| L(s) = 1 | + (−7.89 + 13.9i)2-s + (−80.5 − 33.3i)3-s + (−131. − 219. i)4-s + (492. − 203. i)5-s + (1.10e3 − 857. i)6-s + (729. + 729. i)7-s + (4.09e3 − 90.2i)8-s + (740. + 740. i)9-s + (−1.05e3 + 8.46e3i)10-s + (−6.35e3 + 2.63e3i)11-s + (3.23e3 + 2.20e4i)12-s + (−3.94e3 − 1.63e3i)13-s + (−1.59e4 + 4.39e3i)14-s − 4.64e4·15-s + (−3.10e4 + 5.76e4i)16-s − 6.23e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.493 + 0.869i)2-s + (−0.994 − 0.412i)3-s + (−0.512 − 0.858i)4-s + (0.787 − 0.326i)5-s + (0.849 − 0.661i)6-s + (0.304 + 0.304i)7-s + (0.999 − 0.0220i)8-s + (0.112 + 0.112i)9-s + (−0.105 + 0.846i)10-s + (−0.433 + 0.179i)11-s + (0.156 + 1.06i)12-s + (−0.138 − 0.0572i)13-s + (−0.414 + 0.114i)14-s − 0.918·15-s + (−0.474 + 0.880i)16-s − 0.746i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0802490 + 0.375181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0802490 + 0.375181i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (7.89 - 13.9i)T \) |
| good | 3 | \( 1 + (80.5 + 33.3i)T + (4.63e3 + 4.63e3i)T^{2} \) |
| 5 | \( 1 + (-492. + 203. i)T + (2.76e5 - 2.76e5i)T^{2} \) |
| 7 | \( 1 + (-729. - 729. i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (6.35e3 - 2.63e3i)T + (1.51e8 - 1.51e8i)T^{2} \) |
| 13 | \( 1 + (3.94e3 + 1.63e3i)T + (5.76e8 + 5.76e8i)T^{2} \) |
| 17 | \( 1 + 6.23e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + (1.74e4 - 4.21e4i)T + (-1.20e10 - 1.20e10i)T^{2} \) |
| 23 | \( 1 + (3.32e5 - 3.32e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + (1.95e5 - 4.71e5i)T + (-3.53e11 - 3.53e11i)T^{2} \) |
| 31 | \( 1 - 1.18e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-1.70e5 + 7.04e4i)T + (2.48e12 - 2.48e12i)T^{2} \) |
| 41 | \( 1 + (-6.63e5 - 6.63e5i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (-2.55e6 + 1.05e6i)T + (8.26e12 - 8.26e12i)T^{2} \) |
| 47 | \( 1 + 6.35e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + (-2.67e6 - 6.44e6i)T + (-4.40e13 + 4.40e13i)T^{2} \) |
| 59 | \( 1 + (-4.44e6 - 1.07e7i)T + (-1.03e14 + 1.03e14i)T^{2} \) |
| 61 | \( 1 + (6.02e6 - 1.45e7i)T + (-1.35e14 - 1.35e14i)T^{2} \) |
| 67 | \( 1 + (1.35e7 + 5.59e6i)T + (2.87e14 + 2.87e14i)T^{2} \) |
| 71 | \( 1 + (4.65e5 + 4.65e5i)T + 6.45e14iT^{2} \) |
| 73 | \( 1 + (2.59e7 + 2.59e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + 4.68e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (2.46e7 - 5.94e7i)T + (-1.59e15 - 1.59e15i)T^{2} \) |
| 89 | \( 1 + (-6.67e7 + 6.67e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 - 3.13e7T + 7.83e15T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.85778588510774177947710657618, −14.42134200317492840111612060140, −13.21440740648999795173914689367, −11.78622982139415603626219066789, −10.29544331740365078345082739808, −9.022767556807870970943342357557, −7.40576909502272013878058779337, −5.96500525047244813216073828873, −5.16856138288258987305703377945, −1.47483210994367575717849414608,
0.22164056033077987474676295378, 2.23069628584545998900634192077, 4.40225344410574622425169883728, 6.04129375165580020143027282445, 8.092107650165351575031919229981, 9.860779283626702468120986538103, 10.63676991830361610619697989706, 11.59918017091593605274122428435, 12.99215774815773390512552712464, 14.23829377360051709532580518609
