Buggy python submission (verdict: Time Limit Exceeded):
import sys

sys.setrecursionlimit(10 ** 5)
S = input()
Ls = [1, 10, 9, 12, 3, 4]  # 10^n % 13, n % 6でループ
C = 10 ** 9 + 7

L = len(S) - 1
mem = [[-1 for _ in range(13)] for _ in range(len(S))]


def calc(i, m):  # i桁目以下で余りがmとなるようにする
    if mem[i][m] >= 0:
        return mem[i][m]
    t = S[L - i]
    if t == '?':
        if i == 0:
            if m < 10:
                mem[i][m] = 1
                return 1
            else:
                mem[i][m] = 0
                return 0
        else:
            s = 0
            for j in range(10):
                tm = m - ((Ls[i % 6] * j) % 13)
                if tm < 0:
                    tm += 13
                s += calc(i - 1, tm)
            s %= C
            mem[i][m] = s
            return s
    else:
        tn = int(t)
        if i == 0:
            if m == tn:
                mem[i][m] = 1
                return 1
            else:
                mem[i][m] = 0
                return 0
        else:
            tm = m - ((Ls[i % 6] * tn) % 13)
            if tm < 0:
                tm += 13
            k = calc(i - 1, tm) % C
            mem[i][m] = k
            return k


print(calc(len(S) - 1, 5))

Fixed submission:
S = input()
Ls = [1, 10, 9, 12, 3, 4]  # 10^n % 13, n % 6でループ
C = 10 ** 9 + 7

L = len(S) - 1
mem = [[0 for _ in range(13)] for _ in range(len(S))]
if S[L] == '?':
    for m in range(10):
        mem[0][m] = 1
else:
    mem[0][int(S[L])] = 1

for i in range(1, len(S)):
    t = S[L - i]
    for m in range(13):
        if t == '?':
            s = 0
            for j in range(10):
                tm = m - ((Ls[i % 6] * j) % 13)
                if tm < 0:
                    tm += 13
                s += mem[i - 1][tm]
            mem[i][m] = s % C
        else:
            tm = m - ((Ls[i % 6] * int(t)) % 13)
            if tm < 0:
                tm += 13
            mem[i][m] = mem[i - 1][tm] % C
print(mem[len(S) - 1][5])

Buggy python submission (verdict: Time Limit Exceeded):
S = input()
Ls = [1, 10, 9, 12, 3, 4]  # 10^n % 13, n % 6でループ

L = len(S) - 1
mem = [[0 for _ in range(13)] for _ in range(len(S))]
if S[L] == '?':
    for m in range(10):
        mem[0][m] = 1
else:
    mem[0][int(S[L])] = 1

for i in range(1, len(S)):
    t = S[L - i]
    for m in range(13):
        if t == '?':
            s = 0
            for j in range(10):
                tm = m - ((Ls[i % 6] * j) % 13)
                if tm < 0:
                    tm += 13
                s += mem[i - 1][tm]
            mem[i][m] = s
        else:
            tm = m - ((Ls[i % 6] * int(t)) % 13)
            if tm < 0:
                tm += 13
            mem[i][m] = mem[i - 1][tm]
print(mem[len(S) - 1][5] % (10 ** 9 + 7))

Fixed submission:
S = input()
Ls = [1, 10, 9, 12, 3, 4]  # 10^n % 13, n % 6でループ
C = 10 ** 9 + 7

L = len(S) - 1
mem = [[0 for _ in range(13)] for _ in range(len(S))]
if S[L] == '?':
    for m in range(10):
        mem[0][m] = 1
else:
    mem[0][int(S[L])] = 1

for i in range(1, len(S)):
    t = S[L - i]
    for m in range(13):
        if t == '?':
            s = 0
            for j in range(10):
                tm = m - ((Ls[i % 6] * j) % 13)
                if tm < 0:
                    tm += 13
                s += mem[i - 1][tm]
            mem[i][m] = s % C
        else:
            tm = m - ((Ls[i % 6] * int(t)) % 13)
            if tm < 0:
                tm += 13
            mem[i][m] = mem[i - 1][tm] % C
print(mem[len(S) - 1][5])

Buggy python submission (verdict: Runtime Error):
sys.setrecursionlimit(1000000)
S = input()
Ls = [1, 10, 9, 12, 3, 4]  # 10^n % 13, n % 6でループ

