| L(s) = 1 | + (−0.592 − 3.36i)2-s + (−5.15 − 0.614i)3-s + (−3.42 + 1.24i)4-s + (−8.41 − 7.05i)5-s + (0.992 + 17.7i)6-s + (4.14 + 1.50i)7-s + (−7.42 − 12.8i)8-s + (26.2 + 6.34i)9-s + (−18.7 + 32.4i)10-s + (44.1 − 37.0i)11-s + (18.4 − 4.32i)12-s + (6.80 − 38.6i)13-s + (2.61 − 14.8i)14-s + (39.0 + 41.5i)15-s + (−61.1 + 51.3i)16-s + (−44.7 + 77.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.209 − 1.18i)2-s + (−0.992 − 0.118i)3-s + (−0.428 + 0.155i)4-s + (−0.752 − 0.631i)5-s + (0.0675 + 1.20i)6-s + (0.223 + 0.0813i)7-s + (−0.328 − 0.568i)8-s + (0.972 + 0.234i)9-s + (−0.592 + 1.02i)10-s + (1.21 − 1.01i)11-s + (0.443 − 0.104i)12-s + (0.145 − 0.823i)13-s + (0.0498 − 0.282i)14-s + (0.672 + 0.715i)15-s + (−0.956 + 0.802i)16-s + (−0.638 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.144224 - 0.710289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.144224 - 0.710289i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.15 + 0.614i)T \) |
| good | 2 | \( 1 + (0.592 + 3.36i)T + (-7.51 + 2.73i)T^{2} \) |
| 5 | \( 1 + (8.41 + 7.05i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (-4.14 - 1.50i)T + (262. + 220. i)T^{2} \) |
| 11 | \( 1 + (-44.1 + 37.0i)T + (231. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-6.80 + 38.6i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (44.7 - 77.4i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-29.6 - 51.4i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-21.4 + 7.81i)T + (9.32e3 - 7.82e3i)T^{2} \) |
| 29 | \( 1 + (10.2 + 58.0i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (-313. + 114. i)T + (2.28e4 - 1.91e4i)T^{2} \) |
| 37 | \( 1 + (47.4 - 82.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (62.8 - 356. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-361. + 303. i)T + (1.38e4 - 7.82e4i)T^{2} \) |
| 47 | \( 1 + (222. + 81.0i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 - 391.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-429. - 360. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (158. + 57.7i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (94.5 - 536. i)T + (-2.82e5 - 1.02e5i)T^{2} \) |
| 71 | \( 1 + (97.0 - 168. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (80.6 + 139. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-63.1 - 358. i)T + (-4.63e5 + 1.68e5i)T^{2} \) |
| 83 | \( 1 + (98.1 + 556. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-187. - 324. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (446. - 374. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−16.51090354761320894810358685319, −15.36240245711233726513369485056, −13.17532280454961416937206895749, −12.00995356787666590487155397342, −11.42682065278233527575734097557, −10.19780165562194632737292945885, −8.431376420183442030142998978930, −6.21388934444744618113793531438, −4.03325196703072359429278301910, −0.941049926643177377366575243940,
4.64736029416186619764215571201, 6.64190141612858334722241827442, 7.25239608185441677333853431128, 9.283191562313091629147197717449, 11.24835414794378669312802790621, 11.95155998677233385220224852843, 14.18071235871343662939649572348, 15.34926261309240068898435120914, 16.09196439505006584084233837628, 17.30503945324959783049729899976
