| L(s) = 1 | + (14.3 + 5.21i)2-s + (−40.6 − 23.0i)3-s + (80.3 + 67.4i)4-s + (−74.8 + 424. i)5-s + (−462. − 543. i)6-s + (−616. + 517. i)7-s + (−176. − 305. i)8-s + (1.12e3 + 1.87e3i)9-s + (−3.28e3 + 5.69e3i)10-s + (808. + 4.58e3i)11-s + (−1.71e3 − 4.59e3i)12-s + (−2.90e3 + 1.05e3i)13-s + (−1.15e4 + 4.20e3i)14-s + (1.28e4 − 1.55e4i)15-s + (−3.26e3 − 1.85e4i)16-s + (2.75e3 − 4.77e3i)17-s + ⋯ |
| L(s) = 1 | + (1.26 + 0.461i)2-s + (−0.869 − 0.493i)3-s + (0.627 + 0.526i)4-s + (−0.267 + 1.51i)5-s + (−0.874 − 1.02i)6-s + (−0.679 + 0.570i)7-s + (−0.121 − 0.211i)8-s + (0.512 + 0.858i)9-s + (−1.03 + 1.80i)10-s + (0.183 + 1.03i)11-s + (−0.285 − 0.767i)12-s + (−0.367 + 0.133i)13-s + (−1.12 + 0.409i)14-s + (0.982 − 1.18i)15-s + (−0.199 − 1.13i)16-s + (0.136 − 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.712i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.617588 + 1.47603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.617588 + 1.47603i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (40.6 + 23.0i)T \) |
| good | 2 | \( 1 + (-14.3 - 5.21i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (74.8 - 424. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (616. - 517. i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (-808. - 4.58e3i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (2.90e3 - 1.05e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-2.75e3 + 4.77e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.56e4 + 2.70e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-6.02e4 - 5.05e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-8.61e4 - 3.13e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-2.10e5 - 1.76e5i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (1.14e5 - 1.98e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (8.03e5 - 2.92e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-7.09e4 - 4.02e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-6.03e5 + 5.06e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 - 1.66e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-4.32e5 + 2.45e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (1.38e6 - 1.16e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-2.49e6 + 9.06e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (2.54e6 - 4.40e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-7.60e5 - 1.31e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.61e6 - 5.88e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-4.95e6 - 1.80e6i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (2.46e6 + 4.26e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.41e5 - 7.99e5i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−15.68184730076444649990533379773, −15.05090533041918260035060999221, −13.73993310691936252985007964922, −12.53821938181621364280283435192, −11.56978974437640517130846805172, −10.02788744512048613396453382585, −7.00757055677168596107868468884, −6.56974021812526165206606932290, −4.91704435122080914612294170923, −2.91149954854783736046768989865,
0.59835390881416702479734810127, 3.75439551189768253048208566404, 4.83394972478472380741290112071, 6.08360941709166745955150377106, 8.713454600937711267604322577256, 10.45591799711369371762687154681, 11.91382272678366287910247666461, 12.61899287248153280048874455762, 13.64640255680190517697293437658, 15.31885636046063129132030403272
