L(s) = 1 | + (0.866 + 0.5i)2-s + i·3-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − i·8-s − 9-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s − 0.999i·14-s + (0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)21-s − 0.999·22-s + (−0.866 − 0.5i)23-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + i·3-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − i·8-s − 9-s + (−0.866 + 0.5i)11-s + (−0.5 + 0.866i)13-s − 0.999i·14-s + (0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + (0.5 − 0.866i)19-s + (0.866 − 0.5i)21-s − 0.999·22-s + (−0.866 − 0.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9148019240\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9148019240\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 67 | \( 1 + T \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.06396372837765419088243433711, −12.06300494364004050974075833294, −10.58392425370295767773492326349, −10.06209924086214089659810549732, −9.105257492640903248615743272686, −7.50380806710069990355949851386, −6.44965633872900794959990974483, −5.16449676002361227003296168042, −4.43329433667706348483214198311, −3.25837212886460724700325050683,
2.45801410805512419283023089293, 3.33240206634039053429583645498, 5.35801835167399409429081676615, 5.86237701988828560970330625284, 7.63639696742692720184301075209, 8.244543117062685754216292349168, 9.631241605773465811531850408459, 11.03612746028804515040337747260, 12.03619230109981386841385096944, 12.52850858340813686363197555039