that (L) shouldnt be there,id double check.
Assuming it is not an L its likely a 1960 heres the breakdown but theres no way to know for sure until you confirm the serial number.
1960 SCHWINN
10/03 ------------------ K011111 ----------------- K013763
10/04 ------------------ K013764 ----------------- K016536
10/05 ------------------ K016537 ----------------- K019141
10/06 ------------------ K019142 ----------------- K021244
10/07 ------------------ K021245 ----------------- K023420
10/10 ------------------ K023421 ----------------- K026422
10/11 ------------------ K026423 ----------------- K028852
10/12 ------------------ K028853 ----------------- K031796
10/13 ------------------ K031797 ----------------- K034172
10/14 ------------------ K034173 ----------------- K036950
10/17 ------------------ K036951 ----------------- K039877
10/18 ------------------ K039898 ----------------- K042654
10/19 ------------------ K042655 ----------------- K045454
10/20 ------------------ K045455 ----------------- K047434
10/21 ------------------ K047435 ----------------- K050195
10/24 ------------------ K050196 ----------------- K051887
10/25 ------------------ K051998 ----------------- K053798
10/26 ------------------ K053799 ----------------- K055448
10/27 ------------------ K055449 ----------------- K057648
10/28 ------------------ K057649 ----------------- K059023
10/31 ------------------ K059024 ----------------- K06122
if there is one less digit it backs up to 1953 so check on that L,its likely a one or something.
1953 (K0xxxxx) serial numbers
05/05 ------------------ K00001 ----------------- K01100
05/06 to 05/26 ------- K01101 ----------------- K09635
05/26 ------------------ K09636 ----------------- K12483
05/27 ------------------ K12484 ----------------- K13523
Heres a 1960 Starlet brochure to look at