| L(s) = 1 | + (4.72 + 3.96i)2-s + (−42.6 + 19.1i)3-s + (−15.6 − 88.6i)4-s + (139. + 50.6i)5-s + (−277. − 78.8i)6-s + (26.9 − 152. i)7-s + (671. − 1.16e3i)8-s + (1.45e3 − 1.63e3i)9-s + (456. + 790. i)10-s + (5.84e3 − 2.12e3i)11-s + (2.36e3 + 3.48e3i)12-s + (2.57e3 − 2.15e3i)13-s + (731. − 614. i)14-s + (−6.91e3 + 497. i)15-s + (−3.04e3 + 1.10e3i)16-s + (−1.68e3 − 2.91e3i)17-s + ⋯ |
| L(s) = 1 | + (0.417 + 0.350i)2-s + (−0.912 + 0.408i)3-s + (−0.122 − 0.692i)4-s + (0.498 + 0.181i)5-s + (−0.523 − 0.149i)6-s + (0.0296 − 0.168i)7-s + (0.463 − 0.803i)8-s + (0.666 − 0.745i)9-s + (0.144 + 0.250i)10-s + (1.32 − 0.481i)11-s + (0.394 + 0.582i)12-s + (0.324 − 0.272i)13-s + (0.0712 − 0.0598i)14-s + (−0.528 + 0.0380i)15-s + (−0.186 + 0.0677i)16-s + (−0.0830 − 0.143i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.61787 - 0.448509i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61787 - 0.448509i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (42.6 - 19.1i)T \) |
| good | 2 | \( 1 + (-4.72 - 3.96i)T + (22.2 + 126. i)T^{2} \) |
| 5 | \( 1 + (-139. - 50.6i)T + (5.98e4 + 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-26.9 + 152. i)T + (-7.73e5 - 2.81e5i)T^{2} \) |
| 11 | \( 1 + (-5.84e3 + 2.12e3i)T + (1.49e7 - 1.25e7i)T^{2} \) |
| 13 | \( 1 + (-2.57e3 + 2.15e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (1.68e3 + 2.91e3i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-9.88e3 + 1.71e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (5.80e3 + 3.28e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (6.12e4 + 5.13e4i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-6.38e3 - 3.62e4i)T + (-2.58e10 + 9.40e9i)T^{2} \) |
| 37 | \( 1 + (-2.83e5 - 4.90e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (5.41e4 - 4.54e4i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-7.43e5 + 2.70e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-1.01e5 + 5.78e5i)T + (-4.76e11 - 1.73e11i)T^{2} \) |
| 53 | \( 1 + 1.36e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-2.02e6 - 7.38e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (-3.79e4 + 2.15e5i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (1.98e6 - 1.66e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (1.46e6 + 2.54e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.51e6 - 4.35e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (3.00e6 + 2.51e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-5.97e6 - 5.01e6i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (7.18e5 - 1.24e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (1.10e7 - 4.01e6i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−15.64145924958046794192784119264, −14.48374422937802808391582772213, −13.38323543596772063818438398548, −11.68426331125364944525091887726, −10.45968552356206572996725597251, −9.338183657764772491244002065484, −6.67768128021471648440950774366, −5.75923511419666016909637358450, −4.23005938331436682455619297404, −0.944265477844711083516004046103,
1.70652067804673638922116785915, 4.14073625445488190109045946823, 5.83284560747673040041676830745, 7.45582020380594041586347651673, 9.322511788273999587559476979593, 11.17764895118254799227963048963, 12.11670853816355572307133306530, 13.06911458236136273509920364281, 14.25981474365741513435289960611, 16.23103851001883603121217026332
