L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.978 + 0.207i)3-s + (−0.809 + 0.587i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.381 − 0.169i)7-s + (0.809 + 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.978 − 0.207i)10-s + (−0.275 − 2.62i)11-s + (0.669 − 0.743i)12-s + (−1.91 − 2.13i)13-s + (−0.0436 + 0.415i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−0.588 + 5.59i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.564 + 0.120i)3-s + (−0.404 + 0.293i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.144 − 0.0641i)7-s + (0.286 + 0.207i)8-s + (0.304 − 0.135i)9-s + (−0.309 − 0.0657i)10-s + (−0.0832 − 0.791i)11-s + (0.193 − 0.214i)12-s + (−0.532 − 0.590i)13-s + (−0.0116 + 0.110i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (−0.142 + 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0184i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00497169 - 0.539312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00497169 - 0.539312i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.978 - 0.207i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (5.56 - 0.0902i)T \) |
good | 7 | \( 1 + (0.381 + 0.169i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.275 + 2.62i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (1.91 + 2.13i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.588 - 5.59i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.97 + 2.19i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-2.85 - 2.07i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.59 + 7.97i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (3.43 + 5.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (10.6 + 2.27i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-6.56 + 7.29i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (3.01 - 9.27i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (3.06 - 1.36i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (1.66 - 0.354i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + 9.40T + 61T^{2} \) |
| 67 | \( 1 + (0.288 - 0.499i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.53 - 1.12i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.450 - 4.29i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (0.594 - 5.65i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (9.39 + 1.99i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (-3.08 + 2.24i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-6.33 + 4.60i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.751646517418998135857273704650, −9.008007670925050000058438888985, −8.135355384861944689208991560607, −7.18698743956641144258128696509, −5.93104999350488249520469286563, −5.31192150529869796771042882360, −4.16420378991550734586625062691, −3.17721357845145487398729153812, −1.75271053036983652171537243322, −0.29246576216084647595337838560,
1.67533263761856253298673862321, 3.18221444200056560098976089047, 4.73244168004836060877838958164, 5.21891659227444139268975604553, 6.42853488925509597254429198140, 7.04763373897734168814833685417, 7.62027917362900509089199015141, 8.942098649100682870447928945242, 9.602175361273664324559155302377, 10.31297521513584344001153236908