  A quadratic twist of the L-function associated with a modular form is known
to satisfy a functional equation, which may be even or odd. A result due to
Gross and Zagier explicitly computes the central value of the L-function or its
derivative. In prime level when the functional equation is even, Michel and
Ramakrishnan have used an averaging method to prove several consequences of the
Gross-Zagier formulae, including a non-vanishing result. The present research
concerns L-functions arising from newforms in prime-squared level, which
necessarily have odd functional equations. Such an L-function has a central
value of zero; the Gross-Zagier formulae compute the central value of its
derivative. Using the Michel-Ramakrishnan averaging method, we compute the
average value of these derivatives over different L-functions. In particular,
we show that under suitable conditions there exists an L-function with only a
simple zero at the center of symmetry. The proof requires us to bound the
contribution of oldforms from level $p$.
