CURLcode curl_easy_setopt(CURL *handle, CURLOPT_TIMEVALUE_LARGE, curl_off_t val);
The difference between this option and CURLOPT_TIMEVALUE(3) is the type of the argument. On systems where 'long' is only 32 bit wide, this option has to be used to set dates beyond the year 2038.
CURL *curl = curl_easy_init();
if(curl) {
  curl_easy_setopt(curl, CURLOPT_URL, "https://example.com");
  /* January 1, 2020 is 1577833200 */
  curl_easy_setopt(curl, CURLOPT_TIMEVALUE_LARGE, (curl_off_t)1577833200);
  /* If-Modified-Since the above time stamp */
  curl_easy_setopt(curl, CURLOPT_TIMECONDITION, CURL_TIMECOND_IFMODSINCE);
  /* Perform the request */
  curl_easy_perform(curl);
}