Reference for the asymptotic $\sum_p p\,e^{-\varepsilon p}$ as $\varepsilon\to0^+$

This is different from asking whether the Prime Number Theorem implies the asymptotic. I am specifically asking whether this asymptotic has an explicit published reference (journal article, book, or monograph), rather than whether it follows from standard methods.

I am looking for a literature reference rather than a proof verification.

Consider the exponentially smoothed prime sum

$$ S(\varepsilon)=\sum_{p} p\,e^{-\varepsilon p}, \qquad \varepsilon\to0^+. $$

Using Abel summation together with the Prime Number Theorem, one is naturally led to the asymptotic

$$ S(\varepsilon) = \frac{1}{\varepsilon^2\log(1/\varepsilon)} + O\!\left( \frac{1}{\varepsilon^2\log^2(1/\varepsilon)} \right). $$

My question is purely bibliographic.

Has this asymptotic (or an equivalent formulation) already appeared in the published literature?

If so, I would appreciate a reference (journal article, book, or monograph), preferably with a theorem number, proposition, or page number.

I am not asking whether the derivation is correct. I am only trying to determine whether this asymptotic is already known.

References obtained through Abel summation, Laplace transforms of the prime-counting function, Tauberian methods, Mellin transforms, or related analytic number theory techniques are all welcome.