What is the probability that a large random matrix has no real eigenvalues?

We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2ntimes 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k}=lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,0}= -frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ where $zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{ngeq 1}$, $$lim_{nrightarrow infty}frac {1}{sqrt{2n}} log p_{2n,2k_n}=-frac{1}{sqrt{2pi}}zetaleft(frac{3}{2}right),$$ provided $lim_{nrightarrow infty} left(n^{-1/2}log(n)right) k_{n}=0$.