| L(s) = 1 | + (−0.913 − 0.406i)2-s + (0.978 + 0.207i)3-s + (0.669 + 0.743i)4-s + (0.0177 − 0.168i)5-s + (−0.809 − 0.587i)6-s + (2.46 + 0.969i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (−0.0847 + 0.146i)10-s + (0.324 + 3.30i)11-s + (0.499 + 0.866i)12-s + (0.621 − 0.451i)13-s + (−1.85 − 1.88i)14-s + (0.0523 − 0.161i)15-s + (−0.104 + 0.994i)16-s + (−3.72 + 1.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 0.287i)2-s + (0.564 + 0.120i)3-s + (0.334 + 0.371i)4-s + (0.00792 − 0.0753i)5-s + (−0.330 − 0.239i)6-s + (0.930 + 0.366i)7-s + (−0.109 − 0.336i)8-s + (0.304 + 0.135i)9-s + (−0.0268 + 0.0464i)10-s + (0.0979 + 0.995i)11-s + (0.144 + 0.249i)12-s + (0.172 − 0.125i)13-s + (−0.495 − 0.504i)14-s + (0.0135 − 0.0416i)15-s + (−0.0261 + 0.248i)16-s + (−0.904 + 0.402i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.38013 + 0.183449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38013 + 0.183449i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.913 + 0.406i)T \) |
| 3 | \( 1 + (-0.978 - 0.207i)T \) |
| 7 | \( 1 + (-2.46 - 0.969i)T \) |
| 11 | \( 1 + (-0.324 - 3.30i)T \) |
| good | 5 | \( 1 + (-0.0177 + 0.168i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (-0.621 + 0.451i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.72 - 1.65i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-2.42 + 2.69i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.717 - 1.24i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.02 + 3.15i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.541 - 5.15i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (3.94 - 0.837i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (1.29 + 3.99i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 3.56T + 43T^{2} \) |
| 47 | \( 1 + (-5.60 + 6.22i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.491 - 4.67i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-2.22 - 2.47i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.627 + 5.96i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-2.70 + 4.68i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.83 + 2.05i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.41 - 1.56i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (13.0 + 5.79i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (13.2 + 9.62i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.842 + 1.45i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (12.4 - 9.06i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.93974092664890347969742947394, −10.18740553033053355507061821817, −9.069918830878252962954993177132, −8.653328541147411457149475967602, −7.59517926130861568513298192809, −6.81201182803564958083507720449, −5.21333132075527752313260468419, −4.20034143020001888294596773545, −2.71176056158779059439385009986, −1.59665675815775508843609049143,
1.19268839401717517292346854993, 2.69854345833482901066283301270, 4.14998683137625231818221907003, 5.42097220837767411886867746519, 6.63417345085445189804782719778, 7.50692594402473957567275101755, 8.419297665622960507241622303998, 8.919351274353882589068483352869, 10.05292401896293947878418880938, 10.98528100934106948068708852205
