The focus of our work is to find and classify special structure in the zeta functions of quartic surface pencils. Our methods draw on hypergeometric functions over finite fields, K3 surfaces, and computational methods. This is joint work with Rachel Davis, Jessamyn Dukes, Thais Gomes Ribeiro, Eli Orvis, Adriana Salerno, and Ursula Whitcher which started in June 2023 as part of Rethinking Number Theory 4, an AIM Research Community.