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 | \( 1 - 1.43e3iT - 9.12e5T^{2} \) |
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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
−14.55763312990860560818706587702, −13.46135490029264151640893569077, −12.51716603042738158992335649175, −11.01174079882754920464175818625, −10.70373797128864100642872704438, −9.482908196067092387043079038770, −7.32923975659664106240264411862, −4.79973349865088929418985526904, −3.25494107712853535816675485353, −0.76420387168859795498402828227,
4.94541335610298128272522422069, 5.73783415054090777247628409908, 7.19687916901521482137035701600, 8.642078835470311126680523677009, 9.594210753095342392406161335151, 12.05709749225251416359284179325, 12.95640407556248556421816509067, 14.72601711292589396071019036931, 15.21837334083607554071081209374, 16.49823495160036583839356295788