| L(s) = 1 | + (−22.3 + 26.6i)2-s + (195. + 144. i)3-s + (−32.2 − 183. i)4-s + (−1.50e3 + 4.13e3i)5-s + (−8.21e3 + 1.98e3i)6-s + (1.91e3 − 1.08e4i)7-s + (−2.52e4 − 1.45e4i)8-s + (1.74e4 + 5.64e4i)9-s + (−7.65e4 − 1.32e5i)10-s + (6.62e4 + 1.82e5i)11-s + (2.01e4 − 4.04e4i)12-s + (−4.16e5 + 3.49e5i)13-s + (2.46e5 + 2.93e5i)14-s + (−8.91e5 + 5.91e5i)15-s + (1.13e6 − 4.11e5i)16-s + (2.10e5 − 1.21e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 + 0.832i)2-s + (0.804 + 0.593i)3-s + (−0.0315 − 0.178i)4-s + (−0.481 + 1.32i)5-s + (−1.05 + 0.255i)6-s + (0.113 − 0.644i)7-s + (−0.770 − 0.444i)8-s + (0.294 + 0.955i)9-s + (−0.765 − 1.32i)10-s + (0.411 + 1.13i)11-s + (0.0807 − 0.162i)12-s + (−1.12 + 0.941i)13-s + (0.457 + 0.545i)14-s + (−1.17 + 0.779i)15-s + (1.07 − 0.392i)16-s + (0.148 − 0.0855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.538 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.491727 - 0.898237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.491727 - 0.898237i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-195. - 144. i)T \) |
| good | 2 | \( 1 + (22.3 - 26.6i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (1.50e3 - 4.13e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (-1.91e3 + 1.08e4i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (-6.62e4 - 1.82e5i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (4.16e5 - 3.49e5i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (-2.10e5 + 1.21e5i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-2.24e6 + 3.88e6i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (4.22e6 - 7.45e5i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (-9.23e6 + 1.10e7i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (8.02e5 + 4.54e6i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (2.93e7 + 5.09e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (1.48e7 + 1.76e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (2.10e8 - 7.67e7i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (-4.34e8 - 7.66e7i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 - 6.71e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (4.36e8 - 1.20e9i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (1.00e7 - 5.70e7i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (1.60e9 - 1.34e9i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (-4.01e8 + 2.31e8i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (1.66e8 - 2.87e8i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (1.78e9 + 1.49e9i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (1.91e9 - 2.27e9i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (3.27e9 + 1.88e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (8.33e9 - 3.03e9i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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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
−15.63914464121776930522698717419, −14.91435992251018776176954618122, −13.95813140313094420037540842802, −11.81348955120942690455676431599, −10.19495402194067723343095915623, −9.207937541139483244414962957302, −7.39616404949685749910936021544, −7.12543941450311102554951781163, −4.23724250051316144271734641199, −2.69724974542550971278539841372,
0.45222518740054256085516770778, 1.57555474596846138536770417342, 3.23209201193285244459587757783, 5.58376941303205561660569361222, 8.076469380483933315380479162441, 8.746172121369300280450928864141, 9.957459928054298962627544510208, 11.99478287229820320881897956273, 12.36573059626211745084244511598, 14.08914324648988399036859242141
