| L(s) = 1 | + (0.894 − 1.06i)2-s + (123. + 209. i)3-s + (177. + 1.00e3i)4-s + (−802. + 2.20e3i)5-s + (333. + 55.1i)6-s + (−2.83e3 + 1.60e4i)7-s + (2.46e3 + 1.42e3i)8-s + (−2.84e4 + 5.17e4i)9-s + (1.63e3 + 2.82e3i)10-s + (−5.72e4 − 1.57e5i)11-s + (−1.88e5 + 1.61e5i)12-s + (4.30e5 − 3.60e5i)13-s + (1.46e4 + 1.74e4i)14-s + (−5.60e5 + 1.05e5i)15-s + (−9.79e5 + 3.56e5i)16-s + (1.25e6 − 7.26e5i)17-s + ⋯ |
| L(s) = 1 | + (0.0279 − 0.0332i)2-s + (0.509 + 0.860i)3-s + (0.173 + 0.982i)4-s + (−0.256 + 0.705i)5-s + (0.0428 + 0.00708i)6-s + (−0.168 + 0.957i)7-s + (0.0752 + 0.0434i)8-s + (−0.481 + 0.876i)9-s + (0.0163 + 0.0282i)10-s + (−0.355 − 0.977i)11-s + (−0.757 + 0.649i)12-s + (1.15 − 0.972i)13-s + (0.0271 + 0.0323i)14-s + (−0.738 + 0.138i)15-s + (−0.934 + 0.340i)16-s + (0.886 − 0.511i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.979 - 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.184249 + 1.83168i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.184249 + 1.83168i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-123. - 209. i)T \) |
| good | 2 | \( 1 + (-0.894 + 1.06i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (802. - 2.20e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (2.83e3 - 1.60e4i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (5.72e4 + 1.57e5i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (-4.30e5 + 3.60e5i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (-1.25e6 + 7.26e5i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (7.05e5 - 1.22e6i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (6.23e6 - 1.09e6i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (4.27e6 - 5.08e6i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (-3.16e6 - 1.79e7i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (1.98e7 + 3.43e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (-1.11e8 - 1.33e8i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (6.16e7 - 2.24e7i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (-1.95e8 - 3.44e7i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 + 4.79e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-3.67e8 + 1.00e9i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (2.39e8 - 1.35e9i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (1.60e9 - 1.34e9i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (6.64e8 - 3.83e8i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (1.18e9 - 2.05e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (1.39e9 + 1.16e9i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (-1.23e9 + 1.46e9i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (-9.37e9 - 5.41e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-1.55e8 + 5.65e7i)T + (5.64e19 - 4.74e19i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.79284948446085891558815480948, −14.50316854012082856252774554414, −13.13844027645600748775414552569, −11.62662446096907778843047686817, −10.51812099388959444908860020556, −8.778721976012874907953395895732, −7.87123418246838307442581533187, −5.74938061138494181714575831685, −3.56283150066070517248281535848, −2.82622551784117856751309876130,
0.66201725582969910966467450385, 1.81388799558718979403597396596, 4.21619640283787232242036270635, 6.18904758304278382618349738173, 7.48518910423712592786895276468, 9.013580617202115928397126077002, 10.44157708142745033395570019037, 12.04700589377427913470792743390, 13.35436788996253964457501722445, 14.23037812465005982519135029996
