  Let $k$ be the rational function field over the finite field of $q$ elements
and $\bar{k}$ its fixed algebraic closure. In this paper, we study algebraic
relations over $\bar{k}$ among the fundamental period $\widetilde{\pi}$ of the
Carlitz module and the positive characteristic multizeta values $\zeta(n)$ and
$\zeta(n,n)$ for an "odd" integer $n$, where we say that $n$ is "odd" if $q-1$
does not divide $n$. We prove that either they are algebraically independent
over $\bar{k}$ or satisfy some simple relation over $k$. We also prove that if
$2n$ is "odd" then they are algebraically independent over $\bar{k}$.
