diff -r c6fd0f3aadc4 Lib/test/test_long.py
--- a/Lib/test/test_long.py	Mon Aug 29 15:59:48 2016 +0300
+++ b/Lib/test/test_long.py	Mon Aug 29 15:39:24 2016 +0100
@@ -878,6 +878,21 @@
             self.check_truediv(-x, y)
             self.check_truediv(-x, -y)
 
+    def test_lshift_of_zero(self):
+        self.assertEqual(0 << 0, 0)
+        self.assertEqual(0 << 10, 0)
+        with self.assertRaises(ValueError):
+            0 << -1
+
+    @support.cpython_only
+    def test_huge_lshift_of_zero(self):
+        # Shouldn't try to allocate memory for a huge shift. See issue #27870.
+        # Other implementations may have a different boundary for overflow,
+        # or not raise at all.
+        self.assertEqual(0 << sys.maxsize, 0)
+        with self.assertRaises(OverflowError):
+            0 << (sys.maxsize + 1)
+
     def test_small_ints(self):
         for i in range(-5, 257):
             self.assertIs(i, i + 0)
diff -r c6fd0f3aadc4 Objects/longobject.c
--- a/Objects/longobject.c	Mon Aug 29 15:59:48 2016 +0300
+++ b/Objects/longobject.c	Mon Aug 29 15:39:24 2016 +0100
@@ -4276,6 +4276,11 @@
         PyErr_SetString(PyExc_ValueError, "negative shift count");
         return NULL;
     }
+
+    if (Py_SIZE(a) == 0) {
+        return PyLong_FromLong(0);
+    }
+
     /* wordshift, remshift = divmod(shiftby, PyLong_SHIFT) */
     wordshift = shiftby / PyLong_SHIFT;
     remshift  = shiftby - wordshift * PyLong_SHIFT;