We consider an exactly solvable two-chain quantum spin-1/2 model in a generalized form with enhanced spin frustration. This model is exactly diagonalized via Bethe's ansatz. The low-temperature specific heat of the system is obtained with and without a magnetic field, using the thermodynamic Bethe ansatz equations. We also calculate the magnetic susceptibility in a sufficiently weak field, yielding typical logarithmic corrections. The spin frustration affects only the amplitudes in the magnetic susceptibility and the specific heat. This extends the previous results for a simple two-chain quantum spin model to the generalized one.