| L(s) = 1 | + (−13.9 − 5.08i)2-s + (−44.6 + 13.9i)3-s + (71.4 + 59.9i)4-s + (−48.7 + 276. i)5-s + (694. + 31.8i)6-s + (−1.11e3 + 934. i)7-s + (258. + 447. i)8-s + (1.79e3 − 1.24e3i)9-s + (2.08e3 − 3.61e3i)10-s + (−907. − 5.14e3i)11-s + (−4.02e3 − 1.67e3i)12-s + (−1.22e4 + 4.45e3i)13-s + (2.03e4 − 7.39e3i)14-s + (−1.68e3 − 1.30e4i)15-s + (−3.40e3 − 1.93e4i)16-s + (9.92e3 − 1.71e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.23 − 0.449i)2-s + (−0.954 + 0.298i)3-s + (0.558 + 0.468i)4-s + (−0.174 + 0.989i)5-s + (1.31 + 0.0601i)6-s + (−1.22 + 1.02i)7-s + (0.178 + 0.308i)8-s + (0.821 − 0.570i)9-s + (0.660 − 1.14i)10-s + (−0.205 − 1.16i)11-s + (−0.672 − 0.280i)12-s + (−1.54 + 0.562i)13-s + (1.97 − 0.720i)14-s + (−0.129 − 0.996i)15-s + (−0.207 − 1.17i)16-s + (0.489 − 0.848i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0952 + 0.995i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0952 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.124841 - 0.113468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.124841 - 0.113468i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (44.6 - 13.9i)T \) |
| good | 2 | \( 1 + (13.9 + 5.08i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (48.7 - 276. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (1.11e3 - 934. i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (907. + 5.14e3i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (1.22e4 - 4.45e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-9.92e3 + 1.71e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.63e4 - 2.82e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (892. + 749. i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-1.14e5 - 4.17e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (1.35e5 + 1.13e5i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (-2.09e5 + 3.62e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-2.15e5 + 7.82e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (2.19e4 + 1.24e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (3.25e5 - 2.72e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 - 6.51e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-3.13e4 + 1.77e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (1.87e6 - 1.57e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-6.34e5 + 2.30e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-9.91e4 + 1.71e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.58e6 - 2.74e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.88e6 + 6.86e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (5.41e6 + 1.97e6i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (3.84e6 + 6.65e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-6.44e5 - 3.65e6i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−16.04944656616964756038219403793, −14.44842902346729645952531047492, −12.28812069447279424215222007018, −11.32456622179595813000835112255, −10.10481798653275633926258431654, −9.300135421737245063457782473448, −7.24652738853338937844538214513, −5.68214990621113184623133161165, −2.84378147389782381592393657134, −0.20177873315153876511001413273,
0.888633883071739300657557951584, 4.70128361722649610811235205287, 6.80648442253801444258952391015, 7.71063055971530692767323266676, 9.646552748661854670907449998973, 10.30179391215914618249013507220, 12.42926084108595331221758015444, 13.04273281290735780803491417464, 15.53892166355737679752982091909, 16.63584575122103206069527331586
