a0 = 1

l_0 = 1
l_1 = 7
l_2 = 5*7*13/2
l_3 = 5*7*11*13*19/2
l_4 = 5*7*11*13*17*19*25/8
l_5 = 5*7*11*13*17*19*23*5*31/8
l_6 = 5*7*11*13*17*19*23*5*29*31*37/16

13 .. 35

t = 1.0/sqrt(1-z*z)
a1 = 1.0         * u_2(t)
   + (7/3) * z32 * u_1(t) / 16
   + (5*7*13/9) * z62 * u_0(t) / 256
 
a2 =           1.0                  * u_4(t)
   + (        7/3) * (zeta^-( 3/2)) * u_3(t) / 16
   + (      455/9) * (zeta^-( 6/2)) * u_2(t) / 512
   + (   95095/27) * (zeta^-( 9/2)) * u_1(t) / 8192
   + (40415375/81) * (zeta^-(12/2)) * u_0(t) / 524288

a2 =        1/243)       * u_4(t) / 32768
   + (      7/243) * z32 * u_3(t) / 16384
   + (     455/81) * z62 * u_2(t) / 65536
   + (   95095/81) * z92 * u_1(t) / 65536
   + (40415375/81) * z122 / 524288

/* should the last one there be 1/2187???? */ 
a3 = u_6(t)
   + (7/3)             * z32 * u_5(t) / 16
   + (5*7*13/9)        * z62 * u_4(t) / 512
   + (5*7*11*13*19/27) * z92 * u_3(t) / 8192 
   + (5*7*11*13*17*19*25/81) * z122 * u_2(t) / 524288
   + (5*7*11*13*17*19*23*5*31/243) * z152 * u_1(t) / 8388608
   + (5*7*11*13*17*19*23*5*29*31*37/2187) * z182 * u_0(t) / 3*268435456 /* 2^28... */