + (−0.174 − 0.468i)12-s + (−0.0800 + 0.454i)13-s + (−0.0464 + 0.263i)14-s + (−0.959 + 1.15i)15-s + (0.191 − 0.160i)16-s + (−0.787 + 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21624 + 0.626753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21624 + 0.626753i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (-0.0120 - 1.73i)T \) |
| 7 | \( 1 + (0.939 + 0.342i)T \) |
| good | 5 | \( 1 + (-2.57 - 2.16i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-3.50 + 2.94i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.288 - 1.63i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (3.24 - 5.62i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.61 - 6.26i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.38 - 1.95i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.17 + 6.69i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.09 + 2.94i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (1.49 - 2.58i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.427 - 2.42i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.16 + 6.01i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (3.60 + 1.31i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + (0.262 + 0.220i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.91 - 1.78i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.25 + 7.12i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.99 + 8.65i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.30 + 10.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.931 - 5.28i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.04 + 11.5i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (3.77 + 6.53i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.21 - 4.37i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−11.33385090820828308715262123902, −10.33539493855588551384533927581, −9.951889981159950861636929015345, −9.165489540901277782280739152091, −8.113201014500249471249167560556, −6.24706993113895463708757149072, −5.94120123693089291490747798729, −4.14189987449637822565587124445, −3.37499785895821840265339964716, −2.01436005752283909463394293739,
1.04067618346151303499055817806, 2.51847555501095664349928624417, 4.69817366918939020047886317133, 5.55326971304491867080638815733, 6.62178442477954360889239324474, 7.17741360823195040316350556248, 8.556188902432649906279090251227, 9.222824174080217837813155579278, 9.841646720026871330733677768237, 11.46255352138540298481164668168
