  Let $\zeta_2(\cdot)$ be the Kubota-Leopoldt $2$-adic zeta function. We prove
that, for every nonnegative integer $s$, there exists an odd integer $j$ in the
interval $[s+3,3s+5]$ such that $\zeta_2(j)$ is irrational. In particular, at
least one of $\zeta_2(7),\zeta_2(9),\zeta_2(11),\zeta_2(13)$ is irrational.
  Our approach is inspired by the recent work of Sprang. We construct explicit
rational functions. The Volkenborn integrals of these rational functions'
(higher-order) derivatives produce good linear combinations of $1$ and $2$-adic
Hurwitz zeta values. The most difficult step is proving that certain Volkenborn
integrals are nonzero, which is resolved by carefully manipulating the binomial
coefficients.
