The zeta-regularized product of odious numbers

What is the product of all {\em odious} integers, i.e., of all integers whose binary expansion contains an odd number of $1$'s? Or more precisely, how to define a product of these integers which is not infinite, but still has a "reasonable" definition? We will answer this question by proving that this product is equal to $\pi^{1/4} \sqrt{2 \varphi e^{-\gamma}}$, where $\gamma$ and $\varphi$ are respectively the Euler-Mascheroni and the Flajolet-Martin constants.