| L(s) = 1 | + (80.7 − 6.21i)3-s + (282. − 162. i)5-s + (−2.40e3 + 34.7i)7-s + (6.48e3 − 1.00e3i)9-s + (−1.36e4 − 7.89e3i)11-s − 1.93e4·13-s + (2.17e4 − 1.49e4i)15-s + (−8.05e4 − 4.65e4i)17-s + (−5.27e4 − 9.14e4i)19-s + (−1.93e5 + 1.77e4i)21-s + (6.54e4 − 3.78e4i)23-s + (−1.42e5 + 2.46e5i)25-s + (5.17e5 − 1.21e5i)27-s + 4.54e5i·29-s + (3.74e5 − 6.48e5i)31-s + ⋯ |
| L(s) = 1 | + (0.997 − 0.0767i)3-s + (0.451 − 0.260i)5-s + (−0.999 + 0.0144i)7-s + (0.988 − 0.152i)9-s + (−0.934 − 0.539i)11-s − 0.675·13-s + (0.429 − 0.294i)15-s + (−0.964 − 0.557i)17-s + (−0.405 − 0.701i)19-s + (−0.995 + 0.0911i)21-s + (0.234 − 0.135i)23-s + (−0.364 + 0.630i)25-s + (0.973 − 0.228i)27-s + 0.642i·29-s + (0.405 − 0.702i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 + 0.523i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.301440 - 1.06537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.301440 - 1.06537i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-80.7 + 6.21i)T \) |
| 7 | \( 1 + (2.40e3 - 34.7i)T \) |
| good | 5 | \( 1 + (-282. + 162. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.36e4 + 7.89e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + 1.93e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (8.05e4 + 4.65e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (5.27e4 + 9.14e4i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-6.54e4 + 3.78e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 4.54e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-3.74e5 + 6.48e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (1.21e6 + 2.11e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.22e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.00e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (4.46e6 - 2.57e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (9.67e6 + 5.58e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.69e7 + 9.78e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (6.75e6 + 1.16e7i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-8.48e6 + 1.47e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 7.35e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (2.35e6 - 4.08e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-7.14e6 - 1.23e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 1.20e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (3.45e7 - 1.99e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.04e8T + 7.83e15T^{2} \) |
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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
−12.68744120913768336087418144322, −10.90928759602203947583595311231, −9.655657331564131304965806849982, −9.005695199333055975935698675113, −7.66796692403498444145047762548, −6.48451706306654399868931938764, −4.86838380693897214989241066329, −3.21734737562335746331209759193, −2.20000013128179434047535073940, −0.25731995639316073373003977355,
2.01202432399909501523578372003, 3.02235685941993789257356164470, 4.50455672728801107844323673870, 6.27712905110995663992745214809, 7.43874283073716213578572913887, 8.669745791987439130202886598197, 9.884566445152063445030110395275, 10.39488538472092976437594923765, 12.37079833601418947890524123154, 13.16821048679277262748490571833
