The shifted convolution L-function for Maass forms

Let \(\Phi_1,\Phi_2\) be Maass forms for \(\text{SL}(2,\mathbb Z)\) with Fourier coefficients \(C_1(n),C_2(n)\). For a positive integer \(h\) the meromorphic continuation and growth in \(s\in\mathbb C\) (away from poles) of the shifted convolution L-function $$L_h(s,{\Phi_1,\Phi_2})\, := \sum_{n \neq 0,-h} {C_1(n) C_2(n + h)} \cdot \big|n(n + h)\big|^{-\frac{1}{2}s}$$ is obtained. For \({\rm Re}(s) > 0\) it is shown that the only poles are possible simple poles at \(\frac{1}{2} \pm ir_k\), where \(\tfrac14+r_k^2\) are eigenvalues of the Laplacian. As an application we obtain, for \(T\to\infty\), the asymptotic formula \begin{align*} & \underset{n \neq 0,-h}{\sum_{\sqrt{|n (n + h)|}