L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.866 − 0.499i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.866 + 0.5i)11-s + i·13-s − 0.999·14-s + (0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + (0.866 + 0.499i)22-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.499i)24-s − 25-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.866 − 0.499i)6-s + (0.5 − 0.866i)7-s − 8-s + (−0.866 + 0.5i)11-s + i·13-s − 0.999·14-s + (0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + (0.866 + 0.499i)22-s + (−0.866 + 0.5i)23-s + (−0.866 + 0.499i)24-s − 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8506803330\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8506803330\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 - iT \) |
| 31 | \( 1 + iT \) |
good | 2 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 2iT - T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 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
−11.26796448480203553455967327815, −10.12367844768355126479479256389, −9.703931011239355378131816532714, −8.419997929838757551138900962459, −7.80136424471324208933566855871, −6.80157891457166250840806925404, −5.34181723325164253808521418908, −3.87440947248701690403815744092, −2.53886931670367344245009654332, −1.60641294567980630786349282588,
2.66549256429074363496068972055, 3.48762579600775443326674568583, 5.32693212338431208144727800978, 5.98085371872081457354262675499, 7.50834843972665177469714238890, 8.216775647198376174303104780645, 8.640817716924347589576077843595, 9.675764230718966348357525302388, 10.50276671636804297753257264728, 11.97706412431208658778877458540