| L(s) = 1 | + (−0.422 − 0.906i)2-s + (1.13 − 1.30i)3-s + (−0.642 + 0.766i)4-s + (2.03 − 0.917i)5-s + (−1.66 − 0.478i)6-s + (−1.13 − 4.23i)7-s + (0.965 + 0.258i)8-s + (−0.412 − 2.97i)9-s + (−1.69 − 1.46i)10-s + (1.81 − 1.04i)11-s + (0.269 + 1.71i)12-s + (−3.84 + 2.69i)13-s + (−3.35 + 2.81i)14-s + (1.12 − 3.70i)15-s + (−0.173 − 0.984i)16-s + (1.56 + 3.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.298 − 0.640i)2-s + (0.656 − 0.754i)3-s + (−0.321 + 0.383i)4-s + (0.912 − 0.410i)5-s + (−0.679 − 0.195i)6-s + (−0.428 − 1.59i)7-s + (0.341 + 0.0915i)8-s + (−0.137 − 0.990i)9-s + (−0.535 − 0.461i)10-s + (0.548 − 0.316i)11-s + (0.0778 + 0.493i)12-s + (−1.06 + 0.746i)13-s + (−0.896 + 0.752i)14-s + (0.289 − 0.957i)15-s + (−0.0434 − 0.246i)16-s + (0.380 + 0.815i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.503979 - 1.52214i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.503979 - 1.52214i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.422 + 0.906i)T \) |
| 3 | \( 1 + (-1.13 + 1.30i)T \) |
| 5 | \( 1 + (-2.03 + 0.917i)T \) |
| 19 | \( 1 + (1.31 - 4.15i)T \) |
| good | 7 | \( 1 + (1.13 + 4.23i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.81 + 1.04i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.84 - 2.69i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.56 - 3.36i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-6.58 + 0.576i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (7.59 - 2.76i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.67 + 2.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.16 + 7.16i)T - 37iT^{2} \) |
| 41 | \( 1 + (7.18 - 1.26i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.953 + 0.0834i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-11.1 - 5.17i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-1.37 + 0.120i)T + (52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (4.99 + 1.81i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.57 + 1.32i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (4.17 - 8.95i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-6.19 - 7.38i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-5.21 + 7.45i)T + (-24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (-5.64 + 0.995i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.61 + 6.03i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-1.06 + 6.04i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-6.95 + 3.24i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.22242601908462193324778528029, −9.511122030866582494144910670508, −8.873353597646945724945710774029, −7.68819237088633442087883210360, −7.00923767559460142912723891390, −5.99370846937657423422687636668, −4.35669007631640596563607133499, −3.43643939937054504893552852601, −2.02207519785104897827941963518, −0.981264686730673896922864323742,
2.31700418973311755676483106545, 3.06968537026797699427622817206, 4.95783996945719319325027978843, 5.43522183916706369987159253353, 6.58422551773481088648017321875, 7.55045137982306982693052300799, 8.775504160132043887767425441325, 9.364105062225442360369187401466, 9.732811727755854524102206324320, 10.75999780803409593716351072644
