L(s) = 1 | + 1.73·5-s + (−2 + 1.73i)7-s − 3i·11-s − 3.46i·13-s + 6.92·17-s + 1.73i·19-s − 3i·23-s − 2.00·25-s + 6i·29-s − 5.19i·31-s + (−3.46 + 2.99i)35-s + 7·37-s − 12.1·41-s + 2·43-s + 3.46·47-s + ⋯ |
L(s) = 1 | + 0.774·5-s + (−0.755 + 0.654i)7-s − 0.904i·11-s − 0.960i·13-s + 1.68·17-s + 0.397i·19-s − 0.625i·23-s − 0.400·25-s + 1.11i·29-s − 0.933i·31-s + (−0.585 + 0.507i)35-s + 1.15·37-s − 1.89·41-s + 0.304·43-s + 0.505·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.860063375\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.860063375\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 3iT - 71T^{2} \) |
| 73 | \( 1 - 3.46iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−8.511855443474387378043284028358, −8.081196730621368771428425221743, −7.05028294406455455019080858581, −5.99199822920316607045620948768, −5.82034701978059741727132604678, −5.02061166237691540840470232124, −3.51703100978515882937397451218, −3.09301302687798773058509012169, −1.98201862986420093383132169337, −0.64600425920769073754122259475,
1.13870564884757769888179378592, 2.15135714466086062282842259098, 3.23407553226875657623405817705, 4.09788127557668054129975893460, 4.97298416771052609221957228324, 5.90286518911750377260681662107, 6.52216793672346536270127912157, 7.33822563170294052931255173514, 7.86665951037412811681622372157, 9.160623462650610666172497464772