diff --git a/Lib/test/test_hmac.py b/Lib/test/test_hmac.py
index 1bbf201..ea00367 100644
--- a/Lib/test/test_hmac.py
+++ b/Lib/test/test_hmac.py
@@ -312,10 +312,15 @@
                 self.fail('Expected warning about small block_size')
 
     def test_with_digestmod_no_default(self):
-        with self.assertRaises(ValueError):
+        """The digestmod parameter is required as of Python 3.8."""
+        with self.assertRaisesRegex(TypeError, r'required.*digestmod'):
             key = b"\x0b" * 16
             data = b"Hi There"
             hmac.HMAC(key, data, digestmod=None)
+        with self.assertRaisesRegex(TypeError, r'required.*digestmod'):
+            hmac.new(key, data)
+        with self.assertRaisesRegex(TypeError, r'required.*digestmod'):
+            hmac.HMAC(key, msg=data, digestmod='')
 
 
 class ConstructorTestCase(unittest.TestCase):