Boom

Finished

             return -dirichlet_eta(s)
         elif z == 0:
             return 0
+        # Special case: polylog(2, 1/2) = -log(2)**2/2 + pi**2/12
+        elif s == 2 and z == S(1)/2:
+            return -log(2)**2/2 + pi**2/12
     def fdiff(self, argindex=1):
         s, z = self.args
         from sympy import log, expand_mul, Dummy, exp_polar, I
         s, z = self.args
         if s == 1:
+            return -log(1 + exp_polar(-I*pi)*z)
+            return -log(1 - z)
         if s.is_Integer and s <= 0:
             u = Dummy('u')
             start = u/(1 - u)