In the future, I'd recommend also using "fdisk -lu" to list the partition table and printing the results. The first partition will start at an offset. Some other partitions may not start on a partition boundary as well sometimes. Listing the partition table using 512 byte sectors will eliminate rounding errors.
If the partitions were created in Linux, the first one probably starts on sector 63. I'd recommend using losetup to attach a loop device at this offset and seeing if the filesystem is still OK.
Code:
sudo /sbin/losetup -fs /dev/sda -o $((63*512))
# will respond with which loop device was used. If your losetup doesn't have the -s option, you can use "sudo /sbin/losetup -a" to list the used loop devices.
sudo file -s /dev/loop0
# loop0 is used as an example. It could be loop1.
# if file returns a filesystem, you can attempt to mount it. The /boot partition is often ext2 or ext3.
sudo mount /dev/loop0 -t <filesystem> /boot
sudo df --block-size=512 /boot
I'd recommend sticking with 512 byte sectors and block sizes on all your utilities during your recovery. It keeps things consistant and accurate. The df will give you an indication of the size of the filesystem, which you to use to learn what the start of the second is. This assumes the filesystems weren't wrecked already of course. Next, add the size in sectors of /boot to 63 to obtain what may be the offset of the next partition.
Also consider getting a large external drive to create a backup image (using dd) of your drive. This image backup can be used to reverse unsuccessful attempts, or you might use it to try to recover the partitions on this copy instead of the original. (using losetup with a file instead of device)
PS. I just noticed that third partition was empty. If it wasn't formated, then you can recover it, mount it to learn its size to find the start sector of the third partition. Otherwise if the second or third partitions did start on cylinder boundaries, and the cylinder size indicated by fdisk is the same, you can try perhaps look at multiples of 16064 sectors for the start of the third partition.
