As [before](https://mathoverflow.net/questions/214873/bateman-horn-conjecture-continued), consider the "singular series", which shows up in the Bateman-Horn conjecture: for an irreducible polynomial $f,$ this is equal to


$$
s(f) = \prod\_p \frac{1-\frac{n\_f(p)}p}{1-\frac1p},
$$
which is obviously the value at $1$ of the function


$$
L\_f(s) = \prod\_p \frac{1-\frac{n\_f(p)}{p^s}}{1-\frac1{p^s}}.
$$


The question is: is there some alternative way to evaluate these objects (without multiplying out the Euler product) - an integral formula, or some such?



Bateman-Horn, continued even further
See the following two papers of Nobushige Kurokawa, both appearing in Proc. Japan Acad. Ser. A:


"On Some Euler Products II" (volume 60, 1984, 365-368, esp. Proposition 1)


"Special Values of Euler Products and Hardy-Littlewood Constants" (volume 62, 1986, 25-28), where he writes $Z(s,f)$ for your $L\_f(s)$ and $C(f)$ for your $s(f)$, allowing reducible $f$. In this paper, Theorem A2 rewrites $C(f)$ as rapidly convergent product involving Artin $L$-functions associated to the number field cut out by roots of $f$ and having a cutoff parameter $M$ that can be adjusted.


