 Solution

For the sawtooth Ising chain, each apex spin occurs in only one triangle. Put $K=\beta J$. Summing that spin first gives the effective transfer matrixIn the ordered basis $s=+1,-1$ this isIts symmetric and antisymmetric eigenvectors are $(1,1)/\sqrt{2}$ and $(1,-1)/\sqrt{2}$, with eigenvaluesConsequentlyAt $J=0$, the result is $Z_L=4^L$, correctly counting $2L$ independent Ising spins. As $T\to 0$, $f_{\infty }\to -3J/2$, the energy of three satisfied ferromagnetic bonds per two sites.

The printed summand lacks grouping brackets. The result above uses the translation-invariant common-$J$ interpretation. If base and apex bonds instead have independent couplings $J_b,J_a$, the same derivation gives ${\lambda }_{\pm }=2e^{\beta J_b}\cosh (2\beta J_a)\pm 2e^{-\beta J_b}$; specifying those couplings is essential before giving a different answer.