= Solution
For $K=\mathbb Q(\sqrt{-7})$,
$$
\zeta_K(s)=\zeta(s)L(s,\chi_3).
$$
The character is odd, $\chi_3(2)=1$, and
$$
\sum_{\substack{1\leq k<7/2\\(k,7)=1}}\chi_3(k)
=\chi_3(1)+\chi_3(2)+\chi_3(3)=1+1-1=1.
$$
The supplied odd-character formula gives
$$
L(1,\chi_3)=\frac{\pi}{\sqrt7}.
$$
The <analytic class number formula> for this imaginary quadratic field is
$$
\operatorname*{Res}_{s=1}\zeta_K(s)
=\frac{2\pi h_K}{w_K\sqrt{|d_K|}}
=\frac{\pi h_K}{\sqrt7},
$$
because $w_K=2$ and $d_K=-7$. Since $\operatorname*{Res}_{s=1}\zeta(s)=1$, comparison with $L(1,\chi_3)$ yields $\boxed{h_K=1}$.
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