In multilayer quantum Hall systems, when the layer separation is sufficiently small, the interplay between intralayer and interlayer Coulomb interactions can lead to exotic, multi-component, many-body states. A particular example is the even-denominator fractional quantum Hall state (FQHS) at total filling factor ν=1/2 in bilayer systems with negligible interlayer tunneling. This state can be understood as a generalized Laughlin state described by the Halperin-Laughlin Ψ₃₃₁ wavefunction. While such correlated states have been extensively explored in bilayers, little is known about their counterparts in systems with more than two layers. Here, we investigate a trilayer two-dimensional electron system confined to ultrahigh-quality GaAs triple quantum wells. We observe an exotic FQHS at total filling factor ν=5/7, evinced by a deep longitudinal resistance minimum and a quantized Hall plateau, when the side layers have a density larger than the middle layer and d/l_B ≃ 2.2 (d is the interlayer distance and l_B the magnetic length). This state is naturally interpreted as the long-predicted trilayer, generalized Laughlin (Ψ₃₃₃₁₁) state, characterized by ν=1/3-like intralayer correlation within each layer and ν=1-like interlayer correlation between neighboring layers. Our observation establishes a new member of the Halperin-Laughlin many-body states that extends interlayer coherence to three coupled layers.