| L(s) = 1 | + (0.911 − 2.32i)2-s + (−0.200 + 2.67i)3-s + (−3.09 − 2.86i)4-s + (1.93 + 1.31i)5-s + (6.02 + 2.90i)6-s + (1.00 − 2.44i)7-s + (−4.98 + 2.39i)8-s + (−4.14 − 0.624i)9-s + (4.81 − 3.28i)10-s + (−0.988 + 0.149i)11-s + (8.28 − 7.69i)12-s + (3.12 + 3.91i)13-s + (−4.76 − 4.56i)14-s + (−3.90 + 4.89i)15-s + (0.400 + 5.34i)16-s + (5.99 + 1.84i)17-s + ⋯ |
| L(s) = 1 | + (0.644 − 1.64i)2-s + (−0.115 + 1.54i)3-s + (−1.54 − 1.43i)4-s + (0.863 + 0.588i)5-s + (2.45 + 1.18i)6-s + (0.380 − 0.924i)7-s + (−1.76 + 0.848i)8-s + (−1.38 − 0.208i)9-s + (1.52 − 1.03i)10-s + (−0.298 + 0.0449i)11-s + (2.39 − 2.22i)12-s + (0.866 + 1.08i)13-s + (−1.27 − 1.21i)14-s + (−1.00 + 1.26i)15-s + (0.100 + 1.33i)16-s + (1.45 + 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98775 - 0.842044i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98775 - 0.842044i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.00 + 2.44i)T \) |
| 11 | \( 1 + (0.988 - 0.149i)T \) |
| good | 2 | \( 1 + (-0.911 + 2.32i)T + (-1.46 - 1.36i)T^{2} \) |
| 3 | \( 1 + (0.200 - 2.67i)T + (-2.96 - 0.447i)T^{2} \) |
| 5 | \( 1 + (-1.93 - 1.31i)T + (1.82 + 4.65i)T^{2} \) |
| 13 | \( 1 + (-3.12 - 3.91i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-5.99 - 1.84i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (1.01 + 1.76i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-8.12 + 2.50i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-0.431 - 1.89i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (2.67 - 4.63i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.66 + 1.54i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (7.68 - 3.70i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (7.61 + 3.66i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-3.64 + 9.29i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (0.00629 + 0.00584i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (7.09 - 4.83i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (4.40 - 4.09i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.71 + 4.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.55 + 11.1i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (0.478 + 1.21i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (-3.43 - 5.95i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.97 - 2.47i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (13.4 + 2.03i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−10.67090514595019675413608701146, −10.26771373555835762178758701927, −9.526474383814738011025990255496, −8.678045478556141489415331200414, −6.78519437826583578815509316537, −5.42239690389146867962575385040, −4.72853330653840332505423999974, −3.77798328559228931901346932010, −3.07683911048962384411320525577, −1.53489724152488802040047631994,
1.38166100747397138821751312794, 3.09113279678517817226185310397, 5.11270146447234253465105983997, 5.64579522427900867896192690194, 6.15173483974684337940973956001, 7.27223942886313203777456088173, 8.010319893576397419747094557857, 8.551285706828481482298158416675, 9.601436624832095103210096835791, 11.28544655350795604278555823611
