| L(s) = 1 | + (19.4 + 11.2i)2-s + (124 + 214. i)4-s + (194. − 112. i)5-s + (875 − 1.51e3i)7-s − 179. i·8-s + 5.04e3·10-s + (6.02e3 + 3.47e3i)11-s + (−1.28e4 − 2.22e4i)13-s + (3.40e4 − 1.96e4i)14-s + (3.37e4 − 5.84e4i)16-s − 7.48e4i·17-s + 1.89e4·19-s + (4.82e4 + 2.78e4i)20-s + (7.81e4 + 1.35e5i)22-s + (4.07e5 − 2.35e5i)23-s + ⋯ |
| L(s) = 1 | + (1.21 + 0.701i)2-s + (0.484 + 0.838i)4-s + (0.311 − 0.179i)5-s + (0.364 − 0.631i)7-s − 0.0438i·8-s + 0.504·10-s + (0.411 + 0.237i)11-s + (−0.450 − 0.780i)13-s + (0.885 − 0.511i)14-s + (0.515 − 0.892i)16-s − 0.896i·17-s + 0.145·19-s + (0.301 + 0.173i)20-s + (0.333 + 0.577i)22-s + (1.45 − 0.840i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(4.23941 - 0.370900i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.23941 - 0.370900i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-19.4 - 11.2i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (-194. + 112. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-875 + 1.51e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-6.02e3 - 3.47e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (1.28e4 + 2.22e4i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + 7.48e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.89e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-4.07e5 + 2.35e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.99e5 - 2.30e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-1.75e5 - 3.04e5i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.33e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (1.62e6 - 9.37e5i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-1.76e6 + 3.05e6i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (3.53e6 + 2.04e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 - 6.60e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.18e7 - 6.85e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (3.76e5 - 6.52e5i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.13e6 + 1.96e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + 1.70e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.76e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (-1.14e7 + 1.99e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (4.01e7 + 2.31e7i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 + 7.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (7.36e7 - 1.27e8i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−13.00534605381094582548179579284, −11.99551078317499361564867036095, −10.57277286322107734813214278892, −9.299781405669793551454171233404, −7.62195777862165552643249647601, −6.71115482857305649068374291986, −5.33127829980017769955031612550, −4.51073960504125088675098730060, −3.03518218341646688378629520754, −0.926557038514433326816351671310,
1.62248091786135455156413435329, 2.77602423495116183537920251159, 4.15667044861523077177629295744, 5.32021233636220806539590870002, 6.46355422506733528401487359863, 8.267995110519380986976500324262, 9.599396667430512481707036164480, 11.03689247866100293077548145214, 11.77417310857594419372963832643, 12.71630035574025924738163233605
