| L(s) = 1 | + (−10.9 + 13.0i)2-s + (−112. + 215. i)3-s + (127. + 724. i)4-s + (735. − 2.02e3i)5-s + (−1.56e3 − 3.81e3i)6-s + (2.73e3 − 1.55e4i)7-s + (−2.58e4 − 1.49e4i)8-s + (−3.36e4 − 4.85e4i)9-s + (1.82e4 + 3.16e4i)10-s + (9.94e3 + 2.73e4i)11-s + (−1.70e5 − 5.41e4i)12-s + (1.86e5 − 1.56e5i)13-s + (1.72e5 + 2.05e5i)14-s + (3.52e5 + 3.86e5i)15-s + (−2.30e5 + 8.37e4i)16-s + (1.75e4 − 1.01e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.341 + 0.406i)2-s + (−0.464 + 0.885i)3-s + (0.124 + 0.707i)4-s + (0.235 − 0.646i)5-s + (−0.201 − 0.491i)6-s + (0.162 − 0.923i)7-s + (−0.790 − 0.456i)8-s + (−0.569 − 0.822i)9-s + (0.182 + 0.316i)10-s + (0.0617 + 0.169i)11-s + (−0.684 − 0.217i)12-s + (0.503 − 0.422i)13-s + (0.320 + 0.381i)14-s + (0.463 + 0.508i)15-s + (−0.219 + 0.0798i)16-s + (0.0123 − 0.00712i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.13048 - 0.0621933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13048 - 0.0621933i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (112. - 215. i)T \) |
| good | 2 | \( 1 + (10.9 - 13.0i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (-735. + 2.02e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (-2.73e3 + 1.55e4i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (-9.94e3 - 2.73e4i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (-1.86e5 + 1.56e5i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (-1.75e4 + 1.01e4i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-1.83e6 + 3.17e6i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (3.86e6 - 6.81e5i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (-1.38e7 + 1.64e7i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (-8.01e6 - 4.54e7i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (1.76e7 + 3.04e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (7.40e7 + 8.82e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (-3.25e7 + 1.18e7i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (2.96e8 + 5.23e7i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 + 4.52e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-3.81e8 + 1.04e9i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (-1.46e8 + 8.29e8i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (-1.21e9 + 1.01e9i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (-5.54e8 + 3.20e8i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.14e9 + 1.98e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (-2.90e9 - 2.43e9i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (-1.75e9 + 2.09e9i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (5.78e9 + 3.34e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (7.09e9 - 2.58e9i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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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
−15.52763944934210447515453258872, −13.75782156189566982073771545235, −12.35222437295619685793158146863, −11.02429348337974297105942677988, −9.602235045629673418284750311750, −8.387391187359372809696167178564, −6.77236085269288354465791942271, −4.96035505863746608308071605833, −3.49330647168583715094142267533, −0.57303121838760926283488054343,
1.29157520647412814776186510041, 2.53213974338616309557363585010, 5.58733677499626149479659884571, 6.50999598123736916955912684274, 8.383006695454702499635518053394, 10.01185834484927279497141952625, 11.28496606068181115596537154825, 12.12734679433514237807208917431, 13.83888591530557404433762545022, 14.82502178271655348354467283414
