  We give an explicit formula for the second variation of the logarithm of the
Selberg zeta function, $Z(s)$, on Teichm\"uller space. We then use this formula
to determine the asymptotic behavior as $\text{Re} (s) \to \infty$ of the
second variation. As a consequence, for $m \in \mathbb{N}$, we obtain the
complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t,
\mathcal K_t)\to t\in \mathcal T$ of holomorphic m-differentials over the
Teichm\"uller space $\mathcal T$, for $m$ large. Moreover, we show that this
curvature agrees with the Quillen curvature up to a term of exponential decay,
$O(m^2 e^{-l_0 m}),$ where $l_0$ is the length of the shortest closed
hyperbolic geodesic.
