| L(s) = 1 | + (−1.78 − 10.1i)2-s + (14.4 + 5.88i)3-s + (−69.4 + 25.2i)4-s + (−55.2 − 46.3i)5-s + (33.8 − 156. i)6-s + (−159. − 58.1i)7-s + (215. + 373. i)8-s + (173. + 169. i)9-s + (−371. + 642. i)10-s + (363. − 305. i)11-s + (−1.15e3 − 43.7i)12-s + (96.5 − 547. i)13-s + (−303. + 1.72e3i)14-s + (−524. − 994. i)15-s + (1.58e3 − 1.33e3i)16-s + (251. − 436. i)17-s + ⋯ |
| L(s) = 1 | + (−0.315 − 1.79i)2-s + (0.926 + 0.377i)3-s + (−2.17 + 0.789i)4-s + (−0.988 − 0.829i)5-s + (0.383 − 1.77i)6-s + (−1.23 − 0.448i)7-s + (1.19 + 2.06i)8-s + (0.714 + 0.699i)9-s + (−1.17 + 2.03i)10-s + (0.906 − 0.760i)11-s + (−2.30 − 0.0877i)12-s + (0.158 − 0.899i)13-s + (−0.414 + 2.34i)14-s + (−0.602 − 1.14i)15-s + (1.55 − 1.30i)16-s + (0.211 − 0.366i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.190464 + 0.927789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.190464 + 0.927789i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-14.4 - 5.88i)T \) |
| good | 2 | \( 1 + (1.78 + 10.1i)T + (-30.0 + 10.9i)T^{2} \) |
| 5 | \( 1 + (55.2 + 46.3i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (159. + 58.1i)T + (1.28e4 + 1.08e4i)T^{2} \) |
| 11 | \( 1 + (-363. + 305. i)T + (2.79e4 - 1.58e5i)T^{2} \) |
| 13 | \( 1 + (-96.5 + 547. i)T + (-3.48e5 - 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-251. + 436. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-1.51 - 2.62i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (563. - 205. i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (952. + 5.40e3i)T + (-1.92e7 + 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-3.21e3 + 1.16e3i)T + (2.19e7 - 1.84e7i)T^{2} \) |
| 37 | \( 1 + (2.39e3 - 4.15e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-1.54e3 + 8.78e3i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (1.38e4 - 1.16e4i)T + (2.55e7 - 1.44e8i)T^{2} \) |
| 47 | \( 1 + (-1.87e4 - 6.83e3i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 - 1.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-8.07e3 - 6.77e3i)T + (1.24e8 + 7.04e8i)T^{2} \) |
| 61 | \( 1 + (1.00e4 + 3.65e3i)T + (6.46e8 + 5.42e8i)T^{2} \) |
| 67 | \( 1 + (-307. + 1.74e3i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-2.18e4 + 3.79e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-4.31e4 - 7.46e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.18e4 + 6.72e4i)T + (-2.89e9 + 1.05e9i)T^{2} \) |
| 83 | \( 1 + (1.34e4 + 7.63e4i)T + (-3.70e9 + 1.34e9i)T^{2} \) |
| 89 | \( 1 + (2.14e4 + 3.70e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (1.42e4 - 1.19e4i)T + (1.49e9 - 8.45e9i)T^{2} \) |
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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
−15.70605889167978025440781974654, −13.74956004084457585799555032434, −12.87169261758940435738777061474, −11.73142464414885467364672963685, −10.23852421908980440591446288180, −9.209150147674125122280516227572, −8.131682311414884135528107844140, −4.13198030158491636925424168143, −3.20396653364686749699878634270, −0.65704749912084290301768990796,
3.81800245622238279686850801901, 6.57999427870400223397316241573, 7.17194349061868518650543909708, 8.641836847993982287179755633652, 9.685200016691444811508353143606, 12.35201325972361890555762584317, 13.90593009075962336514643553490, 14.89098056428337741254164552554, 15.53675289679715806472289876717, 16.63185735973400507268381970826
