L(s) = 1 | + (0.5 − 0.866i)3-s + (1 − 1.73i)5-s − 4.56i·7-s + (−0.499 − 0.866i)9-s + 5.65i·11-s + (3.94 − 2.28i)13-s + (−0.999 − 1.73i)15-s + (1 − 1.73i)17-s + (−4 − 1.73i)19-s + (−3.94 − 2.28i)21-s + (4.89 − 2.82i)23-s + (0.500 + 0.866i)25-s − 0.999·27-s + (−1.89 + 1.09i)29-s − 7.89·31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.447 − 0.774i)5-s − 1.72i·7-s + (−0.166 − 0.288i)9-s + 1.70i·11-s + (1.09 − 0.632i)13-s + (−0.258 − 0.447i)15-s + (0.242 − 0.420i)17-s + (−0.917 − 0.397i)19-s + (−0.861 − 0.497i)21-s + (1.02 − 0.589i)23-s + (0.100 + 0.173i)25-s − 0.192·27-s + (−0.352 + 0.203i)29-s − 1.41·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.07388 - 1.47278i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07388 - 1.47278i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 4.56iT - 7T^{2} \) |
| 11 | \( 1 - 5.65iT - 11T^{2} \) |
| 13 | \( 1 + (-3.94 + 2.28i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.89 + 2.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.89 - 1.09i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 - 1.09iT - 37T^{2} \) |
| 41 | \( 1 + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 + 2.59i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.89 - 4.56i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.8 + 6.29i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.89 + 10.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.94 - 8.57i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.39 - 7.61i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.39 + 5.88i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0505 + 0.0874i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (1.89 - 1.09i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.79 - 2.19i)T + (48.5 + 84.0i)T^{2} \) |
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
−9.858964516409451178959619789957, −8.987221057597605976159685533897, −8.129756951016983935313009208445, −7.12644765650082727453210837734, −6.80094498976867752599546096643, −5.33357154523524747085356035441, −4.47444932438434458342485648162, −3.51721116315194639738394114639, −1.90115389892061495964697069483, −0.872516246157931636739796613406,
1.93643253687695127209053452757, 3.01314693549474092486097689440, 3.75693176323031576014553950595, 5.39947936124376128569081077526, 5.92165958749310425289356495072, 6.65129570242739504943078724530, 8.221566371010360894571433251210, 8.747083216857767842613762301103, 9.287991649771734045987046120210, 10.42986159460993829976112726461