| L(s) = 1 | + (0.126 − 1.40i)2-s + (−0.460 − 1.66i)3-s + (−1.96 − 0.355i)4-s + (1.43 − 3.94i)5-s + (−2.41 + 0.437i)6-s + (0.984 + 0.173i)7-s + (−0.749 + 2.72i)8-s + (−2.57 + 1.53i)9-s + (−5.37 − 2.51i)10-s + (−3.58 + 1.30i)11-s + (0.311 + 3.45i)12-s + (−3.97 + 3.33i)13-s + (0.368 − 1.36i)14-s + (−7.24 − 0.581i)15-s + (3.74 + 1.40i)16-s + (2.36 − 1.36i)17-s + ⋯ |
| L(s) = 1 | + (0.0892 − 0.996i)2-s + (−0.265 − 0.964i)3-s + (−0.984 − 0.177i)4-s + (0.641 − 1.76i)5-s + (−0.983 + 0.178i)6-s + (0.372 + 0.0656i)7-s + (−0.265 + 0.964i)8-s + (−0.858 + 0.512i)9-s + (−1.69 − 0.796i)10-s + (−1.07 + 0.393i)11-s + (0.0900 + 0.995i)12-s + (−1.10 + 0.926i)13-s + (0.0986 − 0.364i)14-s + (−1.87 − 0.150i)15-s + (0.936 + 0.350i)16-s + (0.572 − 0.330i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.219 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.539533 + 0.674115i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.539533 + 0.674115i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.126 + 1.40i)T \) |
| 3 | \( 1 + (0.460 + 1.66i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (-1.43 + 3.94i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (3.58 - 1.30i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.97 - 3.33i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.36 + 1.36i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.19 - 0.688i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.969 + 5.49i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.38 + 2.83i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.74 - 0.660i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.0414 + 0.0718i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.58 + 1.88i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.11 + 8.54i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.70 + 9.66i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 10.6iT - 53T^{2} \) |
| 59 | \( 1 + (-5.10 - 1.85i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.88 - 10.6i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (4.50 + 5.37i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.232 - 0.402i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.29 + 7.43i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.91 + 3.47i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.08 - 4.26i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (7.03 + 4.06i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.61 - 0.950i)T + (74.3 - 62.3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.791432413294547342356638444325, −8.887023092263916604800022972679, −8.277411362955580064320630214340, −7.31840909796719467690640632131, −5.72879966429048910445502999180, −5.12666561641047585547632719678, −4.45814718112834222433697349109, −2.44299289805711709823541815456, −1.74442259525763212954650383918, −0.42851544546601491520471543172,
2.82903234057616701908451909008, 3.52607456852172702755342686481, 5.08813754300992339899852526087, 5.56407770990119860145128355616, 6.46453943504177822409632179360, 7.50599860687354529338419913469, 8.086942780177172335351031317489, 9.563795742657750457947056386883, 9.989248251726750701205575632268, 10.67656201783836751407392727094
