  We present results from the AzTEC/ASTE 1.1-mm imaging survey of 35 Ly-alpha
blobs (LABs) found in the SSA22 protocluster at z = 3.1. These 1.1-mm data
reach an r.m.s. noise level of 0.7-1 mJy/beam, making this the largest
millimetre-wave survey of LABs to date. No significant (> 3.5-sigma) emission
is found in any of individual 35 LABs, and from this, we estimate 3-sigma upper
limits on the far-infrared luminosity of L_FIR < 2e+12 Lsun. Stacking analysis
reveals that the 1.1-mm flux density averaged over the LABs is S(1.1mm) < 0.40
mJy (3-sigma), which places a constraint of L_FIR < 4.5e+11 Lsun. This
indicates that earlier 850-um measurements of the LABs may have overestimated
their flux densities. Our results suggest that LABs on average have little
ultra-luminous obscured star-formation, in contrast to a long-believed picture
that LABs undergo an intense episode of dusty star-formation activities with
star-formation rates of ~ 10^3 Msun/yr. Observations with ALMA are needed to
directly study the obscured part of star-formation activity in the LABs.
