L(s) = 1 | + (0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (2.02 + 0.943i)5-s + 0.999i·6-s + (3.31 + 1.91i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.196 + 2.22i)10-s + 1.67·11-s + (−0.866 + 0.499i)12-s + (−1.56 + 2.70i)13-s + 3.82i·14-s + (1.28 + 1.83i)15-s + (−0.5 − 0.866i)16-s + (−2.75 − 4.78i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.906 + 0.421i)5-s + 0.408i·6-s + (1.25 + 0.722i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.0622 + 0.704i)10-s + 0.505·11-s + (−0.249 + 0.144i)12-s + (−0.432 + 0.749i)13-s + 1.02i·14-s + (0.331 + 0.472i)15-s + (−0.125 − 0.216i)16-s + (−0.669 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.949291870\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.949291870\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-2.02 - 0.943i)T \) |
| 37 | \( 1 + (-1.06 + 5.98i)T \) |
good | 7 | \( 1 + (-3.31 - 1.91i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 1.67T + 11T^{2} \) |
| 13 | \( 1 + (1.56 - 2.70i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.75 + 4.78i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.27 + 0.737i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 2.97iT - 29T^{2} \) |
| 31 | \( 1 + 5.65iT - 31T^{2} \) |
| 41 | \( 1 + (-4.89 + 8.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 11.2iT - 47T^{2} \) |
| 53 | \( 1 + (2.49 - 1.44i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.20 + 3.00i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.55 + 3.20i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.933 + 0.539i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.807 + 1.39i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 12.7iT - 73T^{2} \) |
| 79 | \( 1 + (1.38 + 0.796i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.79 - 2.19i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.36 - 4.24i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−9.695424736489807643518518293806, −9.217435133432999461196897332779, −8.507676707268562564061943027912, −7.50755045623094468759657704738, −6.75070051658052502584006313141, −5.76676023313539434970390597119, −4.93952243134693192184678415841, −4.19913961223018866502418592787, −2.69106731557146668371017539616, −1.94565684454789757067892379234,
1.26942102629027862004960899261, 1.91559124302874806521068402778, 3.22408216806101164452690135370, 4.42269917105161031752587328131, 5.04365148685359050652124703128, 6.15578217743918672397890981471, 7.07922842963591821678221960320, 8.317856712838750343222726401254, 8.621011402625035752358206298355, 9.816197926743745671919006032340