| L(s) = 1 | + (0.939 − 0.342i)2-s + (0.560 + 1.63i)3-s + (0.766 − 0.642i)4-s + (−0.343 + 0.408i)5-s + (1.08 + 1.34i)6-s + (−0.716 − 1.24i)7-s + (0.500 − 0.866i)8-s + (−2.37 + 1.83i)9-s + (−0.182 + 0.501i)10-s + (−1.25 − 0.725i)11-s + (1.48 + 0.895i)12-s + (2.94 − 0.519i)13-s + (−1.09 − 0.921i)14-s + (−0.862 − 0.333i)15-s + (0.173 − 0.984i)16-s + (−1.89 − 5.20i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.323 + 0.946i)3-s + (0.383 − 0.321i)4-s + (−0.153 + 0.182i)5-s + (0.443 + 0.550i)6-s + (−0.270 − 0.469i)7-s + (0.176 − 0.306i)8-s + (−0.790 + 0.611i)9-s + (−0.0577 + 0.158i)10-s + (−0.378 − 0.218i)11-s + (0.427 + 0.258i)12-s + (0.817 − 0.144i)13-s + (−0.293 − 0.246i)14-s + (−0.222 − 0.0860i)15-s + (0.0434 − 0.246i)16-s + (−0.459 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47358 + 0.228830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47358 + 0.228830i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.560 - 1.63i)T \) |
| 19 | \( 1 + (4.35 - 0.143i)T \) |
| good | 5 | \( 1 + (0.343 - 0.408i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.716 + 1.24i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.25 + 0.725i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.94 + 0.519i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.89 + 5.20i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (0.396 + 0.472i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.97 - 1.81i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (4.28 - 2.47i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.41iT - 37T^{2} \) |
| 41 | \( 1 + (-1.37 + 7.78i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.88 - 4.10i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (4.37 - 12.0i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-1.41 + 1.18i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.75 + 0.639i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-9.02 + 7.57i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.17 + 8.71i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-9.59 - 8.05i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (2.80 - 15.9i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (7.87 + 1.38i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (4.29 - 2.48i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.832 - 4.71i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.83 + 7.79i)T + (-74.3 + 62.3i)T^{2} \) |
| 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
−13.75695012889357659384729044750, −12.81131747946544507413866005545, −11.27499887930921755490790631046, −10.73406726696070135831329099766, −9.591630466152862653813821550869, −8.367460288139544652940953528185, −6.82360277699750174198047103049, −5.33215266708148953990686542443, −4.10043568733547689167143994689, −2.91502997952076377588230203428,
2.28623865492916084271664319142, 3.99607249726272877028051624872, 5.83716591438116435545938257717, 6.67088666348768164618709432096, 8.066501187589335718387642192366, 8.828832193518175604137415152494, 10.64572354727250896849205523116, 11.87678703868961681293467673735, 12.77401969970058547178175102271, 13.31047155061318555952473959642
