L(s) = 1 | − 0.906i·2-s + 1.17·4-s + (0.370 + 2.20i)5-s − 2.59i·7-s − 2.88i·8-s + (1.99 − 0.336i)10-s − 0.741·11-s − 3.78i·13-s − 2.35·14-s − 0.258·16-s − 3.16i·17-s + 19-s + (0.436 + 2.59i)20-s + 0.672i·22-s − 0.570i·23-s + ⋯ |
L(s) = 1 | − 0.641i·2-s + 0.588·4-s + (0.165 + 0.986i)5-s − 0.981i·7-s − 1.01i·8-s + (0.632 − 0.106i)10-s − 0.223·11-s − 1.05i·13-s − 0.629·14-s − 0.0647·16-s − 0.768i·17-s + 0.229·19-s + (0.0975 + 0.580i)20-s + 0.143i·22-s − 0.119i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43716 - 1.21581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43716 - 1.21581i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.370 - 2.20i)T \) |
| 19 | \( 1 - T \) |
good | 2 | \( 1 + 0.906iT - 2T^{2} \) |
| 7 | \( 1 + 2.59iT - 7T^{2} \) |
| 11 | \( 1 + 0.741T + 11T^{2} \) |
| 13 | \( 1 + 3.78iT - 13T^{2} \) |
| 17 | \( 1 + 3.16iT - 17T^{2} \) |
| 23 | \( 1 + 0.570iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 5.83T + 31T^{2} \) |
| 37 | \( 1 - 1.40iT - 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 2.59iT - 43T^{2} \) |
| 47 | \( 1 - 5.08iT - 47T^{2} \) |
| 53 | \( 1 - 0.160iT - 53T^{2} \) |
| 59 | \( 1 - 8.35T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 - 14.8iT - 67T^{2} \) |
| 71 | \( 1 + 3.64T + 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 - 4.19iT - 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 - 3.78iT - 97T^{2} \) |
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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
−10.32468240158380320972306879611, −9.603501358918820012999657483820, −8.094663516375591439809703847252, −7.30763250343128350611339042101, −6.70846533133356913665145977568, −5.72675792090165696003312939635, −4.31798559317634937561398688010, −3.17644940240857250388479928074, −2.58514641659851077324678392742, −0.961246311443921929651030259647,
1.63373839324785596726698187873, 2.67431927381163926755411113721, 4.32289491246926012575726101773, 5.29521076006401585573314191413, 6.02149632613426477450951275997, 6.78761352205866707793526320130, 7.969544884928370259141898104442, 8.540503832087814769025826280019, 9.281037133292102684033768338719, 10.27103432802632473901376876810