L(s) = 1 | + (0.592 − 1.62i)3-s + (−0.673 + 1.16i)5-s + (0.5 + 0.866i)7-s + (−2.29 − 1.92i)9-s + (0.826 + 1.43i)11-s + (1.68 − 2.91i)13-s + (1.5 + 1.78i)15-s + 0.467·17-s + 3.22·19-s + (1.70 − 0.300i)21-s + (4.47 − 7.74i)23-s + (1.59 + 2.75i)25-s + (−4.5 + 2.59i)27-s + (−3.13 − 5.42i)29-s + (4.61 − 7.99i)31-s + ⋯ |
L(s) = 1 | + (0.342 − 0.939i)3-s + (−0.301 + 0.521i)5-s + (0.188 + 0.327i)7-s + (−0.766 − 0.642i)9-s + (0.249 + 0.431i)11-s + (0.467 − 0.809i)13-s + (0.387 + 0.461i)15-s + 0.113·17-s + 0.740·19-s + (0.372 − 0.0656i)21-s + (0.932 − 1.61i)23-s + (0.318 + 0.551i)25-s + (−0.866 + 0.499i)27-s + (−0.582 − 1.00i)29-s + (0.829 − 1.43i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.752427678\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.752427678\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.592 + 1.62i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.673 - 1.16i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 1.43i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.68 + 2.91i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.467T + 17T^{2} \) |
| 19 | \( 1 - 3.22T + 19T^{2} \) |
| 23 | \( 1 + (-4.47 + 7.74i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.13 + 5.42i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.61 + 7.99i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.23T + 37T^{2} \) |
| 41 | \( 1 + (1.70 - 2.95i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.20 + 3.82i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.67 - 8.10i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 0.573T + 53T^{2} \) |
| 59 | \( 1 + (5.19 - 9.00i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.81 + 6.61i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.298 + 0.516i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.554T + 71T^{2} \) |
| 73 | \( 1 - 2.04T + 73T^{2} \) |
| 79 | \( 1 + (1.20 + 2.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.52 + 13.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9.08T + 89T^{2} \) |
| 97 | \( 1 + (-0.949 - 1.64i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.695830157164762560921827237431, −8.893685683607703970605930678634, −7.939239139855474721399359620038, −7.48946892044556602368701499303, −6.46379956206924139791086327308, −5.80433528669956558524515373121, −4.47930790051873363644347740068, −3.18910708892034705590491890274, −2.42286205617761853709066996569, −0.912704431766251368778684771770,
1.31806526928197970526707385381, 3.06546615981756344861273065414, 3.86373547275153675005352696058, 4.79418099229561396168914779269, 5.52288350965617460420720435936, 6.78385447338640196712519830077, 7.78748688638382499064473593768, 8.662350212946692807922224606391, 9.187955173060430422785849399364, 9.984869053922974154749324489116