Let $(u,v)\in\mathbb{Z}^2\setminus(0,0)$. Let $p$ be a prime number congruent to 1 modulo 4 and $\alpha\geq 2$. Show that $L(u,v,p^\alpha)=0$ as soon as $p^{\alpha-1}$ does not divide $u^2+v^2$. Deduce that if $\alpha\geq 3$, then $L_{\mathrm{prim}}(u,v,p^\alpha)=0$ as soon as $p^{\alpha-2}$ does not divide $u^2+v^2$.