| L(s) = 1 | + (7.34 + 6.16i)2-s + (−14.8 + 4.66i)3-s + (10.4 + 59.0i)4-s + (6.05 + 2.20i)5-s + (−138. − 57.4i)6-s + (−22.9 + 130. i)7-s + (−133. + 232. i)8-s + (199. − 138. i)9-s + (30.9 + 53.5i)10-s + (23.3 − 8.49i)11-s + (−430. − 829. i)12-s + (791. − 664. i)13-s + (−970. + 814. i)14-s + (−100. − 4.53i)15-s + (−611. + 222. i)16-s + (1.01e3 + 1.75e3i)17-s + ⋯ |
| L(s) = 1 | + (1.29 + 1.08i)2-s + (−0.954 + 0.299i)3-s + (0.325 + 1.84i)4-s + (0.108 + 0.0394i)5-s + (−1.56 − 0.650i)6-s + (−0.176 + 1.00i)7-s + (−0.740 + 1.28i)8-s + (0.820 − 0.571i)9-s + (0.0977 + 0.169i)10-s + (0.0581 − 0.0211i)11-s + (−0.862 − 1.66i)12-s + (1.29 − 1.09i)13-s + (−1.32 + 1.11i)14-s + (−0.115 − 0.00520i)15-s + (−0.597 + 0.217i)16-s + (0.852 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.939562 + 1.96604i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.939562 + 1.96604i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (14.8 - 4.66i)T \) |
| good | 2 | \( 1 + (-7.34 - 6.16i)T + (5.55 + 31.5i)T^{2} \) |
| 5 | \( 1 + (-6.05 - 2.20i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (22.9 - 130. i)T + (-1.57e4 - 5.74e3i)T^{2} \) |
| 11 | \( 1 + (-23.3 + 8.49i)T + (1.23e5 - 1.03e5i)T^{2} \) |
| 13 | \( 1 + (-791. + 664. i)T + (6.44e4 - 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-1.01e3 - 1.75e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (549. - 951. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (633. + 3.59e3i)T + (-6.04e6 + 2.20e6i)T^{2} \) |
| 29 | \( 1 + (-346. - 290. i)T + (3.56e6 + 2.01e7i)T^{2} \) |
| 31 | \( 1 + (513. + 2.91e3i)T + (-2.69e7 + 9.79e6i)T^{2} \) |
| 37 | \( 1 + (6.80e3 + 1.17e4i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (4.04e3 - 3.39e3i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-4.56e3 + 1.66e3i)T + (1.12e8 - 9.44e7i)T^{2} \) |
| 47 | \( 1 + (3.12e3 - 1.77e4i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 - 1.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.35e4 - 4.93e3i)T + (5.47e8 + 4.59e8i)T^{2} \) |
| 61 | \( 1 + (-4.24e3 + 2.40e4i)T + (-7.93e8 - 2.88e8i)T^{2} \) |
| 67 | \( 1 + (-1.44e3 + 1.20e3i)T + (2.34e8 - 1.32e9i)T^{2} \) |
| 71 | \( 1 + (-2.20e4 - 3.81e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-2.83e4 + 4.91e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (5.78e4 + 4.85e4i)T + (5.34e8 + 3.03e9i)T^{2} \) |
| 83 | \( 1 + (3.94e4 + 3.30e4i)T + (6.84e8 + 3.87e9i)T^{2} \) |
| 89 | \( 1 + (9.98e3 - 1.72e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-2.44e4 + 8.88e3i)T + (6.57e9 - 5.51e9i)T^{2} \) |
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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
−16.28884568610745458152199909039, −15.53982532968742371894650867844, −14.56577710024788351490838811300, −12.87899462627122529245037160322, −12.25050757177902521462846698069, −10.50206591973999733913433669662, −8.225642091232091819027489042013, −6.17572948514132762411316901628, −5.69584869991413352514317732970, −3.87396431006438702050130063310,
1.31078226097650099196315246060, 3.84720457918765849518408114154, 5.33530373404458583952545705846, 6.91108402177506271626464121286, 9.955653260645023101017260726465, 11.22187692484788791809735311616, 11.87684577566564533719913983433, 13.47237473959759668176564057384, 13.76199004791018263193155796228, 15.77819824107083260277055443418
