  Recent work of Bettin and Conrey on the period functions of Eisenstein series
naturally gave rise to the Dedekind-like sum \[
  c_{a}\left(\frac{h}{k}\right) \ = \
  k^{a}\sum_{m=1}^{k-1}\cot\left(\frac{\pi
mh}{k}\right)\zeta\left(-a,\frac{m}{k}\right), \] where $a\in\mathbb{C}$, $h$
and $k$ are positive coprime integers, and $\zeta(a,x)$ denotes the Hurwitz
zeta function. We derive a new reciprocity theorem for these Bettin--Conrey
sums, which in the case of an odd negative integer $a$ can be explicitly given
in terms of Bernoulli numbers. This, in turn, implies explicit formulas for the
period functions appearing in Bettin--Conrey's work. We study generalizations
of Bettin--Conrey sums involving zeta derivatives and multiple cotangent
factors and relate these to special values of the Estermann zeta function.
