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