| 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 | \( 1 + (2.83 - 7.79i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−13.31047155061318555952473959642, −12.77401969970058547178175102271, −11.87678703868961681293467673735, −10.64572354727250896849205523116, −8.828832193518175604137415152494, −8.066501187589335718387642192366, −6.67088666348768164618709432096, −5.83716591438116435545938257717, −3.99607249726272877028051624872, −2.28623865492916084271664319142,
2.91502997952076377588230203428, 4.10043568733547689167143994689, 5.33215266708148953990686542443, 6.82360277699750174198047103049, 8.367460288139544652940953528185, 9.591630466152862653813821550869, 10.73406726696070135831329099766, 11.27499887930921755490790631046, 12.81131747946544507413866005545, 13.75695012889357659384729044750
