Repository: sympy/sympy

Issue:
zeta(0, n) where n is negative is wrong
```Python
>>> n=sympy.Symbol("n", negative=True)
>>> sympy.zeta(0, n)
-n - 1/2
```
This should be `-n+1/2`

Discussion:

Patch:
diff --git a/sympy/functions/special/zeta_functions.py b/sympy/functions/special/zeta_functions.py
index 561c3e0cab..d5240777eb 100644
--- a/sympy/functions/special/zeta_functions.py
+++ b/sympy/functions/special/zeta_functions.py
@@ -433,10 +433,7 @@ def eval(cls, z, a_=None):
             elif z is S.Infinity:
                 return S.One
             elif z is S.Zero:
-                if a.is_negative:
-                    return S.Half - a - 1
-                else:
-                    return S.Half - a
+                return S.Half - a
             elif z is S.One:
                 return S.ComplexInfinity
             elif z.is_Integer:


Test patch:
diff --git a/sympy/functions/special/tests/test_zeta_functions.py b/sympy/functions/special/tests/test_zeta_functions.py
index de2007fe16..6f246db8e2 100644
--- a/sympy/functions/special/tests/test_zeta_functions.py
+++ b/sympy/functions/special/tests/test_zeta_functions.py
@@ -6,6 +6,7 @@
 
 x = Symbol('x')
 a = Symbol('a')
+b = Symbol('b', negative=True)
 z = Symbol('z')
 s = Symbol('s')
 
@@ -17,6 +18,7 @@ def test_zeta_eval():
 
     assert zeta(0) == Rational(-1, 2)
     assert zeta(0, x) == Rational(1, 2) - x
+    assert zeta(0, b) == Rational(1, 2) - b
 
     assert zeta(1) == zoo
     assert zeta(1, 2) == zoo
@@ -52,10 +54,10 @@ def test_zeta_eval():
     assert zeta(-4, -8) == 8772
 
     assert zeta(0, 1) == -Rational(1, 2)
-    assert zeta(0, -1) == Rational(1, 2)
+    assert zeta(0, -1) == Rational(3, 2)
 
     assert zeta(0, 2) == -Rational(3, 2)
-    assert zeta(0, -2) == Rational(3, 2)
+    assert zeta(0, -2) == Rational(5, 2)
 
     assert zeta(
         3).evalf(20).epsilon_eq(Float("1.2020569031595942854", 20), 1e-19)