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We calculate a $\mathrm{G}_2$-period of a Fourier coefficient of a cuspidalEisenstein series on the split simply-connected group $\mathrm{E}_6$, andrelate this period to the Ginzburg-Rallis period of cusp forms on$\mathrm{GL}_6$. This gives us a relation between the Ginzburg-Rallis periodand the central value of the exterior cube L-function of $\mathrm{GL}_6$

A $\mathrm{G}_2$-period of a Fourier coefficient of an Eisenstein series on $\mathrm{E}_6$