LAUSR

Abstract We establish explicit Ichino’s formulae for the central values of the triple product L-functions with emphasis on the calculations for the real place. The key ingredient for our computations is Proposition 8 which generalizes a result in [P. Michel and A. Venkatesh, The subconvexity problem for GL2{\rm GL}_{2}, Publ. Math. Inst. Hautes Études Sci. 111 2010, 171–271]. As an application we prove the optimal upper bound of a sum of restricted L2{L^{2}}-norms of the L2{L^{2}}-normalized newforms on certain quadratic extensions with prime level and bounded spectral parameter following the methods in [V. Blomer, On the 4-norm of an automorphic form, J. Eur. Math. Soc. (JEMS) 15 2013, 5, 1825–1852].