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In this paper, for any $\\gamma \\in [0, \\lambda^{\\mathsf N} (\\Omega) )$, we establish the following improved Moser-Trudinger inequality \\[ \\sup_{u} \\int_{\\Omega} e^{2\\pi u^2} dx < +\\infty \\] for arbitrary functions $u$ in $H^1(\\Omega)$ satisfying $\\int_\\Omega u dx =0$ and $\\|\\nabla u\\|_2^2 -\\alpha \\|u\\|_2^2 \\leqslant 1$. Furthermore, this supremum is attained by some function $u^*\\in H^1(\\Omega)$. 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