In this paper we introduce elliptic Bernoulli functions and numbers, which are related to special Jacobi forms of two variables and study their properties. More importantly, we state and prove elliptic analogues to the following important theorems: i) The Dedekind reciprocity law for Dedekind classical sums, here we introduce enhanced multiple elliptic Dedekind sums and study their reciprocity law. ii) The congruence of Clausen-von-Staudt and Kummer for Bernoulli numbers, here we state and prove it for elliptic Bernoulli numbers. iii) We obtain Damerell's type result concerning the algebraicity of the special values of the Hecke L-function related to our Jacobi forms. iv) As a corollary, we connect these elliptic Bernoulli numbers (explicitly computed) to the special values of Hecke L-functions of imaginary quadratic number field and associated to some Grössencharacter.