| L(s) = 1 | + (−0.112 − 0.782i)2-s + (−0.909 − 0.415i)3-s + (1.31 − 0.387i)4-s + (−1.72 + 1.99i)5-s + (−0.222 + 0.758i)6-s + (−1.86 + 1.87i)7-s + (−1.10 − 2.42i)8-s + (0.654 + 0.755i)9-s + (1.75 + 1.12i)10-s + (2.41 + 0.347i)11-s + (−1.36 − 0.195i)12-s + (3.57 − 5.55i)13-s + (1.67 + 1.24i)14-s + (2.40 − 1.09i)15-s + (0.539 − 0.346i)16-s + (4.61 + 1.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.0795 − 0.553i)2-s + (−0.525 − 0.239i)3-s + (0.659 − 0.193i)4-s + (−0.773 + 0.892i)5-s + (−0.0909 + 0.309i)6-s + (−0.704 + 0.710i)7-s + (−0.391 − 0.858i)8-s + (0.218 + 0.251i)9-s + (0.555 + 0.356i)10-s + (0.727 + 0.104i)11-s + (−0.392 − 0.0564i)12-s + (0.990 − 1.54i)13-s + (0.448 + 0.333i)14-s + (0.620 − 0.283i)15-s + (0.134 − 0.0866i)16-s + (1.11 + 0.328i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.17668 - 0.394530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17668 - 0.394530i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (1.86 - 1.87i)T \) |
| 23 | \( 1 + (3.22 - 3.55i)T \) |
| good | 2 | \( 1 + (0.112 + 0.782i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (1.72 - 1.99i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-2.41 - 0.347i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-3.57 + 5.55i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.61 - 1.35i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-8.15 + 2.39i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-5.41 - 1.59i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.351 - 0.160i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (3.16 - 2.73i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.75 + 3.25i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-7.91 - 3.61i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 + (4.81 + 7.49i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-0.711 + 1.10i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.14 - 4.70i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (5.63 - 0.810i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.314 + 2.18i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.31 + 7.87i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-3.38 + 5.26i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (2.37 + 2.73i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (0.867 - 1.90i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (3.63 - 4.19i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.97945702326269372182509692260, −10.25618000408802575176922273781, −9.442699735298979177310558375315, −7.954124950715491504158977775889, −7.17628179307559022861436381479, −6.21779749418300036805679212489, −5.54298386380533246521547252892, −3.40562705809976338717227252004, −3.09472704360690997891455050260, −1.13787052369920132628975292809,
1.15787152762512599659232591662, 3.46102460081986589291933221850, 4.27700904596835459205500065553, 5.62073722115042856444740092760, 6.54180087753547096443249393550, 7.29317497806388781272404107864, 8.257202639066018772064968942646, 9.198980476819640068698714066438, 10.16035871466274309786867434569, 11.31564762823476927992165800444
