  Let $1\le N<M$ with $N$ and $M$ coprime and square-free. Through classical
analytic methods we estimate the first moment of central $L$-values $
L(1/2,f\times g) $ where $f\in S^*_k(N)$ runs over primitive holomorphic forms
of level $N$ and trivial nebentypus and $g$ is a given form of level $M$. As a
result, we recover the bound $ L(1/2,f\times g) \ll_\varepsilon (N + \sqrt{M})
N^\varepsilon M^\varepsilon $ when $g$ is dihedral. The first moment method
also applies to the special derivative $L'(1/2,f\times g)$ under the assumption
that it is non-negative for all $f\in S^*_k(N)$.
