  Langlands has introduced a formula for a specific product of orbital
integrals in $\mbox{GL}(2, \mathbb{Q})$. Altu\u{g} employs this formula to
manipulate the regular elliptic part of the trace formula, with the aim of
eliminating the contribution of the trivial representation from the spectral
side.
  Arthur predicts that this formula coincides with a product of polynomials
associated with zeta functions of orders developed by Zhiwei Yun. In a previous
paper, the author determined the explicit polynomials for the relevant
quadratic orders. This paper demonstrates how these polynomials can effectively
generalize Langlands' formula to $GL(2, K)$, for general algebraic number
fields $K$.
  Furthermore, we also use this formula to extend a well-known formula of
Zagier to any algebraic number field and explain its applications in the
contexts of the strategy of Beyond Endoscopy proposed by Langlands.
