Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by

$$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) \rangle \sin(n\pi x).$$

For $t>0, y\in L^1(0,1),$ we have $\|S(t)y\|_{L^\infty(0,1)}\le C t^{-\frac{1}{2}} \|y\|_{L^1(0,1)}$.

My question is about an explicit value of the constant $C$.

Thanks!