Cohomology of Restricted Filiform Lie Algebras ${\mathfrak{m}}_2^{\lambda }(p)$

For the $p$-dimensional filiform Lie algebra ${\mathfrak{m}}_2(p)$ over a field $\mathbb{F}$ of prime characteristic $p\ge 5$ with nonzero Lie brackets $[e_1,e_i]=e_{i+1}$ for $1<i<p$ and $[e_2,e_i]=e_{i+2}$ for $2<i<p-1$, we show that there is a family ${\mathfrak{m}}_2^{\lambda }(p)$ of restricted Lie algebra structures parameterized by elements $\lambda \in {\mathbb{F}}^p$. We explicitly describe bases for the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and $[p]$-operations in the corresponding restricted one-dimensional central extensions.