| L(s) = 1 | + (1.37 + 1.20i)2-s + (0.923 − 1.46i)3-s + (0.169 + 1.33i)4-s + (−2.39 − 0.348i)5-s + (3.03 − 0.893i)6-s + (−0.297 − 3.28i)7-s + (0.675 − 0.996i)8-s + (−1.29 − 2.70i)9-s + (−2.86 − 3.37i)10-s + (−0.959 + 0.976i)11-s + (2.10 + 0.981i)12-s + (0.0464 + 0.00167i)13-s + (3.55 − 4.86i)14-s + (−2.72 + 3.19i)15-s + (4.72 − 1.22i)16-s + (2.24 − 1.03i)17-s + ⋯ |
| L(s) = 1 | + (0.969 + 0.854i)2-s + (0.533 − 0.845i)3-s + (0.0845 + 0.665i)4-s + (−1.07 − 0.155i)5-s + (1.23 − 0.364i)6-s + (−0.112 − 1.24i)7-s + (0.238 − 0.352i)8-s + (−0.431 − 0.902i)9-s + (−0.906 − 1.06i)10-s + (−0.289 + 0.294i)11-s + (0.607 + 0.283i)12-s + (0.0128 + 0.000465i)13-s + (0.950 − 1.29i)14-s + (−0.703 + 0.823i)15-s + (1.18 − 0.305i)16-s + (0.543 − 0.251i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.03295 - 0.943477i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03295 - 0.943477i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.923 + 1.46i)T \) |
| 59 | \( 1 + (-3.12 - 7.01i)T \) |
| good | 2 | \( 1 + (-1.37 - 1.20i)T + (0.252 + 1.98i)T^{2} \) |
| 5 | \( 1 + (2.39 + 0.348i)T + (4.79 + 1.42i)T^{2} \) |
| 7 | \( 1 + (0.297 + 3.28i)T + (-6.88 + 1.25i)T^{2} \) |
| 11 | \( 1 + (0.959 - 0.976i)T + (-0.198 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.0464 - 0.00167i)T + (12.9 + 0.938i)T^{2} \) |
| 17 | \( 1 + (-2.24 + 1.03i)T + (11.0 - 12.9i)T^{2} \) |
| 19 | \( 1 + (2.41 + 0.531i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (-7.67 - 2.89i)T + (17.2 + 15.2i)T^{2} \) |
| 29 | \( 1 + (-0.659 - 3.27i)T + (-26.7 + 11.2i)T^{2} \) |
| 31 | \( 1 + (-0.164 + 0.519i)T + (-25.3 - 17.8i)T^{2} \) |
| 37 | \( 1 + (-2.81 - 4.15i)T + (-13.6 + 34.3i)T^{2} \) |
| 41 | \( 1 + (-2.06 - 1.69i)T + (8.08 + 40.1i)T^{2} \) |
| 43 | \( 1 + (-2.08 + 0.538i)T + (37.6 - 20.8i)T^{2} \) |
| 47 | \( 1 + (2.33 - 0.338i)T + (45.0 - 13.3i)T^{2} \) |
| 53 | \( 1 + (-4.63 + 5.45i)T + (-8.57 - 52.3i)T^{2} \) |
| 61 | \( 1 + (7.16 + 6.31i)T + (7.68 + 60.5i)T^{2} \) |
| 67 | \( 1 + (0.922 + 1.90i)T + (-41.5 + 52.5i)T^{2} \) |
| 71 | \( 1 + (4.97 - 12.4i)T + (-51.5 - 48.8i)T^{2} \) |
| 73 | \( 1 + (-4.14 - 0.451i)T + (71.2 + 15.6i)T^{2} \) |
| 79 | \( 1 + (-0.201 + 11.1i)T + (-78.9 - 2.85i)T^{2} \) |
| 83 | \( 1 + (1.06 - 0.538i)T + (49.0 - 66.9i)T^{2} \) |
| 89 | \( 1 + (-13.2 - 4.44i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (-5.03 - 6.88i)T + (-29.3 + 92.4i)T^{2} \) |
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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
−10.89617789078270192854939285132, −9.778237894991008483213239838987, −8.498171578323315064316671365600, −7.43226979894544709452741887483, −7.33559679585221533988929540253, −6.33737525362395186608604661928, −5.00143313455736492626726052514, −4.03970847953103809065923496434, −3.20262478261404008782460036582, −0.974502022448113305353838360298,
2.41468749116777131326091807606, 3.17069117175899284591911264633, 4.06559634472887501258614501758, 4.96160142150901121650084633424, 5.88555255149266114949244973167, 7.64263505560230013112181739802, 8.418351513915907553994487752394, 9.192120053474823447307495226423, 10.46610758056415102851252280054, 11.08285711261455695663628321025
