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 + (3.87 − 1.72i)7-s + (0.809 − 0.587i)8-s + (0.913 + 0.406i)9-s + (−0.978 + 0.207i)10-s + (−0.300 + 2.85i)11-s + (0.669 + 0.743i)12-s + (2.45 − 2.72i)13-s + (0.442 + 4.21i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.133 − 1.27i)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 + (1.46 − 0.651i)7-s + (0.286 − 0.207i)8-s + (0.304 + 0.135i)9-s + (−0.309 + 0.0657i)10-s + (−0.0904 + 0.860i)11-s + (0.193 + 0.214i)12-s + (0.680 − 0.755i)13-s + (0.118 + 1.12i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.0324 − 0.308i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35894 + 0.150624i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35894 + 0.150624i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-5.42 + 1.23i)T \) |
good | 7 | \( 1 + (-3.87 + 1.72i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (0.300 - 2.85i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-2.45 + 2.72i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.133 + 1.27i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (5.60 + 6.22i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.267 - 0.194i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.60 + 4.94i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-0.905 + 1.56i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.07 + 1.50i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-6.91 - 7.68i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.82 - 8.70i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (9.23 + 4.10i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (0.142 + 0.0302i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 - 4.05T + 61T^{2} \) |
| 67 | \( 1 + (0.978 + 1.69i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.96 - 3.10i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (0.627 - 5.96i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (1.10 + 10.4i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (0.858 - 0.182i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-7.85 - 5.70i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 7.82i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.19437255713287817056304700565, −9.219422517617480417995806226836, −8.095795271795025131977689204140, −7.62469259154739201821479227833, −6.69661204497375219981649136774, −5.91869154178530091105169920001, −4.76072295187391046216351829438, −4.33757081840226425683601897536, −2.36395890832106140792060277825, −0.920081683926832819337262153608,
1.24495951018514744837589507201, 2.17268225171612407652607311648, 3.82526855382872596857514738238, 4.66547235598220363203432340796, 5.60648723142989816342206456304, 6.35430785870364045016243002515, 7.897007506844845889133749845210, 8.544500156316896055714254778103, 9.054403878659586499647540499801, 10.36594113142116003747862041499