Summary: \(p\)-adic strings are important objects of string theory, as well as of \(p\)-adic mathematical physics and nonlocal cosmology. By a concept of adelic string one can unify and simultaneously study various aspects of ordinary and \(p\)-adic strings. By this way, one can consider adelic strings as a very useful instrument in the further investigation of modern string theory. It is remarkable that for some scalar \(p\)-adic strings exist effective Lagrangians, which are based on real instead of \(p\)-adic numbers and describe not only four-point scattering amplitudes but also all higher ones at the tree level. In this work, starting from \(p\)-adic Lagrangians, we consider some approaches to construction of effective field Lagrangians for \(p\)-adic sector of adelic strings. It yields Lagrangians for nonlinear and nonlocal scalar field theory, where spacetime nonlocality is determined by an infinite number of derivatives contained in the operator-valued Riemann zeta function. Owing to the Riemann zeta function in the dynamics of these scalar field theories, obtained Lagrangians are also interesting in themselves.