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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4163892498\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4163892498\) |
\(L(1)\) |
|
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
−12.77128818781416612137199332616, −12.11494600182646605770811471076, −10.68161793487289808136445453498, −9.829723607099874443952264537757, −9.028142894516585149340121942550, −8.419824572179928034669012973985, −7.00874152878573842366442728339, −5.86192943407996183961397074661, −4.43400830213849649691932529700, −3.06862958808679531741982289930,
1.15003496418143021220767078789, 2.86926749676441456327653966422, 4.86565227648055727757135798497, 6.56762273525719702520866453237, 7.18369849732048065400585758462, 8.656001310041740246012097114734, 9.216889091870554163749756343538, 10.41683092778607604561645014316, 11.41071828953632805737585955614, 12.00635294644551483926887931755