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The classical Lerch's formula states the following normalized product:$$\prod_{n=0}^\infty(x+n)=\frac{\sqrt{2\pi}}{\Gamma(x)},\quad \textrm{Re}(x)>0,$$ where $\Gamma(x)$ is the Euler gamma function.  In this note, by using Barnes' multiple zeta function and its alternatingform, we obtain two kinds of generalizations of Lerch's formula, which implythe product $$ \prod_{n=1}^\infty n=\sqrt{2\pi} $$ (in the sense of zetaregularization) and the product $$\frac{2\cdot2}{1\cdot 3}\frac{4\cdot4}{3\cdot5}\frac{6\cdot6}{5\cdot 7}\cdots=\frac{\pi}{2}$$ (Wallis' formula in 1656),respectively.

Generalizations of Lerch's formula by Barnes' multiple zeta functions