diff --git a/Lib/test/test_long.py b/Lib/test/test_long.py
index f0dd0749f77..4d293f24e02 100644
--- a/Lib/test/test_long.py
+++ b/Lib/test/test_long.py
@@ -878,6 +878,21 @@ class LongTest(unittest.TestCase):
             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 --git a/Misc/NEWS b/Misc/NEWS
index 9873abbab8e..5ce3c2c9437 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -10,6 +10,9 @@ What's New in Python 3.6.0 beta 1
 Core and Builtins
 -----------------
 
+- Issue #27870: A left shift of zero by a large integer no longer attempts
+  to allocate large amounts of memory.
+
 - Issue #25402: In int-to-decimal-string conversion, improve the estimate
   of the intermediate memory required, and remove an unnecessarily strict
   overflow check. Patch by Serhiy Storchaka.
diff --git a/Objects/longobject.c b/Objects/longobject.c
index ba235995359..9d6474c6bde 100644
--- a/Objects/longobject.c
+++ b/Objects/longobject.c
@@ -4281,6 +4281,11 @@ long_lshift(PyObject *v, PyObject *w)
         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;