L(s) = 1 | + (0.445 − 0.895i)2-s + (−0.602 − 0.798i)4-s + (−0.243 − 0.489i)5-s + (−0.982 + 0.183i)8-s + (0.0922 − 0.995i)9-s − 0.547·10-s + (−1.12 − 0.435i)13-s + (−0.273 + 0.961i)16-s + (1.93 + 0.361i)17-s + (−0.850 − 0.526i)18-s + (−0.243 + 0.489i)20-s + (0.422 − 0.558i)25-s + (−0.890 + 0.811i)26-s + (−0.510 + 1.79i)29-s + (0.739 + 0.673i)32-s + ⋯ |
L(s) = 1 | + (0.445 − 0.895i)2-s + (−0.602 − 0.798i)4-s + (−0.243 − 0.489i)5-s + (−0.982 + 0.183i)8-s + (0.0922 − 0.995i)9-s − 0.547·10-s + (−1.12 − 0.435i)13-s + (−0.273 + 0.961i)16-s + (1.93 + 0.361i)17-s + (−0.850 − 0.526i)18-s + (−0.243 + 0.489i)20-s + (0.422 − 0.558i)25-s + (−0.890 + 0.811i)26-s + (−0.510 + 1.79i)29-s + (0.739 + 0.673i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9703735842\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9703735842\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.445 + 0.895i)T \) |
| 137 | \( 1 + (-0.932 + 0.361i)T \) |
good | 3 | \( 1 + (-0.0922 + 0.995i)T^{2} \) |
| 5 | \( 1 + (0.243 + 0.489i)T + (-0.602 + 0.798i)T^{2} \) |
| 7 | \( 1 + (-0.739 - 0.673i)T^{2} \) |
| 11 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 13 | \( 1 + (1.12 + 0.435i)T + (0.739 + 0.673i)T^{2} \) |
| 17 | \( 1 + (-1.93 - 0.361i)T + (0.932 + 0.361i)T^{2} \) |
| 19 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 23 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 29 | \( 1 + (0.510 - 1.79i)T + (-0.850 - 0.526i)T^{2} \) |
| 31 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 37 | \( 1 - 0.891T + T^{2} \) |
| 41 | \( 1 + 1.70T + T^{2} \) |
| 43 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 47 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 53 | \( 1 + (-1.67 - 1.03i)T + (0.445 + 0.895i)T^{2} \) |
| 59 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 61 | \( 1 + (0.0505 + 0.544i)T + (-0.982 + 0.183i)T^{2} \) |
| 67 | \( 1 + (-0.739 - 0.673i)T^{2} \) |
| 71 | \( 1 + (0.273 + 0.961i)T^{2} \) |
| 73 | \( 1 + (-0.172 + 0.0666i)T + (0.739 - 0.673i)T^{2} \) |
| 79 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 83 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 89 | \( 1 + (-0.658 - 1.32i)T + (-0.602 + 0.798i)T^{2} \) |
| 97 | \( 1 + (1.20 + 1.59i)T + (-0.273 + 0.961i)T^{2} \) |
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
−10.67108184900301690926338528395, −9.961439132859558333188833058191, −9.222343287329648639713378699914, −8.258182821502703212030161394343, −7.06680760943304423113463121455, −5.76585659945057369149871331576, −4.99813297707642116203511269558, −3.83100561696843605075067149370, −2.91760769099016069518977105394, −1.16322710598427185871476176298,
2.56912780228779511222112919657, 3.78725607926130540990708264525, 4.95954063626901973371513896438, 5.66958712981339164935687806437, 6.98473140183762384605112220964, 7.55551467341900377102845177665, 8.264600285440680910004515677825, 9.573841384639247646455020209284, 10.22408453654852227501946026030, 11.59303170169096129078575047125