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We then express the period $\\langle{\\mathrm{I}_n(f), \\mathrm{I}_n(f)}\\rangle$ of $\\mathrm{I}_n(f)$ for Hecke eigenforms $f$ in terms of special values of some types of $L$-functions of $f$. 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Let $n$ be a positive integer. We denote by $\\mathrm{I}_n$ the lift of the Hermitian modular form of weight $(w_{\\lambda})_{\\lambda}$ with level 1 to unramified automorphic forms defined on the unitary group constructed by Yamana. We then express the period $\\langle{\\mathrm{I}_n(f), \\mathrm{I}_n(f)}\\rangle$ of $\\mathrm{I}_n(f)$ for Hecke eigenforms $f$ in terms of special values of some types of $L$-functions of $f$. 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