  We study the luminosity function and formation rate of long gamma-ray bursts
(GRBs) by using a maximum likelihood method. This is the first time this method
is applied to a well-defined sample of GRBs that is complete in redshift. The
sample is composed of 99 bursts detected by the $Swift$ satellite, 81 of them
with measured redshift and luminosity for a completeness level of $82\%$. We
confirm that a strong redshift evolution in luminosity (with an evolution index
of $\delta=2.22^{+0.32}_{-0.31}$) or in density ($\delta=1.92^{+0.20}_{-0.21}$)
is needed in order to reproduce the observations well. But since the predicted
redshift and luminosity distributions in the two scenarios are very similar, it
is difficult to distinguish between these two kinds of evolutions only on the
basis of the current sample. Furthermore, we also consider an empirical density
case in which the GRB rate density is directly described as a broken power-law
function and the luminosity function is taken to be non-evolving. In this case,
we find that the GRB formation rate rises like $(1+z)^{3.85^{+0.48}_{-0.45}}$
for $z\leq2$ and is proportional to $(1+z)^{-1.07^{+0.98}_{-1.12}}$ for
$z\geq2$. The local GRB rate is $1.49^{+0.63}_{-0.64}$ Gpc$^{-3}$ yr$^{-1}$.
The GRB rate may be consistent with the cosmic star formation rate (SFR) at
$z\leq2$, but shows an enhancement compared to the SFR at $z\geq2$.
