  We study zeta functions enumerating subalgebras or ideals of Lie algebras
over finite field of prime order $\mathbb{F}_p$. We first develop a general
blueprint method for computing zeta functions of $\mathbb{F}_p$-Lie algebras,
and demonstrate its practical applications in detail to obtain explicit
formulas for various interesting new examples that are not covered in any known
literature yet. For nilpotent cases this also provides zeta functions counting
subgroups and normal subgroups of finite $p$-groups of exponent $p$ for almost
all primes via the Lazard correspondence. We investigate their connections to
the study of finite $p$-groups, and discuss what can be deduced from these
finite Dirichlet polynomials.
