L(s) = 1 | + (−1.5 + 2.59i)3-s + (0.5 + 0.866i)5-s + (2 − 1.73i)7-s + (−3 − 5.19i)9-s + (0.5 − 0.866i)11-s + 2·13-s − 3·15-s + (−1.5 + 2.59i)17-s + (−2.5 − 4.33i)19-s + (1.5 + 7.79i)21-s + (1.5 + 2.59i)23-s + (2 − 3.46i)25-s + 9·27-s − 6·29-s + (0.5 − 0.866i)31-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.49i)3-s + (0.223 + 0.387i)5-s + (0.755 − 0.654i)7-s + (−1 − 1.73i)9-s + (0.150 − 0.261i)11-s + 0.554·13-s − 0.774·15-s + (−0.363 + 0.630i)17-s + (−0.573 − 0.993i)19-s + (0.327 + 1.70i)21-s + (0.312 + 0.541i)23-s + (0.400 − 0.692i)25-s + 1.73·27-s − 1.11·29-s + (0.0898 − 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.611727 + 0.406909i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.611727 + 0.406909i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 3 | \( 1 + (1.5 - 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 + 2.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 - 9.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−15.45180652684651068427060106111, −14.74678045598644006759510080436, −13.39493068958989750796753253955, −11.49841620452065568321808967167, −10.90686188159101635745940938538, −10.00406633894279305696986491913, −8.613194565707461501191185933750, −6.56227563216336332560750120500, −5.11112800413170815637287925945, −3.87701761008051489127417062745,
1.76846724099755612032672074698, 5.17063541996666485357012921658, 6.31388810281972817628763161017, 7.64440779996722150762856840388, 8.859041642410459046960176387808, 10.89906967327213489504165293918, 11.86672095023638690750641091131, 12.69366067660129257240872095062, 13.64634290129883345826855146101, 14.95036903226213436645933411100