L(s) = 1 | + (−2.17 − 1.25i)3-s + (0.891 + 0.514i)5-s − 4.18·7-s + (1.66 + 2.87i)9-s + 2.06i·11-s + (2.25 − 1.29i)13-s + (−1.29 − 2.24i)15-s + (−3.49 + 6.04i)17-s + (1.09 − 4.22i)19-s + (9.10 + 5.25i)21-s + (2.14 + 3.71i)23-s + (−1.96 − 3.41i)25-s − 0.806i·27-s + (5.17 − 2.98i)29-s + 2.70·31-s + ⋯ |
L(s) = 1 | + (−1.25 − 0.725i)3-s + (0.398 + 0.230i)5-s − 1.58·7-s + (0.553 + 0.958i)9-s + 0.621i·11-s + (0.624 − 0.360i)13-s + (−0.334 − 0.578i)15-s + (−0.846 + 1.46i)17-s + (0.250 − 0.968i)19-s + (1.98 + 1.14i)21-s + (0.447 + 0.775i)23-s + (−0.393 − 0.682i)25-s − 0.155i·27-s + (0.960 − 0.554i)29-s + 0.485·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7670627313\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7670627313\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.09 + 4.22i)T \) |
good | 3 | \( 1 + (2.17 + 1.25i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.891 - 0.514i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 - 2.06iT - 11T^{2} \) |
| 13 | \( 1 + (-2.25 + 1.29i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.49 - 6.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.14 - 3.71i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.17 + 2.98i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 + 9.93iT - 37T^{2} \) |
| 41 | \( 1 + (2.42 - 4.20i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.57 - 4.94i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.86 + 4.96i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.00 - 2.31i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.382 + 0.220i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.59 + 2.07i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.07 + 0.618i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.69 + 13.3i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.846 + 1.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.13 + 10.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.92iT - 83T^{2} \) |
| 89 | \( 1 + (-5.55 - 9.62i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.11 + 7.12i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.753310850658177894083213343049, −8.964554297018654536133366955935, −7.73063897049410508197108021765, −6.75461274896438586539351016689, −6.30962563179302913959431013546, −5.83263389071382885314674442862, −4.59865973855387011987784202111, −3.37887900630123898118702716210, −2.10916785560634191491837839767, −0.59635754712511935744954954597,
0.77359649722688091005104462971, 2.80943141044063938207125796411, 3.82251964977739076173575060881, 4.86633463958556416928242362347, 5.66397701605444307072405415685, 6.41083673339329234262650903432, 6.89089479426972615425275185094, 8.456487899005276026924056429479, 9.342231394835701556256329518925, 9.853304708741830630764726832871