I'm trying to estimate the product $$\prod_{p\lt q\lt r\lt s}1-\frac{24}{(pqrs)^2}$$ where $p,q,r,s$ are primes.

This is for the purpose of calculating the density of Sloane's A070284 [1]. The idea is that there are 4! congruence classes mod $(pqrs)^2$ such that $n,n+1,n+2,n+3$ are each congruent to 0 mod the square of one of the primes. A number is not the first of four such numbers exactly when it is not in any of these congruence classes for any quadruple of distinct primes.

I feel that there should be some way to accelerate the calculation, possibly using the prime zeta function. At the least, this is useful for the sum approximation which is far easier to calculate with dynamic programming (making it essentially linear rather than quartic, if you can spare $O(n)$ memory). Unfortunately there are enough congruence classes that the cancellation is an important part of the problem, so the sum approximation is poor.

Any suggestions for the calculation, or different approaches to the problem, would be appreciated.

Math. Comp.69 (2000) 407--420], it's not a big deal to compute sums/products over primes up to $10^7$---$10^9$. You don't say how close you wish to be the constant... Do you wish to prove it is 1?! :-) – Wadim Zudilin Jul 23 '10 at 8:12