As before, 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

$$ L_f(s) = s(f) = \prod_p \frac{1-\frac{n_f(p)}{p^s}}{1-\frac1{p^s}}. $$

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