| L(s) = 1 | + (1.96 − 2.26i)3-s + (−0.319 − 0.552i)5-s + (−1.11 − 1.93i)7-s + (−1.28 − 8.90i)9-s + (1.89 + 3.27i)11-s + 9.65i·13-s + (−1.88 − 0.362i)15-s + (11.4 − 19.8i)17-s + (7.12 − 17.6i)19-s + (−6.57 − 1.26i)21-s + 7.93·23-s + (12.2 − 21.2i)25-s + (−22.7 − 14.5i)27-s + (−26.3 − 15.2i)29-s + (−50.4 − 29.1i)31-s + ⋯ |
| L(s) = 1 | + (0.654 − 0.755i)3-s + (−0.0638 − 0.110i)5-s + (−0.159 − 0.276i)7-s + (−0.142 − 0.989i)9-s + (0.171 + 0.297i)11-s + 0.742i·13-s + (−0.125 − 0.0241i)15-s + (0.673 − 1.16i)17-s + (0.375 − 0.926i)19-s + (−0.313 − 0.0602i)21-s + 0.344·23-s + (0.491 − 0.851i)25-s + (−0.841 − 0.539i)27-s + (−0.908 − 0.524i)29-s + (−1.62 − 0.938i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.915640379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.915640379\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.96 + 2.26i)T \) |
| 19 | \( 1 + (-7.12 + 17.6i)T \) |
| good | 5 | \( 1 + (0.319 + 0.552i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (1.11 + 1.93i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-1.89 - 3.27i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 9.65iT - 169T^{2} \) |
| 17 | \( 1 + (-11.4 + 19.8i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 - 7.93T + 529T^{2} \) |
| 29 | \( 1 + (26.3 + 15.2i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (50.4 + 29.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 12.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (50.1 - 28.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 - 29.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-22.2 + 38.5i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (15.1 - 8.76i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (41.2 - 23.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (34.9 - 60.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + 20.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-99.8 - 57.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-66.8 + 115. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 15.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-47.6 - 82.5i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (70.8 - 40.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 45.2iT - 9.40e3T^{2} \) |
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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
−9.584026602826273615111205463873, −9.275591491129854902379644791799, −8.191368493108851089856042863641, −7.23716153392262812661006804933, −6.81548343583740383066043289929, −5.52676995754057846856013532441, −4.28949722048667536007971416687, −3.16839460475323338561916323308, −2.04424535880626914781807436036, −0.63067375845724495208306925839,
1.69187764666295410289747468638, 3.24675028400227845903680572837, 3.70525180934291202984917143200, 5.18935160710923904194533063906, 5.82755551400803923774252028303, 7.28784465400632445129066417360, 8.069627674669579897949831196285, 8.929371067077275550518201965894, 9.603266614912046168343349710303, 10.65657045151048627494729570690
