L(s) = 1 | + (−1.16 − 1.28i)3-s + (−1.92 + 2.29i)5-s + (−0.0588 + 0.161i)7-s + (−0.290 + 2.98i)9-s + (4.37 − 3.67i)11-s + (0.936 − 5.31i)13-s + (5.17 − 0.200i)15-s + (2.85 + 1.64i)17-s + (5.19 − 3.00i)19-s + (0.275 − 0.112i)21-s + (−4.55 + 1.65i)23-s + (−0.685 − 3.88i)25-s + (4.16 − 3.10i)27-s + (−0.654 + 0.115i)29-s + (−2.74 − 7.54i)31-s + ⋯ |
L(s) = 1 | + (−0.671 − 0.740i)3-s + (−0.859 + 1.02i)5-s + (−0.0222 + 0.0611i)7-s + (−0.0968 + 0.995i)9-s + (1.31 − 1.10i)11-s + (0.259 − 1.47i)13-s + (1.33 − 0.0518i)15-s + (0.692 + 0.400i)17-s + (1.19 − 0.688i)19-s + (0.0602 − 0.0246i)21-s + (−0.949 + 0.345i)23-s + (−0.137 − 0.777i)25-s + (0.802 − 0.597i)27-s + (−0.121 + 0.0214i)29-s + (−0.493 − 1.35i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.898456 - 0.418981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.898456 - 0.418981i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.16 + 1.28i)T \) |
good | 5 | \( 1 + (1.92 - 2.29i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.0588 - 0.161i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (-4.37 + 3.67i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.936 + 5.31i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-2.85 - 1.64i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.19 + 3.00i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.55 - 1.65i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.654 - 0.115i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.74 + 7.54i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.25 + 3.90i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.59 - 1.51i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.57 - 1.87i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.36 - 1.95i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 5.08iT - 53T^{2} \) |
| 59 | \( 1 + (3.98 + 3.34i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.78 - 1.37i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (14.1 + 2.50i)T + (62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.47 + 2.54i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.06 + 5.30i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.23 + 1.62i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.641 - 3.63i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (0.248 - 0.143i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.32 + 7.82i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.18911464070045485668218507896, −10.52240296349835651226189475218, −9.187779493700050257297734496671, −7.73651156007429294724457035806, −7.58628679122439019575101791652, −6.18566282614418888114263055510, −5.71729647670707713057618269137, −3.91150943919596632609993515409, −2.92400579255607780675594959283, −0.852917880443753682943194532580,
1.28547976276075373937894769843, 3.84278348744451383122785556633, 4.30369796208126125441116199252, 5.32647020337370557457036058044, 6.57287745872574147214724986192, 7.53417566953354758494956777981, 8.894421958909720536834087180370, 9.383137005837371210436527704912, 10.29447042743790943507879694422, 11.75103406506361528991217277136