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We present another expression to regularize the Euler product representationof the Riemann zeta function. % in this paper. The expression itself isessentially same as the usual Euler product that is the infinite product, butwe define a new one as the limit of the product of some terms derived from theusual Euler product. We also refer to the relation between the Bernoulli numberand $P(z)$, which is an infinite summation of a $z$ power of the inverseprimes. When we apply the same technique to the $L$-function associated to anelliptic curve, we can evaluate the power of the Taylor expansion for thefunction even in the critical strip, which is deeply related to problems knownas the Birch and Swinnerton-Dyer conjecture and the Beilinson conjecture.

Regularized Euler product for the zeta function and the Birch and Swinnerton-Dyer and the Beilinson conjecture