L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.125 + 1.72i)3-s + (−0.499 − 0.866i)4-s + (3.72 + 2.15i)5-s + (−1.55 − 0.755i)6-s + (−2.62 − 0.301i)7-s + 0.999·8-s + (−2.96 + 0.432i)9-s + (−3.72 + 2.15i)10-s + (0.167 + 3.31i)11-s + (1.43 − 0.972i)12-s − 2.22i·13-s + (1.57 − 2.12i)14-s + (−3.24 + 6.70i)15-s + (−0.5 + 0.866i)16-s + (2.17 + 3.76i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.0722 + 0.997i)3-s + (−0.249 − 0.433i)4-s + (1.66 + 0.961i)5-s + (−0.636 − 0.308i)6-s + (−0.993 − 0.113i)7-s + 0.353·8-s + (−0.989 + 0.144i)9-s + (−1.17 + 0.679i)10-s + (0.0505 + 0.998i)11-s + (0.413 − 0.280i)12-s − 0.617i·13-s + (0.420 − 0.568i)14-s + (−0.838 + 1.73i)15-s + (−0.125 + 0.216i)16-s + (0.527 + 0.914i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.286267 + 1.24690i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.286267 + 1.24690i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.125 - 1.72i)T \) |
| 7 | \( 1 + (2.62 + 0.301i)T \) |
| 11 | \( 1 + (-0.167 - 3.31i)T \) |
good | 5 | \( 1 + (-3.72 - 2.15i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 2.22iT - 13T^{2} \) |
| 17 | \( 1 + (-2.17 - 3.76i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.35 + 0.779i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.48 + 0.859i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.52T + 29T^{2} \) |
| 31 | \( 1 + (1.83 + 3.17i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.36 + 5.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 4.66iT - 43T^{2} \) |
| 47 | \( 1 + (-3.24 - 1.87i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.14 - 1.23i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.19 - 1.84i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.02 + 2.32i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.08 + 3.61i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (-12.7 + 7.33i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.48 - 4.89i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 + (-10.8 - 6.24i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.541T + 97T^{2} \) |
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
−10.68823183187703635111841717996, −10.37855557812470174134367180410, −9.652915989253480773827758853026, −9.167952116067149001876270849691, −7.76271737146140945073236566182, −6.47791866440538284271745503314, −6.05699075399304180775717221884, −5.00393168759592806455237396471, −3.48717641444038052515835797758, −2.27072477863625543949312428673,
0.894701498696575090040546965789, 2.10353019464038865124042651781, 3.19208804449260986216564657687, 5.12200714879173622617736411819, 6.06524432604850394926743926365, 6.78742518220215783045298158742, 8.302572675186062288800342285335, 8.984261291318509901373729631716, 9.631422926008930797739998855767, 10.49700420404698472563895245882