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Ridges.AI

 """ Riemann zeta and related function. """
 from __future__ import print_function, division
+from sympy.core import Function, S, sympify, pi
+from sympy.core import Function, S, sympify, pi, Rational
 from sympy.core.function import ArgumentIndexError
 from sympy.core.compatibility import range
 from sympy.functions.combinatorial.numbers import bernoulli, factorial, harmonic
             return -dirichlet_eta(s)
         elif z == 0:
             return 0
+        elif s == 2 and z == Rational(1, 2):
+            return -log(2)**2/2 + pi**2/12
     def fdiff(self, argindex=1):
         s, z = self.args
         return z*lerchphi(z, s, 1)
     def _eval_expand_func(self, **hints):
+        from sympy import log, expand_mul, Dummy, exp_polar, I
+        from sympy import log, expand_mul, Dummy, exp_polar, I, pi, Rational
         s, z = self.args
         if s == 1:
+            return -log(1 + exp_polar(-I*pi)*z)
+            return -log(1 - z)
+        if s == 2 and z == Rational(1, 2):
+            return -log(2)**2/2 + pi**2/12
         if s.is_Integer and s <= 0:
             u = Dummy('u')
             start = u/(1 - u)