| L(s) = 1 | + (−4.30 − 3.60i)2-s + (−7.60 + 46.1i)3-s + (−16.7 − 94.9i)4-s + (201. + 73.4i)5-s + (199. − 171. i)6-s + (−20.5 + 116. i)7-s + (−630. + 1.09e3i)8-s + (−2.07e3 − 701. i)9-s + (−602. − 1.04e3i)10-s + (−4.02e3 + 1.46e3i)11-s + (4.51e3 − 50.4i)12-s + (−8.19e3 + 6.87e3i)13-s + (508. − 426. i)14-s + (−4.92e3 + 8.75e3i)15-s + (−4.94e3 + 1.80e3i)16-s + (6.75e3 + 1.17e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.380 − 0.319i)2-s + (−0.162 + 0.986i)3-s + (−0.130 − 0.742i)4-s + (0.721 + 0.262i)5-s + (0.376 − 0.323i)6-s + (−0.0226 + 0.128i)7-s + (−0.435 + 0.753i)8-s + (−0.947 − 0.320i)9-s + (−0.190 − 0.330i)10-s + (−0.911 + 0.331i)11-s + (0.753 − 0.00843i)12-s + (−1.03 + 0.867i)13-s + (0.0494 − 0.0415i)14-s + (−0.376 + 0.669i)15-s + (−0.302 + 0.109i)16-s + (0.333 + 0.577i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.199721 + 0.528406i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.199721 + 0.528406i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.60 - 46.1i)T \) |
| good | 2 | \( 1 + (4.30 + 3.60i)T + (22.2 + 126. i)T^{2} \) |
| 5 | \( 1 + (-201. - 73.4i)T + (5.98e4 + 5.02e4i)T^{2} \) |
| 7 | \( 1 + (20.5 - 116. i)T + (-7.73e5 - 2.81e5i)T^{2} \) |
| 11 | \( 1 + (4.02e3 - 1.46e3i)T + (1.49e7 - 1.25e7i)T^{2} \) |
| 13 | \( 1 + (8.19e3 - 6.87e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-6.75e3 - 1.17e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.85e4 - 3.21e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (1.64e4 + 9.31e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-7.50e4 - 6.29e4i)T + (2.99e9 + 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-3.54e4 - 2.00e5i)T + (-2.58e10 + 9.40e9i)T^{2} \) |
| 37 | \( 1 + (4.82e4 + 8.35e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.65e5 + 1.38e5i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + (6.16e5 - 2.24e5i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (8.74e3 - 4.96e4i)T + (-4.76e11 - 1.73e11i)T^{2} \) |
| 53 | \( 1 - 1.83e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.56e6 - 5.69e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (3.95e4 - 2.24e5i)T + (-2.95e12 - 1.07e12i)T^{2} \) |
| 67 | \( 1 + (1.05e6 - 8.84e5i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (1.09e6 + 1.89e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.04e6 + 3.54e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (3.86e6 + 3.24e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (4.06e6 + 3.40e6i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (4.64e5 - 8.05e5i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.19e7 + 4.35e6i)T + (6.18e13 - 5.19e13i)T^{2} \) |
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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
−16.29299835411761394647265805945, −14.82653054295686445149731259495, −14.21604019006796414839900952535, −12.16902672306586437128442077765, −10.40365741894066947537768648795, −10.13450084970434671244073280512, −8.704430238076124048420813363256, −6.13720613690911505986241012120, −4.78868134583489707137881948501, −2.28106756403211888439613252936,
0.30550852956295700361305348273, 2.64690133464629683658030478861, 5.48294835436887011925355576298, 7.19249688666119394024988048683, 8.191433451661650059824063705575, 9.743426112428911635512985858122, 11.67545797027731071045694419357, 13.00028967343643517467858244487, 13.55603200015986144259338186108, 15.49635069961791499240831549554
