$F(k,\lambda) = Q(k+1,\lambda)$ (see, for example, here), where $Q(s,x) = \Gamma(s,x)/\Gamma(s)$ is the regularized incomplete gamma function. This is a well-known enough special function that some common software packages have an implementation for its inverse in $x$. For example, Mathematica uses the command InverseGammaRegularized
. Wolfram|Alpha uses the same command and is freely available, so that may be preferred.
As an illustration, suppose $F = 0.5$ and $k = 10$. Wolfram|Alpha says that InverseGammaRegularized[11,0.5] = 10.668522...
, and we can verify this by calculating $e^{-10.6685} \sum_{i=0}^{10} \frac{10.6685^i}{i!},$ which Wolfram|Alpha says is $0.5$.