As [before][1], 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?

  [1]: http://mathoverflow.net/questions/214873/bateman-horn-conjecture-continued