  For a family of real quadratic fields $\{K_n=\FQ(\sqrt{f(n)})\}_{n\in \FN}$,
a Dirichlet character $\chi$ modulo $q$ and prescribed ideals $\{\fb_n\subset
K_n\}$, we investigate the linear behaviour of the special value of partial
Hecke's L-function $L_{K_n}(s,\chi_n:=\chi\circ N_{K_n},\fb_n)$ at $s=0$. We
show that for $n=qk+r$, $L_{K_n}(0,\chi_n,\fb_n)$ can be written as
$$\frac{1}{12q^2}(A_{\chi}(r)+kB_{\chi}(r)),$$ where
$A_{\chi}(r),B_{\chi}(r)\in \FZ[\chi(1),\chi(2),..., \chi(q)]$ if a certain
condition on $\fb_n$ in terms of its continued fraction is satisfied.
Furthermore, we write precisely $A_{\chi}(r)$ and $B_{\chi}(r)$ using values of
the Bernoulli polynomials. We describe how the linearity is used in solving
class number one problem for some families and recover the proofs in some
cases. Finally, we list some families of real quadratic fields with the
linearity.
