  We consider the problem of finding the optimal exponent in the
Moser-Trudinger inequality \[ \sup \left\{\int_\Omega
\exp{\left(\alpha\,|u|^{\frac{N}{N-s}}\right)}\,\bigg|\,u \in
\widetilde{W}^{s,p}_0(\Omega),\,[u]_{W^{s,p}(\mathbb{R}^N)}\leq 1 \right\}< +
\infty.\] Here $\Omega$ is a bounded domain of $\mathbb{R}^N$ ($N\geq 2$), $s
\in (0,1)$, $sp = N$, $\widetilde{W}^{s,p}_0(\Omega)$ is a Sobolev-Slobodeckij
space, and $[\cdot]_{W^{s,p}(\mathbb{R}^N)}$ is the associated Gagliardo
seminorm. We exhibit an explicit exponent $\alpha^*_{s,N}>0$, which does not
depend on $\Omega$, such that the Moser-Trudinger inequality does not hold true
for $\alpha \in (\alpha^*_{s,N},+\infty)$.
