No. For example, OpenSSL let’s the user set this when they create certificates.
See: http://www.openssl.org/docs/apps/x509.html
-set_serial n specifies the serial number to use. This option can be used with either the -signkey or -CA options. If used in conjunction
with the -CA option the serial number file (as specified by the
-CAserial or -CAcreateserial options) is not used.The serial number can be decimal or hex (if preceded by 0x). Negative
serial numbers can also be specified but their use is not recommended.