| L(s) = 1 | + (−0.447 − 0.269i)2-s + (0.298 − 0.750i)3-s + (−0.809 − 1.52i)4-s + (1.69 + 0.184i)5-s + (−0.335 + 0.255i)6-s + (−0.866 − 0.291i)7-s + (−0.105 + 1.94i)8-s + (1.70 + 1.61i)9-s + (−0.710 − 0.540i)10-s + (−0.487 − 0.719i)11-s + (−1.38 + 0.150i)12-s + (−4.62 + 4.38i)13-s + (0.308 + 0.363i)14-s + (0.646 − 1.21i)15-s + (−1.36 + 2.01i)16-s + (3.69 − 1.24i)17-s + ⋯ |
| L(s) = 1 | + (−0.316 − 0.190i)2-s + (0.172 − 0.433i)3-s + (−0.404 − 0.763i)4-s + (0.760 + 0.0826i)5-s + (−0.136 + 0.104i)6-s + (−0.327 − 0.110i)7-s + (−0.0372 + 0.686i)8-s + (0.568 + 0.538i)9-s + (−0.224 − 0.170i)10-s + (−0.147 − 0.216i)11-s + (−0.400 + 0.0435i)12-s + (−1.28 + 1.21i)13-s + (0.0825 + 0.0972i)14-s + (0.166 − 0.314i)15-s + (−0.342 + 0.504i)16-s + (0.895 − 0.301i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.717691 - 0.321559i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.717691 - 0.321559i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-3.06 + 7.04i)T \) |
| good | 2 | \( 1 + (0.447 + 0.269i)T + (0.936 + 1.76i)T^{2} \) |
| 3 | \( 1 + (-0.298 + 0.750i)T + (-2.17 - 2.06i)T^{2} \) |
| 5 | \( 1 + (-1.69 - 0.184i)T + (4.88 + 1.07i)T^{2} \) |
| 7 | \( 1 + (0.866 + 0.291i)T + (5.57 + 4.23i)T^{2} \) |
| 11 | \( 1 + (0.487 + 0.719i)T + (-4.07 + 10.2i)T^{2} \) |
| 13 | \( 1 + (4.62 - 4.38i)T + (0.703 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-3.69 + 1.24i)T + (13.5 - 10.2i)T^{2} \) |
| 19 | \( 1 + (0.0961 - 0.586i)T + (-18.0 - 6.06i)T^{2} \) |
| 23 | \( 1 + (0.357 - 1.28i)T + (-19.7 - 11.8i)T^{2} \) |
| 29 | \( 1 + (-3.09 + 1.86i)T + (13.5 - 25.6i)T^{2} \) |
| 31 | \( 1 + (1.41 + 8.63i)T + (-29.3 + 9.89i)T^{2} \) |
| 37 | \( 1 + (0.111 + 2.06i)T + (-36.7 + 4.00i)T^{2} \) |
| 41 | \( 1 + (-1.09 - 3.92i)T + (-35.1 + 21.1i)T^{2} \) |
| 43 | \( 1 + (3.25 - 4.80i)T + (-15.9 - 39.9i)T^{2} \) |
| 47 | \( 1 + (11.8 - 1.29i)T + (45.9 - 10.1i)T^{2} \) |
| 53 | \( 1 + (6.83 - 5.19i)T + (14.1 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.77 - 3.47i)T + (28.5 + 53.8i)T^{2} \) |
| 67 | \( 1 + (-0.161 + 2.97i)T + (-66.6 - 7.24i)T^{2} \) |
| 71 | \( 1 + (-14.2 + 1.54i)T + (69.3 - 15.2i)T^{2} \) |
| 73 | \( 1 + (-0.549 - 0.647i)T + (-11.8 + 72.0i)T^{2} \) |
| 79 | \( 1 + (4.01 + 10.0i)T + (-57.3 + 54.3i)T^{2} \) |
| 83 | \( 1 + (2.60 + 1.20i)T + (53.7 + 63.2i)T^{2} \) |
| 89 | \( 1 + (-9.10 + 5.47i)T + (41.6 - 78.6i)T^{2} \) |
| 97 | \( 1 + (-5.69 + 6.70i)T + (-15.6 - 95.7i)T^{2} \) |
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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
−14.64432708990996284093597379353, −13.92573587341817973830722464369, −13.00105799009595977391974947269, −11.50996287311861300098662791198, −9.942487805066451006010658381381, −9.607020013866896403046024001633, −7.83312330003602156437155649026, −6.34056197618242390398373654830, −4.85323127059047112483980154222, −2.02077067959410380878480542822,
3.27273340688946711144876858951, 5.07564468400815254552691563808, 6.93154323913491065994371638462, 8.277197969317206235397812468900, 9.647415614215758289831457649193, 10.12307625346326312901749926128, 12.35660274823224849718319458773, 12.84189011209740734526268392465, 14.26911463589774156286393994460, 15.41447413004899420122413409565
