The expected number of elements to generate a finite group with $d$-generated Sylow subgroups

Given a finite group $G,$ let $e(G)$ be expected number of elements of $G$ which have to be drawn at random, with replacement, before a set of generators is found. If all the Sylow subgroups of $G$ can be generated by $d$ elements, then $e(G)\leq d+\kappa$ with $\kappa \sim 2.75239495.$ The number $\kappa$ is explicitly described in terms of the Riemann zeta function and is best possible. If $G$ is a permutation group of degree $n,$ then either $G=S_3$ and $e(G)=2.9$ or $e(G)\leq \lfloor n/2\rfloor+\kappa^*$ with $\kappa^* \sim 1.606695.$