Zeros of a family of approximations of Hecke L-functions associated with cusp forms

We consider a family of approximations of a Hecke L-function L_f(s) attached to a holomorphic cusp form f of positive integral weight k with respect to the full modular group. These families are of the form (Formula presented.), where s= σ+ it is a complex variable and a(n) is a normalized Fourier coefficient of f. From an approximate functional equation, one sees that L_f(X; s) is a good approximation to L_f(s) when X= t/ 2 π. We obtain vertical strips where most of the zeros of L_f(X; s) lie. We study the distribution of zeros of L_f(X; s) when X is independent of t. For X= 1 and 2, we prove that all the complex zeros of L_f(X; s) lie on the critical line σ= 1 / 2. We also show that as T→ ∞ and X= T^o(1), 100% of the complex zeros of L_f(X; s) up to height T lie on the critical line. Here by 100% we mean that the ratio between the number of simple zeros on the critical line and the total number of zeros up to height T approaches 1 as T→ ∞.