In [2]: polylog(s, z).diff(z)
Out[2]: 
polylog(s - 1, z)
─────────────────
        z        

In [3]: polylog(s, 1)
Out[3]: ζ(s)

In [4]: polylog(1.5, 2.5)
Out[4]: 1.18400018990854 - 3.39329470091135⋅ⅈ

In [5]: expand_func(polylog(1, z))
Out[5]: -log(-z + 1)

In [6]: expand_func(polylog(0, z))
Out[6]: 
  z   
──────
-z + 1

In [7]: expand_func(polylog(-3, z))
Out[7]: 
      4           3           2           
   6⋅z        12⋅z         7⋅z        z   
───────── + ───────── + ───────── + ──────
        4           3           2   -z + 1
(-z + 1)    (-z + 1)    (-z + 1)          

In [8]: lerchphi(z, s, a).diff(z)
Out[8]: 
-a⋅lerchphi(z, s, a) + lerchphi(z, s - 1, a)
────────────────────────────────────────────
                     z                      

In [9]: lerchphi(z, s, a).diff(a)
Out[9]: -s⋅lerchphi(z, s + 1, a)

In [10]: lerchphi(1.5, 2.5, 3.5)
Out[10]: 0.0631609107374596 - 0.14761296283615⋅ⅈ

In [11]: lerchphi(1, s, a)
Out[11]: lerchphi(1, s, a)

In [12]: expand_func(lerchphi(1, s, a))
Out[12]: ζ(s, a)

In [13]: expand_func(lerchphi(z, s, 1))
Out[13]: 
polylog(s, z)
─────────────
      z      

In [14]: expand_func(lerchphi(z, s, S(7)/3))
Out[14]: 
              ⎛                     -2⋅ⅈ⋅π        ⎛          2⋅ⅈ⋅π⎞    -4⋅ⅈ⋅π        ⎛          4⋅ⅈ⋅π⎞⎞        
              ⎜                     ──────        ⎜          ─────⎟    ──────        ⎜          ─────⎟⎟        
              ⎜       ⎛   3 ⎽⎽⎽⎞      3           ⎜   3 ⎽⎽⎽    3  ⎟      3           ⎜   3 ⎽⎽⎽    3  ⎟⎟        
        s - 1 ⎜polylog⎝s, ╲╱ z ⎠   ℯ      ⋅polylog⎝s, ╲╱ z ⋅ℯ     ⎠   ℯ      ⋅polylog⎝s, ╲╱ z ⋅ℯ     ⎠⎟        
       3     ⋅⎜───────────────── + ──────────────────────────────── + ────────────────────────────────⎟        
   s          ⎜      3 ⎽⎽⎽                      3 ⎽⎽⎽                              3 ⎽⎽⎽              ⎟      -s
  3           ⎝      ╲╱ z                       ╲╱ z                               ╲╱ z               ⎠   4/3  
- ── + ──────────────────────────────────────────────────────────────────────────────────────────────── - ─────
   2                                                   2                                                    z  
  z                                                   z