Product of nine dice divisible by four

Yungeun throws nine fair six-sided dice at once and multiplies the nine numbers shown. What is the probability that the product is a multiple of $4$?

$\tfrac{505}{512}$. The product fails only if every die is odd (probability $2^{-9}$), or exactly one die is even and that die is $2$ or $6$, contributing a single factor of $2$ (probability $9\cdot\tfrac13\cdot2^{-8}=\tfrac{3}{256}$). So the answer is $1-\tfrac1{512}-\tfrac{6}{512}=\tfrac{505}{512}$.