Explicit Bound of $\mathrm{\mid }\zeta (1+it)\mathrm{\mid }$

In this work, we show that for all $t\ge e$, $$\mathrm{\mid }\zeta (1+it)\mathrm{\mid }\le 0.6443\log t\mathrm{.}$$ The equality is achieved when $t=17.7477$. We also use the Riemann-Siegel formula and numerical computations to show that $$\mathrm{\mid }\zeta (1+it)\mathrm{\mid }\le \frac{1}{2}\log t+0.6633\text{when}t\ge e\mathrm{.}$$When $t\ge 100$, the bound $\frac{1}{2}\log t+0.6633$ is better than the bound $0.6443\log t$.