| L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.766 − 0.642i)6-s + (1.89 − 3.28i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.173 − 0.984i)10-s + (2.21 + 3.84i)11-s + (0.499 − 0.866i)12-s + (0.503 + 0.422i)13-s + (3.56 + 1.29i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.847 + 4.80i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.442 + 0.371i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (−0.312 − 0.262i)6-s + (0.717 − 1.24i)7-s + (−0.176 − 0.306i)8-s + (0.0578 − 0.328i)9-s + (0.0549 − 0.311i)10-s + (0.669 + 1.15i)11-s + (0.144 − 0.249i)12-s + (0.139 + 0.117i)13-s + (0.953 + 0.347i)14-s + (0.242 − 0.0883i)15-s + (0.191 − 0.160i)16-s + (0.205 + 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12646 + 0.731364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12646 + 0.731364i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (-3.94 + 1.84i)T \) |
| good | 7 | \( 1 + (-1.89 + 3.28i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.21 - 3.84i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.503 - 0.422i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.847 - 4.80i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-5.69 + 2.07i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.551 + 3.12i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (4.26 - 7.38i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 + (2.97 - 2.49i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.90 + 2.51i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.45 - 8.26i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-10.9 + 3.98i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.02 - 5.80i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.24 + 1.54i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.92 + 10.8i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.60 - 1.67i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-12.3 + 10.3i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (10.6 - 8.94i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.60 + 9.71i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.87 + 7.44i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.783 + 4.44i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.86377234732440132689670208810, −10.04885832324004672161760175183, −9.099821002728530131107658581715, −8.050892258235592919803481154111, −7.21388144334513745774928972051, −6.56775143866272230705834277804, −5.13354558435002185057764145673, −4.46343632883167769250180429813, −3.65029649533841134684306534357, −1.21706275712681704479241788158,
1.04590662280729359086771797563, 2.56754425844666915642423424097, 3.65998814498861821943136113939, 5.17925128832639877100398144263, 5.64641215719506831528353014559, 6.96186227290198380831964665697, 8.086290998460985868518334765010, 8.864978102360246824677267930382, 9.695567191994844417918245901993, 11.05399083801382865021240153577
