| L(s) = 1 | + (−12.3 − 9.51i)3-s − 26.7i·5-s + 70.6i·7-s + (61.8 + 235. i)9-s − 98.7·11-s − 210.·13-s + (−254. + 330. i)15-s + 524. i·17-s − 1.11e3i·19-s + (672. − 872. i)21-s + 866.·23-s + 2.40e3·25-s + (1.47e3 − 3.48e3i)27-s + 2.16e3i·29-s + 1.72e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.791 − 0.610i)3-s − 0.478i·5-s + 0.545i·7-s + (0.254 + 0.967i)9-s − 0.246·11-s − 0.345·13-s + (−0.292 + 0.379i)15-s + 0.440i·17-s − 0.708i·19-s + (0.332 − 0.431i)21-s + 0.341·23-s + 0.770·25-s + (0.388 − 0.921i)27-s + 0.477i·29-s + 0.322i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8797779332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8797779332\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (12.3 + 9.51i)T \) |
| good | 5 | \( 1 + 26.7iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 70.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 98.7T + 1.61e5T^{2} \) |
| 13 | \( 1 + 210.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 524. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.11e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 866.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.16e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.72e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.23e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 9.13e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.18e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.60e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.09e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.41e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 8.96e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.75e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.00e5iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 2.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 619. iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 7.49e4T + 8.58e9T^{2} \) |
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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
−10.37283695803018910743361109662, −9.203645195839012696948073263085, −8.294669827457779240680683969017, −7.28176599858302745864587037080, −6.33436233229669926371834462723, −5.34127712633419938167075183187, −4.58430200742981983247498357350, −2.78580185208053076625854258734, −1.52059766969120268016990812974, −0.29643591884794886895456736761,
0.998720482230269350502981718102, 2.82165361049884927156564515715, 4.02679335497925899958810491645, 4.96742368425802399295619054597, 6.05932157618232078198194211118, 6.94778969879659850276983420231, 7.935965929207882168195043708267, 9.333392906720815721850977311054, 10.05743378670410789631898237878, 10.83461480174443673794451885040
