  In recent years, Rogers and Zudilin developed a method to write $L$-values
attached to elliptic curves as periods. In order to apply this method to a
broader collection of $L$-values, we study Eisenstein series and determine
their Fourier series at cusps. Subsequently, we write the $L$-values of an
elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate
the value at $k>1$ explicitly as a period. As a side result, we give simple
integral expressions for the generating functions of $L(E,k)$ when even (or
odd) $k$ runs over positive integers.
