| L(s) = 1 | + (3.59 − 4.28i)2-s + (−7.31 − 5.24i)3-s + (−2.66 − 15.1i)4-s + (2.69 − 7.40i)5-s + (−48.8 + 12.5i)6-s + (7.02 − 39.8i)7-s + (3.09 + 1.78i)8-s + (25.9 + 76.7i)9-s + (−22.0 − 38.2i)10-s + (29.8 + 81.9i)11-s + (−59.8 + 124. i)12-s + (181. − 152. i)13-s + (−145. − 173. i)14-s + (−58.5 + 40.0i)15-s + (249. − 90.9i)16-s + (15.5 − 8.95i)17-s + ⋯ |
| L(s) = 1 | + (0.899 − 1.07i)2-s + (−0.812 − 0.582i)3-s + (−0.166 − 0.945i)4-s + (0.107 − 0.296i)5-s + (−1.35 + 0.347i)6-s + (0.143 − 0.813i)7-s + (0.0483 + 0.0279i)8-s + (0.320 + 0.947i)9-s + (−0.220 − 0.382i)10-s + (0.246 + 0.677i)11-s + (−0.415 + 0.865i)12-s + (1.07 − 0.903i)13-s + (−0.743 − 0.886i)14-s + (−0.260 + 0.177i)15-s + (0.975 − 0.355i)16-s + (0.0536 − 0.0309i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.858503 - 1.49455i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.858503 - 1.49455i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.31 + 5.24i)T \) |
| good | 2 | \( 1 + (-3.59 + 4.28i)T + (-2.77 - 15.7i)T^{2} \) |
| 5 | \( 1 + (-2.69 + 7.40i)T + (-478. - 401. i)T^{2} \) |
| 7 | \( 1 + (-7.02 + 39.8i)T + (-2.25e3 - 821. i)T^{2} \) |
| 11 | \( 1 + (-29.8 - 81.9i)T + (-1.12e4 + 9.41e3i)T^{2} \) |
| 13 | \( 1 + (-181. + 152. i)T + (4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-15.5 + 8.95i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (248. - 429. i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (687. - 121. i)T + (2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (735. - 876. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (149. + 846. i)T + (-8.67e5 + 3.15e5i)T^{2} \) |
| 37 | \( 1 + (650. + 1.12e3i)T + (-9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-1.95e3 - 2.33e3i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-344. + 125. i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-80.2 - 14.1i)T + (4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 - 1.62e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-165. + 453. i)T + (-9.28e6 - 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-956. + 5.42e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (319. - 267. i)T + (3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (6.21e3 - 3.58e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (4.04e3 - 7.01e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (3.04e3 + 2.55e3i)T + (6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-5.21e3 + 6.21e3i)T + (-8.24e6 - 4.67e7i)T^{2} \) |
| 89 | \( 1 + (2.70e3 + 1.56e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-6.71e3 + 2.44e3i)T + (6.78e7 - 5.69e7i)T^{2} \) |
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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
−16.37047040385162049816772361883, −14.38853484992535794151107339064, −13.14016999696658616107847337056, −12.52831276492316773047226963482, −11.20198927847272450411455065312, −10.31630938051161474088144370365, −7.71690969685961277045627797548, −5.73263524728805320096629260062, −4.08239633529226382917425988926, −1.47608127571523755904688309714,
4.16880602615993070777665084783, 5.73869609966999200934587611131, 6.61079635902640295921360752409, 8.823248481203960325160907399689, 10.73076145540842750822446755803, 12.01479255793757566121740342265, 13.57170127475096962945487782612, 14.75649631116676605109001770992, 15.77682496730493281232110570245, 16.47172497722973650828880189124
