Let $\pi=a+bi\in \mathbb{Z}[i]$ be a Gaussian prime with $a$ and $b$ nonzero, and $b$ even. For odd rational primes $p=\pi\bar\pi$ and $q\neq p$, define $\pi^{\frac{1}{2}\left(q-\left(\frac{-1}{q}\right)\right)} = \alpha + \beta i$, then: \begin{align} \left(\frac{p}{q}\right)=-1 &\iff q\vert\alpha,\;\mathrm{and}\\ \left(\frac{p}{q}\right)=1 &\iff q\vert\beta. \end{align}

I was able to prove this using expansions of $\alpha\beta$ for $q=3\;\mathrm{mod}\;4$, and $p\alpha\beta$ for $q=1\;\mathrm{mod}\;4$, which is straightforward but not particularly insightful. I have searched the literature for a proof that ties this result to the broader concepts of quadratic residues, but I have not been able to find any mention of it.

Specifically, my question is, is this a known result? If so, is there a reference that ties it into the broader context of quadratic residues and reciprocity laws?