L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−2.23 − 0.133i)5-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.86 − 1.23i)10-s + (−1 − 1.73i)11-s + (0.866 + 0.499i)12-s + 6i·13-s + (−1.99 + i)15-s + (−0.5 + 0.866i)16-s + (−3.46 + 2i)17-s + (0.866 − 0.499i)18-s + (−3 + 5.19i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.998 − 0.0599i)5-s + 0.408·6-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.590 − 0.389i)10-s + (−0.301 − 0.522i)11-s + (0.249 + 0.144i)12-s + 1.66i·13-s + (−0.516 + 0.258i)15-s + (−0.125 + 0.216i)16-s + (−0.840 + 0.485i)17-s + (0.204 − 0.117i)18-s + (−0.688 + 1.19i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.780713655\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.780713655\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (2.23 + 0.133i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + (3.46 - 2i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.92 - 4i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (-6.92 - 4i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 + 3i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4 + 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (12.1 - 7i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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
−9.455055108092579505707769399620, −8.724849730925137251376257936003, −8.103293005636642744177950885229, −7.23753636082842902691949245314, −6.66812242422096873187174868445, −5.67730010778085872709329806581, −4.43479075928924350437424655891, −3.94018299200200011963750316868, −2.94508456549567810897229606887, −1.65518737062262727659278140908,
0.53210190014273122481672975267, 2.50628168821487579101804264974, 3.05085682902169457154547704379, 4.22439321431298591835417070705, 4.75076816766237036679414563070, 5.74008929745213698935488437903, 7.09938408808740370340819829476, 7.45668365910903238163164263319, 8.613158062406501993469818640579, 9.116892051119832143222628697298