L(s) = 1 | + 5.21i·2-s − 3·3-s − 19.2·4-s − 5.83i·5-s − 15.6i·6-s + 31.3i·7-s − 58.6i·8-s + 9·9-s + 30.4·10-s + 16.2i·11-s + 57.7·12-s + (−43.4 + 17.5i)13-s − 163.·14-s + 17.5i·15-s + 152.·16-s + 54·17-s + ⋯ |
L(s) = 1 | + 1.84i·2-s − 0.577·3-s − 2.40·4-s − 0.522i·5-s − 1.06i·6-s + 1.69i·7-s − 2.59i·8-s + 0.333·9-s + 0.963·10-s + 0.446i·11-s + 1.38·12-s + (−0.927 + 0.373i)13-s − 3.11·14-s + 0.301i·15-s + 2.37·16-s + 0.770·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 + 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.150293 - 0.775121i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.150293 - 0.775121i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 + (43.4 - 17.5i)T \) |
good | 2 | \( 1 - 5.21iT - 8T^{2} \) |
| 5 | \( 1 + 5.83iT - 125T^{2} \) |
| 7 | \( 1 - 31.3iT - 343T^{2} \) |
| 11 | \( 1 - 16.2iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 66.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 164.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58.9iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 110. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 55.0iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 113.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 514. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 242.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 265. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 468.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 852. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 165. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 315. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 479.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 574. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 66.7iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.43e3iT - 9.12e5T^{2} \) |
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
−16.49823495160036583839356295788, −15.21837334083607554071081209374, −14.72601711292589396071019036931, −12.95640407556248556421816509067, −12.05709749225251416359284179325, −9.594210753095342392406161335151, −8.642078835470311126680523677009, −7.19687916901521482137035701600, −5.73783415054090777247628409908, −4.94541335610298128272522422069,
0.76420387168859795498402828227, 3.25494107712853535816675485353, 4.79973349865088929418985526904, 7.32923975659664106240264411862, 9.482908196067092387043079038770, 10.70373797128864100642872704438, 11.01174079882754920464175818625, 12.51716603042738158992335649175, 13.46135490029264151640893569077, 14.55763312990860560818706587702