L(s) = 1 | + (0.866 − 0.5i)3-s + (−1.86 + 1.23i)5-s + (1.73 − 2i)7-s + (0.499 − 0.866i)9-s + (−2.5 − 4.33i)11-s − i·13-s + (−1 + 2i)15-s + (−1.73 + i)17-s + (−3.5 + 6.06i)19-s + (0.499 − 2.59i)21-s + (−2.59 − 1.5i)23-s + (1.96 − 4.59i)25-s − 0.999i·27-s + (−3 − 5.19i)31-s + (−4.33 − 2.5i)33-s + ⋯ |
L(s) = 1 | + (0.499 − 0.288i)3-s + (−0.834 + 0.550i)5-s + (0.654 − 0.755i)7-s + (0.166 − 0.288i)9-s + (−0.753 − 1.30i)11-s − 0.277i·13-s + (−0.258 + 0.516i)15-s + (−0.420 + 0.242i)17-s + (−0.802 + 1.39i)19-s + (0.109 − 0.566i)21-s + (−0.541 − 0.312i)23-s + (0.392 − 0.919i)25-s − 0.192i·27-s + (−0.538 − 0.933i)31-s + (−0.753 − 0.435i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7432604123\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7432604123\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (1.86 - 1.23i)T \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 11 | \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.33 - 2.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + (11.2 + 6.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.866 + 0.5i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.19 - 3i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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
−8.620434413420405143045093869614, −8.013832820940433720434349385891, −7.82368365384378837104120755483, −6.66812792004240194767648781543, −5.94769890439093373656069368336, −4.66863527161857467515690422104, −3.78579758902243307153773159391, −3.09659087153969776061909428397, −1.82066084015965360069579524556, −0.24887481811153605779030336358,
1.80674691080792076270632642620, 2.67582961653597843891877120540, 3.96888670374430314587109230220, 4.82084159735300823595300041783, 5.16822638616059567750405277658, 6.70036382886893314687653294752, 7.48794799585419355844455250280, 8.190586521233070695174596934039, 8.884863865857028537872998909190, 9.423277817465540032354621130240