L(s) = 1 | + (−0.382 + 0.923i)3-s + (−2.14 + 0.890i)5-s + (1.10 − 1.10i)7-s + (−0.707 − 0.707i)9-s + (0.999 + 2.41i)11-s + (2.03 + 0.841i)13-s − 2.32i·15-s + 5.68i·17-s + (−6.02 − 2.49i)19-s + (0.595 + 1.43i)21-s + (−3.60 − 3.60i)23-s + (0.293 − 0.293i)25-s + (0.923 − 0.382i)27-s + (−3.82 + 9.23i)29-s − 1.98·31-s + ⋯ |
L(s) = 1 | + (−0.220 + 0.533i)3-s + (−0.961 + 0.398i)5-s + (0.415 − 0.415i)7-s + (−0.235 − 0.235i)9-s + (0.301 + 0.727i)11-s + (0.563 + 0.233i)13-s − 0.600i·15-s + 1.37i·17-s + (−1.38 − 0.572i)19-s + (0.129 + 0.313i)21-s + (−0.751 − 0.751i)23-s + (0.0586 − 0.0586i)25-s + (0.177 − 0.0736i)27-s + (−0.710 + 1.71i)29-s − 0.357·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.111622 + 0.598459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.111622 + 0.598459i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.382 - 0.923i)T \) |
good | 5 | \( 1 + (2.14 - 0.890i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.10 + 1.10i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.999 - 2.41i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (-2.03 - 0.841i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 5.68iT - 17T^{2} \) |
| 19 | \( 1 + (6.02 + 2.49i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (3.60 + 3.60i)T + 23iT^{2} \) |
| 29 | \( 1 + (3.82 - 9.23i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 1.98T + 31T^{2} \) |
| 37 | \( 1 + (5.97 - 2.47i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (4.33 + 4.33i)T + 41iT^{2} \) |
| 43 | \( 1 + (4.39 + 10.6i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 5.32iT - 47T^{2} \) |
| 53 | \( 1 + (0.802 + 1.93i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.97 + 2.47i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (-3.53 + 8.54i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (2.25 - 5.44i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (2.57 - 2.57i)T - 71iT^{2} \) |
| 73 | \( 1 + (-8.01 - 8.01i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (-2.50 - 1.03i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (2.98 - 2.98i)T - 89iT^{2} \) |
| 97 | \( 1 + 5.81T + 97T^{2} \) |
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
−10.78617251687994480517495482929, −10.11454817509818396545234134566, −8.803880376590883658398977100766, −8.294824574919158127966069102028, −7.13605319831255152327307938672, −6.49728190400802619963101844504, −5.18072514689969536768508846050, −4.07657933718153633383932936139, −3.69770460234681381608651815272, −1.86820021173226021460848399412,
0.31044414777762346064858649128, 1.92258889978051956549858659000, 3.44100735458114815574759396721, 4.43718190740988813060498702207, 5.57437819532354249520990478935, 6.36895344344129727646355635616, 7.54939532416592665724141681366, 8.196251951052777597043899200614, 8.803446928135186264276805946283, 9.952667279100527028828570545306