When you're drawing by hand, usually only the asymptotes are required; the effects of zeros, poles, higher ordered zeros and poles, and most of the simple cases are usually tabulated in the first chapter of any controls textbook.
Yes, there's one zero. I'm not sure how far along you are in your complex math or controls course, but s is the 'generic' frequency [tex]s=\sigma+j\omega[/tex]. If you're finding the frequency response (as you are in a Bode plot) you're only concerned with the situation where [tex]\sigma[/tex] is zero and frequency is imaginary (i.e. sinusoidal). That's why some of the previously-linked resources use [tex]j\omega[/tex] instead of s (even though the transfer function is given in terms of s)
Put another way, if your transfer function is called H(s), you're finding |H(jw)| (the magnitude of the frequency response) and the phase