L(s) = 1 | + (−0.173 + 0.984i)2-s + (−1.17 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (1.17 − 0.984i)6-s + (0.5 − 0.866i)8-s + (0.233 + 1.32i)9-s + (0.939 − 1.62i)11-s + (0.766 + 1.32i)12-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s − 1.34·18-s + (1.43 + 1.20i)22-s + (−1.43 + 0.524i)24-s + (0.766 − 0.642i)25-s + (0.266 − 0.460i)27-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)2-s + (−1.17 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (1.17 − 0.984i)6-s + (0.5 − 0.866i)8-s + (0.233 + 1.32i)9-s + (0.939 − 1.62i)11-s + (0.766 + 1.32i)12-s + (0.766 + 0.642i)16-s + (−0.173 + 0.984i)17-s − 1.34·18-s + (1.43 + 1.20i)22-s + (−1.43 + 0.524i)24-s + (0.766 − 0.642i)25-s + (0.266 − 0.460i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6149541849\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6149541849\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 17 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 59 | \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
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
−8.610722784017316713599147020722, −7.979395329010842470685174686176, −7.11786304809919399987813315469, −6.31804651217851625577876624011, −6.15362952992457032304134659536, −5.37905917138244937722781899376, −4.40081446980589270815937468625, −3.38524134251195502051527243661, −1.57122371661012060054120018349, −0.58811031395369308694813600679,
1.20196901327071213545914081562, 2.47474454466266280823803541918, 3.68369643626173809122005502347, 4.46725323643901804253596394269, 4.86605197479917672675218687782, 5.70310730569100461597635185829, 6.81890433077880118429633820384, 7.48508136173940062818496868835, 8.807206526012708317851129704791, 9.354934138570114276385660053727