@SteveLinton's answer explains the situation nicely. I'll just add the complete formulas and the radius of the Hill spheres.
To obtain the distance to L1, find the smallest value of $r$ such that
$$\frac{M_2}{r^2} + \frac{M_1}{R^2} - \frac{r(M_1+M_2)}{R^3} - \frac{M_1}{(R-r)^2} = 0.$$
To obtain the distance to L2, find the smallest value of $r$ such that
$$\frac{M_1}{R^2} + \frac{r(M_1+M_2)}{R^3} - \frac{M_1}{(R+r)^2} - \frac{M_2}{r^2} = 0.$$
Even though Mars is 50% farther from the Sun than the Earth, it's mass is only 11% that of Earth's so while the distances to Earth's Lagrange point are about 1% of that to the Sun for Earth, those of Mars are only about 0.5% of the distance to the Sun for Mars.
In either case, a diagram would show two dots very close to each planet. The diagrams on the internet usually exaggerate this greatly to make it easier to see.
The values for the distance from the planets to the Sun and to their Sun-associated L1 and L2 points look like this.
a_Earth: 149598023 km
Sun-Earth L1: 1491524 km
Sun-Earth L2: 1501504 km
Earth r_Hill: 1496531 km
a_Mars: 227939200 km
Sun-Mars L1: 1082311 km
Sun-Mars L2: 1085748 km
Mars r_Hill: 1084032 km

The Python script based on scipy.optimize's Brentq:
def solve_L1 (r, R, M1, M2):
return M2/r**2 + M1/R**2 - r*(M1 + M2)/R**3 - M1/(R-r)**2
def solve_L2 (r, R, M1, M2):
return M1/R**2 + r*(M1 + M2)/R**3 - M1/(R+r)**2 - M2/r**2
def r_Hill(R, M1, M2):
return R * (M2 / (3.*M1))**(1./3.)
import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import brentq
a_Earth = 149598023. # Earth's semi-major axis (km)
a_Mars = 227939200. # Mars' semi-major axis (km)
r_low = 1000000. # 1.0 million km (lower guess)
r_high = 1600000. # 1.6 million km (upper guess)
M_Sun = 1.9886E+30 # approximate mass (kg)
M_Earth = 5.9724E+24 # approximate mass (kg)
M_Mars = 6.4171E+23 # approximate mass (kg)
r_Hill_Earth = r_Hill(a_Earth, M_Sun, M_Earth)
r_Hill_Mars = r_Hill(a_Mars, M_Sun, M_Mars)
r = np.linspace(r_low, r_high)
if True:
plt.figure()
plt.plot(r, solve_L1(r, a_Earth, M_Sun, M_Earth), '-g')
plt.plot(r, solve_L1(r, a_Mars, M_Sun, M_Mars), '-r')
plt.plot(r, solve_L2(r, a_Earth, M_Sun, M_Earth), '--g')
plt.plot(r, solve_L2(r, a_Mars, M_Sun, M_Mars), '--r')
plt.plot([r_Hill_Earth], [0], 'ok')
plt.plot([r_Hill_Mars ], [0], 'ok')
plt.text(1040000, 1.1E+11, 'L1 Mars L2', fontsize=14)
plt.text(1450000, 3.0E+11, 'L1 Earth L2', fontsize=14)
plt.plot(r, np.zeros_like(r), '-k')
plt.ylim(-4E+11, 4E+11)
plt.show()
# for Mars:
r_L1_Mars = brentq(solve_L1, r_low, r_high, args=(a_Mars, M_Sun, M_Mars))
r_L2_Mars = brentq(solve_L2, r_low, r_high, args=(a_Mars, M_Sun, M_Mars))
# for Earth:
r_L1_Earth = brentq(solve_L1, r_low, r_high, args=(a_Earth, M_Sun, M_Earth))
r_L2_Earth = brentq(solve_L2, r_low, r_high, args=(a_Earth, M_Sun, M_Earth))
print "a_Earth: ", int(a_Earth), " km"
print "Sun-Earth L1: ", int(r_L1_Earth), " km"
print "Sun-Earth L2: ", int(r_L2_Earth), " km"
print "Earth r_Hill: ", int(r_Hill_Earth), " km"
print ''
print "a_Mars: ", int(a_Mars), " km"
print "Sun-Mars L1: ", int(r_L1_Mars), " km"
print "Sun-Mars L2: ", int(r_L2_Mars), " km"
print "Mars r_Hill: ", int(r_Hill_Mars), " km"