  Let $p_{n}$ denote the $n$th prime and for any fixed positive integer $k$ and
$X\geq 2$, put \[ G_{k}(X):=\max _{p _{n+k}\leq X} \min \{ p_{n+1}-p_{n},
\ldots , p_{n+k}-p_{n+k-1} \}. \] Ford, Maynard and Tao proved that there
exists an effective absolute constant $c_{LG}>0$ such that \[ G_{k}(X)\geq
\frac{c_{LG}}{k^{2}}\frac{\log X \log \log X \log \log \log \log X}{\log \log
\log X} \] holds for any sufficiently large $X$. The main purpose of this paper
is to determine the constant $c_{LG}$ above. We see that $c_{LG}$ is determined
by several factors related to analytic number theory, for example, the ratio of
integrals of functions in the multidimensional sieve of Maynard, the
distribution of primes in arithmetic progressions to large moduli, and the
coefficient of upper bound sieve of Selberg. We prove that the above inequality
is valid at least for $c_{LG}\approx 2.0\times 10^{-17}$.
