| L(s) = 1 | + 3.27·5-s − 0.418i·7-s − 6.50i·11-s + 7.27·17-s + 4.35i·19-s + 8.71i·23-s + 5.72·25-s − 1.37i·35-s − 5.67i·43-s − 13.4i·47-s + 6.82·49-s − 21.3i·55-s − 11.2·61-s + 5.82·73-s − 2.72·77-s + ⋯ |
| L(s) = 1 | + 1.46·5-s − 0.158i·7-s − 1.96i·11-s + 1.76·17-s + 0.999i·19-s + 1.81i·23-s + 1.14·25-s − 0.231i·35-s − 0.865i·43-s − 1.96i·47-s + 0.974·49-s − 2.87i·55-s − 1.44·61-s + 0.681·73-s − 0.310·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.609039713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.609039713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 + 0.418iT - 7T^{2} \) |
| 11 | \( 1 + 6.50iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 7.27T + 17T^{2} \) |
| 23 | \( 1 - 8.71iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 5.67iT - 43T^{2} \) |
| 47 | \( 1 + 13.4iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 8.71iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.827569892836983367786813753287, −8.067638229368781897546033807902, −7.28226926142121971952429781873, −6.13604393540101163221459722061, −5.67565317516073225597428295887, −5.32520059819713431039158272755, −3.65133520122593812866803010096, −3.19921106142275687498963307861, −1.87598233346154234226327183218, −0.974372968958819450486375148832,
1.23029007753519793489287854890, 2.21923774169182495897521653860, 2.88442811725028228128903644658, 4.40981939200478667690986524616, 4.97161909276522138717738041736, 5.84513974566926806013730997380, 6.54732097209835806309112001867, 7.29263881756819190159975844470, 8.090338785249253886553008269021, 9.247012055816029834078527307432
