The center of the generic algebra of degree p

Let $F$ be an algebraically closed field of characteristic zero, and let $p$ be an odd prime. We show that the center of the generic division algebra of degree $p$ is stably rational over $F$. Equivalently, if we let $V=M_p(F) \oplus M_p(F)$ and $PGL_p$ act on $V$ by simultaneous conjugation, then we show that the function field of the quotient variety $V/PGL_p$ is stably rational over $F$.