| L(s) = 1 | + (0.385 − 0.103i)2-s + (−3.91 − 3.41i)3-s + (−6.79 + 3.92i)4-s + (−9.27 − 9.27i)5-s + (−1.86 − 0.913i)6-s + (2.08 − 7.79i)7-s + (−4.46 + 4.46i)8-s + (3.61 + 26.7i)9-s + (−4.53 − 2.61i)10-s + (−4.51 − 16.8i)11-s + (39.9 + 7.87i)12-s + (−7.38 − 46.2i)13-s − 3.21i·14-s + (4.57 + 68.0i)15-s + (30.1 − 52.1i)16-s + (43.3 + 75.1i)17-s + ⋯ |
| L(s) = 1 | + (0.136 − 0.0364i)2-s + (−0.752 − 0.658i)3-s + (−0.848 + 0.490i)4-s + (−0.829 − 0.829i)5-s + (−0.126 − 0.0621i)6-s + (0.112 − 0.420i)7-s + (−0.197 + 0.197i)8-s + (0.133 + 0.990i)9-s + (−0.143 − 0.0827i)10-s + (−0.123 − 0.461i)11-s + (0.961 + 0.189i)12-s + (−0.157 − 0.987i)13-s − 0.0614i·14-s + (0.0787 + 1.17i)15-s + (0.470 − 0.814i)16-s + (0.618 + 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.433i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0967518 - 0.424167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0967518 - 0.424167i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.91 + 3.41i)T \) |
| 13 | \( 1 + (7.38 + 46.2i)T \) |
| good | 2 | \( 1 + (-0.385 + 0.103i)T + (6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (9.27 + 9.27i)T + 125iT^{2} \) |
| 7 | \( 1 + (-2.08 + 7.79i)T + (-297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (4.51 + 16.8i)T + (-1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (-43.3 - 75.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (81.1 + 21.7i)T + (5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (4.88 - 8.46i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (218. + 126. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-95.7 + 95.7i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (310. - 83.1i)T + (4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (121. - 32.6i)T + (5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-126. + 72.7i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-373. + 373. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 493. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-539. - 144. i)T + (1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (44.4 + 77.0i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (77.5 + 289. i)T + (-2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-287. + 1.07e3i)T + (-3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (-266. - 266. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 + 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-453. - 453. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (213. + 797. i)T + (-6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (600. + 160. i)T + (7.90e5 + 4.56e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.34864326868516220598015752766, −13.61579551783033998015356371772, −12.75675347784190739320858415488, −11.99274569131387229079865091758, −10.52267999800548968863721811929, −8.490470497573721107381300238725, −7.68692886186672125234899825161, −5.55579049080948841314466018138, −4.10382758998157761889811523996, −0.39456217612182799724935804711,
3.92972095560120933508418861795, 5.29216951406000425526192223157, 6.95216888050856520639046136461, 8.993617104170182135021662683103, 10.19787452204402945585288496960, 11.35467896631218224852795579016, 12.44680097930588770843872927430, 14.29811740172421641328069083852, 15.01891445097584247396370197872, 16.05747793150559149635962075517
