A Gelfand pair is a pair (G; K) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. When G is a locally compact topological group and K is a compact subgroup, (G; K) is a Gelfand pair if and only if the algebra of (K; K)–double invariant compactly supported continuous functions(measures) on G with multiplication defined by convolution is commutative. In studying the concept of Gelfand pairs, the identification of spherical functions is of particular importance. In this paper, the spherical functions of Gelfand pair (G; K) in subspace E1 of L1(G) containing functions of form f ∗ f~ is introduced, where f belonges to Cc(G)(The convolution algebra of continuous, complex-valued functions on G with compact support). Also the characters of E1# have been identified. Finally, by introducing the space Gb# including the bi-K-invariant unitary characters and the space Gd# including bounded spherical functions, the locally compact groups G relatively to Gb#=Gd#, are characterized.