L = len(S) - 1
mem = [[-1 for _ in range(13)] for _ in range(len(S))]


def calc(i, m):  # i桁目以下で余りがmとなるようにする
    if mem[i][m] >= 0:
        return mem[i][m]
    t = S[L - i]
    if t == '?':
        if i == 0:
            if m < 10:
                mem[i][m] = 1
                return 1
            else:
                mem[i][m] = 0
                return 0
        else:
            s = 0
            for j in range(10):
                tm = m - ((Ls[i % 6] * j) % 13)
                if tm < 0:
                    tm += 13
                s += calc(i - 1, tm)
            mem[i][m] = s
            return s
    else:
        tn = int(t)
        if i == 0:
            if m == tn:
                mem[i][m] = 1
                return 1
            else:
                mem[i][m] = 0
                return 0
        else:
            tm = m - ((Ls[i % 6] * tn) % 13)
            if tm < 0:
                tm += 13
            k = calc(i - 1, tm)
            mem[i][m] = k
            return k


print(calc(len(S) - 1, 5) % (10 ** 9 + 7))

Fixed submission:
S = input()
Ls = [1, 10, 9, 12, 3, 4]  # 10^n % 13, n % 6でループ
C = 10 ** 9 + 7

L = len(S) - 1
mem = [[0 for _ in range(13)] for _ in range(len(S))]
if S[L] == '?':
    for m in range(10):
        mem[0][m] = 1
else:
    mem[0][int(S[L])] = 1

for i in range(1, len(S)):
    t = S[L - i]
    for m in range(13):
        if t == '?':
            s = 0
            for j in range(10):
                tm = m - ((Ls[i % 6] * j) % 13)
                if tm < 0:
                    tm += 13
                s += mem[i - 1][tm]
            mem[i][m] = s % C
        else:
            tm = m - ((Ls[i % 6] * int(t)) % 13)
            if tm < 0:
                tm += 13
            mem[i][m] = mem[i - 1][tm] % C
print(mem[len(S) - 1][5])

Buggy python submission (verdict: Runtime Error):
S = input()
Ls = [1, 10, 9, 12, 3, 4]  # 10^n % 13, n % 6でループ

L = len(S) - 1
mem = [[-1 for _ in range(13)] for _ in range(len(S))]


def calc(i, m):  # i桁目以下で余りがmとなるようにする
    if mem[i][m] >= 0:
        return mem[i][m]
    t = S[L - i]
    if t == '?':
        if i == 0:
            if m < 10:
                mem[i][m] = 1
                return 1
            else:
                mem[i][m] = 0
                return 0
        else:
            s = 0
            for j in range(10):
                tm = m - ((Ls[i % 6] * j) % 13)
                if tm < 0:
                    tm += 13
                s += calc(i - 1, tm)
            mem[i][m] = s
            return s
    else:
        tn = int(t)
        if i == 0:
            if m == tn:
                mem[i][m] = 1
                return 1
            else:
                mem[i][m] = 0
                return 0
        else:
            tm = m - ((Ls[i % 6] * tn) % 13)
            if tm < 0:
                tm += 13
            k = calc(i - 1, tm)
            mem[i][m] = k
            return k


print(calc(len(S) - 1, 5) % (10 ** 9 + 7))

Fixed submission:
S = input()
Ls = [1, 10, 9, 12, 3, 4]  # 10^n % 13, n % 6でループ
C = 10 ** 9 + 7

L = len(S) - 1
mem = [[0 for _ in range(13)] for _ in range(len(S))]
if S[L] == '?':
    for m in range(10):
        mem[0][m] = 1
else:
    mem[0][int(S[L])] = 1

for i in range(1, len(S)):
    t = S[L - i]
    for m in range(13):
        if t == '?':
            s = 0
            for j in range(10):
                tm = m - ((Ls[i % 6] * j) % 13)
                if tm < 0:
                    tm += 13
                s += mem[i - 1][tm]
            mem[i][m] = s % C
        else:
            tm = m - ((Ls[i % 6] * int(t)) % 13)
            if tm < 0:
                tm += 13
            mem[i][m] = mem[i - 1][tm] % C
print(mem[len(S) - 1][5])