| L(s) = 1 | + (2.37 − 0.866i)2-s + (3.37 − 2.83i)4-s + (−0.0812 − 0.460i)5-s + (−2.47 − 2.07i)7-s + (3.05 − 5.28i)8-s + (−0.592 − 1.02i)10-s + (0.539 − 3.05i)11-s + (2.05 + 0.747i)13-s + (−7.67 − 2.79i)14-s + (1.15 − 6.53i)16-s + (1.5 + 2.59i)17-s + (−0.0209 + 0.0362i)19-s + (−1.58 − 1.32i)20-s + (−1.36 − 7.74i)22-s + (−4.67 + 3.92i)23-s + ⋯ |
| L(s) = 1 | + (1.68 − 0.612i)2-s + (1.68 − 1.41i)4-s + (−0.0363 − 0.206i)5-s + (−0.934 − 0.783i)7-s + (1.07 − 1.86i)8-s + (−0.187 − 0.324i)10-s + (0.162 − 0.922i)11-s + (0.569 + 0.207i)13-s + (−2.05 − 0.746i)14-s + (0.288 − 1.63i)16-s + (0.363 + 0.630i)17-s + (−0.00480 + 0.00832i)19-s + (−0.353 − 0.296i)20-s + (−0.291 − 1.65i)22-s + (−0.975 + 0.818i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.52166 - 2.67280i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.52166 - 2.67280i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-2.37 + 0.866i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (0.0812 + 0.460i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.47 + 2.07i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.539 + 3.05i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.05 - 0.747i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0209 - 0.0362i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.67 - 3.92i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-6.17 + 2.24i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (4.76 - 4.00i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.79 + 3.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.23 - 2.63i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.102 - 0.579i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-7.40 - 6.20i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + 4.95T + 53T^{2} \) |
| 59 | \( 1 + (-1.48 - 8.40i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (0.971 + 0.815i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (9.40 + 3.42i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.91 - 10.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.11 + 7.13i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.3 - 3.77i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (1.41 - 0.516i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (7.93 - 13.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.23 + 18.3i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.60993484941809614282451440358, −9.655670388346522389186077445917, −8.452274416383808175080923055614, −7.14476928520651747175142878539, −6.21016745354160102399001971201, −5.69609799201081114586181535754, −4.36025548447866173911538772970, −3.70073926471724745582060194903, −2.88251453772076108895226351368, −1.24079584972774686738362277000,
2.40615078735739029379176355235, 3.26996753321113511141156833973, 4.28469084225015372729719388170, 5.26912491901641081810838465170, 6.11474882552887929582592192987, 6.76208138739877573405766192770, 7.58498897922067043656716548860, 8.768336618569824948477490675895, 9.802119571378896860390402737052, 10.85940383750285413675460280550
