| L(s) = 1 | + (−2.18 − 1.40i)2-s + (0.989 − 0.142i)3-s + (1.96 + 4.30i)4-s + (1.99 + 0.584i)5-s + (−2.36 − 1.07i)6-s + (2.37 + 1.16i)7-s + (1.01 − 7.02i)8-s + (0.959 − 0.281i)9-s + (−3.52 − 4.06i)10-s + (−0.546 − 0.849i)11-s + (2.56 + 3.98i)12-s + (−2.07 + 1.80i)13-s + (−3.55 − 5.87i)14-s + (2.05 + 0.295i)15-s + (−5.86 + 6.77i)16-s + (1.29 − 2.84i)17-s + ⋯ |
| L(s) = 1 | + (−1.54 − 0.992i)2-s + (0.571 − 0.0821i)3-s + (0.983 + 2.15i)4-s + (0.889 + 0.261i)5-s + (−0.963 − 0.440i)6-s + (0.897 + 0.440i)7-s + (0.357 − 2.48i)8-s + (0.319 − 0.0939i)9-s + (−1.11 − 1.28i)10-s + (−0.164 − 0.256i)11-s + (0.739 + 1.15i)12-s + (−0.576 + 0.499i)13-s + (−0.948 − 1.57i)14-s + (0.530 + 0.0762i)15-s + (−1.46 + 1.69i)16-s + (0.315 − 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01469 - 0.246723i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01469 - 0.246723i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.989 + 0.142i)T \) |
| 7 | \( 1 + (-2.37 - 1.16i)T \) |
| 23 | \( 1 + (-1.55 - 4.53i)T \) |
| good | 2 | \( 1 + (2.18 + 1.40i)T + (0.830 + 1.81i)T^{2} \) |
| 5 | \( 1 + (-1.99 - 0.584i)T + (4.20 + 2.70i)T^{2} \) |
| 11 | \( 1 + (0.546 + 0.849i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.07 - 1.80i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.29 + 2.84i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.30 - 5.05i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.204 - 0.447i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-6.70 - 0.963i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-0.106 - 0.362i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (1.45 - 4.95i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (8.56 - 1.23i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 12.1iT - 47T^{2} \) |
| 53 | \( 1 + (0.165 + 0.143i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.18 + 2.76i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.30 + 9.06i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.08 + 7.90i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (8.80 + 5.65i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-10.4 + 4.79i)T + (47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (5.51 - 4.77i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-4.57 + 1.34i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-2.41 - 16.8i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-0.968 - 0.284i)T + (81.6 + 52.4i)T^{2} \) |
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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
−10.66015903519764351857420857087, −9.767522051202734308209889596730, −9.457829683558311822779827523422, −8.317969471150226527894421265285, −7.83292299881697524192299063817, −6.71787800677721373532044630911, −5.18671208026037329289820435765, −3.39630427760291425976155954452, −2.32333074531181019489132091882, −1.49831871303425657349761786549,
1.15584904414138431327096925284, 2.40535553313772978203505856502, 4.72486470970711447733747509242, 5.66047499702997702295182977602, 6.82786294036011495081987158068, 7.62288917686188491343807987253, 8.380890399677456328623787315259, 9.087979135176999013719590080073, 10.03266218812070070176163392947, 10.39119099910606110165232127209
