diff --git a/pageserver/src/http/openapi_spec.yml b/pageserver/src/http/openapi_spec.yml
index 566086c527..7ea148971f 100644
--- a/pageserver/src/http/openapi_spec.yml
+++ b/pageserver/src/http/openapi_spec.yml
@@ -212,6 +212,12 @@ paths:
               schema:
                 type: string
                 format: date-time
+        "412":
+          description: No timestamp is found for given LSN, e.g. if there had been no commits till LSN
+          content:
+            application/json:
+              schema:
+                $ref: "#/components/schemas/PreconditionFailedError"
 
   /v1/tenant/{tenant_id}/timeline/{timeline_id}/get_lsn_by_timestamp:
     parameters:
diff --git a/pageserver/src/http/routes.rs b/pageserver/src/http/routes.rs
index bce590016e..e979a70aec 100644
--- a/pageserver/src/http/routes.rs
+++ b/pageserver/src/http/routes.rs
@@ -989,7 +989,7 @@ async fn get_lsn_by_timestamp_handler(
     if !tenant_shard_id.is_shard_zero() {
         // Requires SLRU contents, which are only stored on shard zero
         return Err(ApiError::BadRequest(anyhow!(
-            "Size calculations are only available on shard zero"
+            "Lsn calculations by timestamp are only available on shard zero"
         )));
     }
 
@@ -1064,7 +1064,7 @@ async fn get_timestamp_of_lsn_handler(
     if !tenant_shard_id.is_shard_zero() {
         // Requires SLRU contents, which are only stored on shard zero
         return Err(ApiError::BadRequest(anyhow!(
-            "Size calculations are only available on shard zero"
+            "Timestamp calculations by lsn are only available on shard zero"
         )));
     }
 
@@ -1090,8 +1090,8 @@ async fn get_timestamp_of_lsn_handler(
             .to_string();
             json_response(StatusCode::OK, time)
         }
-        None => Err(ApiError::NotFound(
-            anyhow::anyhow!("Timestamp for lsn {} not found", lsn).into(),
+        None => Err(ApiError::PreconditionFailed(
+            format!("Timestamp for lsn {} not found", lsn).into(),
         )),
     }
 }
diff --git a/pageserver/src/pgdatadir_mapping.rs b/pageserver/src/pgdatadir_mapping.rs
index e3e06ab91a..4c5a07ba57 100644
--- a/pageserver/src/pgdatadir_mapping.rs
+++ b/pageserver/src/pgdatadir_mapping.rs
@@ -691,7 +691,7 @@ impl Timeline {
         Ok(buf.get_u32_le())
     }
 
-    /// Get size of an SLRU segment
+    /// Does the slru segment exist?
     pub(crate) async fn get_slru_segment_exists(
         &self,
         kind: SlruKind,
@@ -844,9 +844,9 @@ impl Timeline {
         .await
     }
 
-    /// Obtain the possible timestamp range for the given lsn.
+    /// Obtain the timestamp for the given lsn.
     ///
-    /// If the lsn has no timestamps, returns None. returns `(min, max, median)` if it has timestamps.
+    /// If the lsn has no timestamps (e.g. no commits), returns None.
     pub(crate) async fn get_timestamp_for_lsn(
         &self,
         probe_lsn: Lsn,
diff --git a/test_runner/regress/test_lsn_mapping.py b/test_runner/regress/test_lsn_mapping.py
index 7280a91a12..c5a1bf0d16 100644
--- a/test_runner/regress/test_lsn_mapping.py
+++ b/test_runner/regress/test_lsn_mapping.py
@@ -276,3 +276,34 @@ def test_ts_of_lsn_api(neon_env_builder: NeonEnvBuilder):
             if i > 1:
                 before_timestamp = tbl[i - step_size][1]
                 assert timestamp >= before_timestamp, "before_timestamp before timestamp"
+
+
+def test_timestamp_of_lsn_empty_branch(neon_env_builder: NeonEnvBuilder):
+    """
+    Test that getting the timestamp of the head LSN of a newly created branch works.
+    This verifies that we don't get a 404 error when trying to get the timestamp
+    of the head LSN of a branch that was just created.
+    We now return a special status code 412 to indicate if there is no timestamp found for lsn.
+
+    Reproducer for https://github.com/neondatabase/neon/issues/11439
+    """
+    env = neon_env_builder.init_start()
+
+    # Create a new branch
+    new_timeline_id = env.create_branch("test_timestamp_of_lsn_empty_branch")
+
+    # Retrieve the commit LSN of the empty branch, which we have never run postgres on
+    detail = env.pageserver.http_client().timeline_detail(
+        tenant_id=env.initial_tenant, timeline_id=new_timeline_id
+    )
+    head_lsn = detail["last_record_lsn"]
+
+    # Verify that we get 412 status code
+    with env.pageserver.http_client() as client:
+        with pytest.raises(PageserverApiException) as err:
+            client.timeline_get_timestamp_of_lsn(
+                env.initial_tenant,
+                new_timeline_id,
+                head_lsn,
+            )
+        assert err.value.status_code == 412