  We introduce the notion of {\it quasi-inner} function and show that the
product $u=\rho_\infty\prod \rho_v$ of $m+1$ ratios of local {$L$-}factors
{$\rho_v(z)=\gamma_v(z)/\gamma_v(1-z)$} over a finite set $F$ of places of the
field of rational numbers {inclusive of} the archimedean place is {quasi-inner}
on the left of the critical line $\Re(z)= \frac 12$ in the following sense. The
off diagonal part $u_{21}$ of the matrix of the multiplication by $u$ in the
orthogonal decomposition of the Hilbert space $L^2$ of square integrable
functions on the critical line into the Hardy space $H^2$ and its orthogonal
complement is a compact operator. When interpreted on the unit disk, the
quasi-inner condition means that the associated Haenkel matrix is compact. We
show that none of the individual non-archimedean ratios $\rho_v$ is quasi-inner
and, in order to prove our main result we use Gauss multiplication theorem to
factor the archimedean ratio $\rho_\infty$ into a product of $m$ quasi-inner
functions whose product with each $\rho_v$ retains the property to be
quasi-inner. Finally we prove that Sonin's space is simply the kernel of the
diagonal part $u_{22}$ for the quasi-inner function $u=\rho_\infty$, and when
$u(F)=\prod_{v\in F} \rho_v$ the kernels of the $u(F)_{22}$ form an inductive
system of infinite dimensional spaces which are the semi-local analogues of
(classical) Sonin's spaces.
