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
$$ L_f(s) = \prod_p \frac{1-\frac{n_f(p)}{p^s}}{1-\frac1{p^s}}. $$
The question is: is there some alternative way to evaluate these objects (without multiplying out the Euler product) - an integral formula, or some such?
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.
I record this here in the manner of usefulness.
The preprint draft http://www.math.u-bordeaux1.fr/~cohen/hardylw.dvi of Cohen gives a method for computing such constants.
The idea is to turn an Euler product into an Euler sum by taking logarithms, transform by Möbius inversion to turn sums over prime powers into logs of $L$-functions at positive integers, and then evaluate the $L$-functions by "standard" methods. To speed the convergence, one can consider the primes up to $A$ (say 30) separately.
In Sections 1 and 3, Cohen considers how to evaluate said $L$-functions (such methods are standard following Hecke/Lavrik, and are implemented by Dokchitser in GP/PARI or Magma), and in sections 4 and 5 he describes the cases of quadratic and cubic fields more fully. Section 2 gives an overview about the manipulations with sums over prime powers.
As a model case to illustrate, consider (Cohen's Section 4) the case of a quadratic Dirichlet character $\chi$. The base objects are the sums $S_m(\chi)=\sum_p \chi(p)/p^m$, and by Möbius inversion these are equal to
$$S_m(\chi)=\sum_{k\ge 1}^\infty {\mu(k)\over k}\log L(km,\chi^k).$$
However, by splitting the $p$-sum at $A$, one in fact by a similar inversion has
$$S_m(\chi)=\sum_{p\lt A}{\chi(p)\over p^m}+\sum_{k\ge 1}^\infty {\mu(k)\over k}\log L_{p\ge A}(km,\chi^k),$$
and this $\log L_{p\ge A}(km,\chi^k)$ is bounded by $O(1/A^{km})$ (Cohen does this in detail in Section 2.1).
To use the above formula, one needs a method to compute $L(km,\chi^k)$, and as was said, for any $L$-function this is standard (by an approximate functional equation with weights given by an incomplete Mellin transform of the $\Gamma$-factors), and takes time proportional to approximately the square root of the conductor of $\chi^k$ (with mild constants dependent on the desired precision, and also the size of $km$).
In order to obtain the Hardy-Littlewood or Bateman-Horn constants from the base objects $S_m(\chi)$, one needs to piece them together in the proper manner, which is largely bookkeeping (see 4.1 and 4.2 of Cohen), involving Dirichlet convolution of arithmetic sequences.
For higher degree polynomials, instead of Dirichlet $L$-functions one will also need Artin $L$-functions, which Cohen points out could be computed by $\zeta$-quotients, though this would be rather inefficient. Instead, a Magma package of Dokchitser allows one to compute with Artin representations rather easily. The time is still proportional to approximately square root of the conductor, though with higher degree $L$-functions the constant factor(s) are not quite so nice.
I am not sure this really answers the question. In this method, one does not "multiply out" the Euler product, except for the small primes (though one could take $A=2$, if treating small primes differently is not pleasing). The bulk contribution is determined by an $L$-function computation, which I suppose could be seen as an "integral" (in terms of a Mellin transform?) if desired.
As it seems to be of some interest, let me record some comments on the convergence rate of the Euler product (which are probably standard).
We consider $$C(f)=\prod_p \frac{1-\eta_f(p)/p}{1-1/p},$$ and as was pointed out, this is kind of like an $L$-function at $s=1$, namely that at good primes the Euler factor $(1-\eta_f(p)/p^s)^{-1}$ gives the same Dirichlet series coefficient at $p$ (though not powers of $p$) as the Dedekind $\zeta$-function of the field associated to $f$.
Everything below will be in terms of relative precision, which should not matter much as $C(f)$ itself is not particularly big nor small. Consider the contribution in the above product from $p\ge X$ and take logs, getting $$\sum_{p\ge X}\log(1-\eta_f(p)/p)-\log(1-1/p)=\sum_{p^k\ge X}\frac{1-\eta_f(p)^k}{p^k}=\sum_{p\ge X}\frac{1-\eta_f(p)}{p}+O(1/X),$$ where the big-Oh constant should be no worse than about the square of the degree of $f$. Under GRH and the Artin conjecture, we will show this $p$-sum tail is $O((\log |\Delta_f|)/\sqrt X)$.
As indicated above, $\eta_f(p)$ relates to Dedekind $\zeta$-function for $f$. Let us write $A(s)=\zeta_f(s)/\zeta(s)$, and this Artin $L$-function will be "nice" if we assume enough hypotheses. In particular, by Perron's formula we have $$\int_{(1)}X^s\frac{A'}{A}(s+1){ds\over 2\pi is}= -\sum_{\|{\frak p}\|^k\le X}{\log\|{\frak p}\|\over \|{\frak p}\|^k} +\sum_{p^k\le X}{\log p\over p^k}.$$ The terms with $k\ge 2$ contribute $B_1+B_2+O(1/X)$, where $B_i$ are the relevant sums extended to infinity (not cut off at $X$), and the prime ideals with $\|{\frak p}\|\neq p$ contribute $O(1/\sqrt X)$.
The prime ideals with $\|{\frak p}\|=p$ are those counted by $\eta_f(p)$ (at least for good primes), so we have $$\int_{(1)}X^s\frac{A'}{A}(s+1){ds\over 2\pi is}=B_1+B_2+\sum_{p\le X}(1-\eta_f(p))\frac{\log p}{p}+O(1/\sqrt X).$$ We then truncate the line of integration at say height $X^3$, with error $O(X/X^3)$ (again the big-Oh depends on the degree of $f$). We then move to contour to the left, say $\sigma=-3/4$. Under GRH the zeros are on $\sigma=-1/2$, and the contribution from $\sigma=-3/4$ is $O((\log N)(\log X)/X^{3/4})$, where $N$ is the conductor, thus the absolute value of the discriminant of $f$. Similarly the zeros themselves contribute $O(\log N)(\log X)/\sqrt X$, by zero-density estimates (namely $\log N$) and a harmonic sum up to height $X^3$ from the $1/s$.
So we get $$\sum_{p\le X}(1-\eta_f(p))\frac{\log p}{p}=B+O((\log N)(\log X)/\sqrt X)$$ and by partial summation $$\sum_{p\le X}\frac{1-\eta_f(p)}{p}=\tilde B+O((\log N)/\sqrt X).$$ In particular we have $$\sum_{p\ge X}\frac{1-\eta_f(p)}{p}=O((\log N)/\sqrt X),$$ which as above gives the convergence rate for the desired constant $C(f)$.
Maybe some analytic number theorist will come around and correct anything (and fix any sign errors I made). Also, the Artin conjecture might not be necessary, just the GRH for $\zeta_f$ and $\zeta$ (poles on the half-line are not bothersome).
