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Let $\pi$ be an irreducible cuspidal automorphic representation of aquasi-split unitary group ${\rm U}_{\mathfrak n}$ defined over a number field$F$. Under the assumption that $\pi$ has a generic global Arthur parameter, weestablish the non-vanishing of the central value of $L$-functions,$L(\frac{1}{2},\pi\times\chi)$, with a certain automorphic character $\chi$ of${\rm U}_1$, for the case of ${\mathfrak n}=2,3,4$, and for the general${\mathfrak n}\geq 5$ by assuming a conjecture on certain refined properties ofglobal Arthur packets. In consequence, we obtain some simultaneousnon-vanishing results for the central $L$-values by means of the theory ofendoscopy.

On the Non-vanishing of the Central Value of Certain L-functions: Unitary Groups