Explicit zero density estimate near unity

We will provide the first explicit zero-density estimate for $\zeta$ of the form $N(\sigma ,T)\le \mathcal{C}{T}^{B(1-\sigma {)}^{3\mathrm{/}2}}(\log T{)}^C$. In particular, we improve $C$ to $10393\mathrm{/}900=11.547\dots \mathrm{.}$