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This article introduces the notion of L$_p$-C$^*$-semi-inner product space, a generalization of the concept of C$^*$-semi-inner product space  introduced by Gamchi et al., where we consider H\"{o}lder's inequality instead of Cauchy Schwartz' inequality. We establish some basic results L$_p$-C$^*$-semi-inner product spaces, analogous to those valid for C$^*$-semi-inner product spaces and Hilbert C$^*$-modules.