  We study atypically large fluctuations of height $H$ in the 1+1-dimensional
Kardar-Parisi-Zhang (KPZ) equation at long times $t$, when starting from a
"droplet" initial condition. We derive exact large deviation function of height
for $\lambda H<0$, where $\lambda$ is the nonlinearity coefficient of the KPZ
equation. This large deviation function describes a crossover from the
Tracy-Widom distribution tail at small $|H|/t$, which scales as $|H|^3/t$, to a
different tail at large $|H|/t$, which scales as $|H|^{5/2}/t^{1/2}$. The
latter tail exists at all times $t>0$. It was previously obtained in the
framework of the optimal fluctuation method. It was also obtained at short
times from exact representation of the complete height statistics. The
crossover between the two tails, at long times, occurs at $|H|\sim t$ as
previously conjectured. Our analytical findings are supported by numerical
evaluations using exact representation of the complete height statistics.
