L(s) = 1 | + (1.10 − 1.91i)2-s + (−1.44 − 2.51i)4-s + (−1.06 + 1.83i)5-s + (2.63 − 0.272i)7-s − 1.98·8-s + (2.34 + 4.06i)10-s + (2.39 + 4.14i)11-s + 13-s + (2.39 − 5.34i)14-s + (0.697 − 1.20i)16-s + (−1.88 − 3.27i)17-s + (1.78 − 3.08i)19-s + 6.15·20-s + 10.5·22-s + (2.23 − 3.87i)23-s + ⋯ |
L(s) = 1 | + (0.782 − 1.35i)2-s + (−0.724 − 1.25i)4-s + (−0.474 + 0.822i)5-s + (0.994 − 0.102i)7-s − 0.703·8-s + (0.742 + 1.28i)10-s + (0.721 + 1.25i)11-s + 0.277·13-s + (0.638 − 1.42i)14-s + (0.174 − 0.301i)16-s + (−0.458 − 0.793i)17-s + (0.409 − 0.708i)19-s + 1.37·20-s + 2.25·22-s + (0.466 − 0.807i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.91928 - 1.62363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91928 - 1.62363i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.63 + 0.272i)T \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + (-1.10 + 1.91i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.06 - 1.83i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.39 - 4.14i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.88 + 3.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.78 + 3.08i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.23 + 3.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.90T + 29T^{2} \) |
| 31 | \( 1 + (-1.88 - 3.26i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.81 - 4.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 3.40T + 43T^{2} \) |
| 47 | \( 1 + (3.55 - 6.15i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.19 + 10.7i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.39 - 4.14i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.60 - 2.77i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.44 - 2.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + (3.85 + 6.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.58 + 4.48i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + (-1.83 + 3.17i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.40T + 97T^{2} \) |
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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
−10.42473865585087695764195159678, −9.565616410728435341618707511860, −8.484220665229477408904868840617, −7.28058366306710341990465052609, −6.67970964625285003736209104925, −4.88139403259339428511881500499, −4.64249509231711905919317394734, −3.42878302711723011385344953584, −2.51562220126975598413937742939, −1.34611435391039071731205465820,
1.34655831532777280177977784130, 3.56522816085926870298947120015, 4.33799121873479505115125097298, 5.21615834105578046507712550938, 5.94089207482769393864247984064, 6.82727775055513084595208973815, 8.007340163298763068208229264406, 8.326160250107991150017411326044, 9.037780087432242006869910727277, 10.55700577074519834006355907372