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We relate non-critical special values $p$-adic $L$-functions associated toalgebraic Hecke characters of an imaginary quadratic number field with classnumber one to $p$-adic Coleman function called the $p$-adicEisenstein-Kronecker series, when the conductors of the algebraic Heckecharacters are divisible by $p$.

p-adic Eisenstein-Kronecker series and non-critical values of p-adic  Hecke L-functions of an Imaginary Quadratic Field when the conductor is  divisible by p