strange non-linear 7x7 equation

Hi can somebody give me some hint about what numerical method ,
I have to use to solve the system of equations below?

p0,p1,p2,p3,p4 E (0.0-1.0)
s>0
alpha>0

eq1:=
1/30720*(
-968192*Gamma[9/2+s]+201600*alpha*Gamma[s+1]*p0+504000*alpha*Gamma[s+1]*p1+
680400*alpha*Gamma[s+1]*p2+819000*alpha*Gamma[s+1]*p3+937125*alpha*Gamma[s+1]*p4+
1894080*alpha*Gamma[s+1]*p4*s+675840*alpha*Gamma[s+1]*p0*s+245760*alpha*Gamma[s+1]*p0*s^3+
291840*alpha*Gamma[s+1]*p2*s^3+991680*alpha*Gamma[s+1]*p2*s^2+1084800*alpha*Gamma[s+1]*p1*s+
660480*alpha*Gamma[s+1]*p0*s^2+30720*alpha*Gamma[s+1]*p3*s^4+314880*alpha*Gamma[s+1]*p3*s^3+
1668720*alpha*Gamma[s+1]*p3*s+30720*alpha*Gamma[s+1]*p0*s^4+30720*alpha*Gamma[s+1]*p4*s^4+
833280*alpha*Gamma[s+1]*p1*s^2+1135680*alpha*Gamma[s+1]*p3*s^2+30720*alpha*Gamma[s+1]*p2*s^4+
1265280*alpha*Gamma[s+1]*p4*s^2+268800*alpha*Gamma[s+1]*p1*s^3+30720*alpha*Gamma[s+1]*p1*s^4+
1407360*alpha*Gamma[s+1]*p2*s+337920*alpha*Gamma[s+1]*p4*s^3)/Gamma[9/2+s]==0


eq2:=
1/2*alpha^2*p0+alpha^2*p2*s+9/2*alpha^2*p2+
13/2*alpha^2*p3+5/2*alpha^2*p1+alpha^2*p3*s+
alpha^2*p4*s+17/2*alpha^2*p4+alpha^2*p0*s+
alpha^2*p1*s-66497/40==0

eq3:=
1/30720*(
-3414027392*Gamma[9/2+s]+10639125*alpha^3*Gamma[2+s]*p4+1713600*alpha^3*Gamma[2+s]*p1+
4107600*alpha^3*Gamma[2+s]*p2+7119000*alpha^3*Gamma[2+s]*p3+245760*alpha^3*Gamma[2+s]*p0*s^3+
476160*alpha^3*Gamma[2+s]*p2*s^3+8549040*alpha^3*Gamma[2+s]*p3*s+5370240*alpha^3*Gamma[2+s]*p2*s+
591360*alpha^3*Gamma[2+s]*p3*s^3+30720*alpha^3*Gamma[2+s]*p3*s^4+2489280*alpha^3*Gamma[2+s]*p2*s^2+
2720640*alpha^3*Gamma[2+s]*p1*s+1524480*alpha^3*Gamma[2+s]*p1*s^2+360960*alpha^3*Gamma[2+s]*p1*s^3+
30720*alpha^3*Gamma[2+s]*p0*s^4+675840*alpha^3*Gamma[2+s]*p0*s+30720*alpha^3*Gamma[2+s]*p2*s^4+
660480*alpha^3*Gamma[2+s]*p0*s^2+30720*alpha^3*Gamma[2+s]*p4*s^4+30720*alpha^3*Gamma[2+s]*p1*s^4+
3554880*alpha^3*Gamma[2+s]*p3*s^2+4721280*alpha^3*Gamma[2+s]*p4*s^2+12181440*alpha^3*Gamma[2+s]*p4*s+
201600*alpha^3*Gamma[2+s]*p0+706560*alpha^3*Gamma[2+s]*p4*s^3)/Gamma[9/2+s]==0

eq4:=
26*alpha^4*p4*s+39/4*alpha^4*p1+3/4*alpha^4*p0+255/4*alpha^4*p3+alpha^4*p4*s^2+2*alpha^4*p0*s+
435/4*alpha^4*p4+alpha^4*p2*s^2+14*alpha^4*p2*s+20*alpha^4*p3*s+alpha^4*p0*s^2+123/4*alpha^4*p2+
alpha^4*p1*s^2+8*alpha^4*p1*s+alpha^4*p3*s^2-1321865381/160==0

eq5:=
1/30720*(
15195600*alpha^5*Gamma[3+s]*p2+3729600*alpha^5*Gamma[3+s]*p1-19942746397472*Gamma[9/2+s]+
201600*alpha^5*Gamma[3+s]*p0+72908325*alpha^5*Gamma[3+s]*p4+37510200*alpha^5*Gamma[3+s]*p3+
5447040*alpha^5*Gamma[3+s]*p1*s+2676480*alpha^5*Gamma[3+s]*p1*s^2+675840*alpha^5*Gamma[3+s]*p0*s+
783360*alpha^5*Gamma[3+s]*p2*s^3+15661440*alpha^5*Gamma[3+s]*p2*s+245760*alpha^5*Gamma[3+s]*p0*s^3+
660480*alpha^5*Gamma[3+s]*p0*s^2+30720*alpha^5*Gamma[3+s]*p3*s^4+30720*alpha^5*Gamma[3+s]*p0*s^4+
14167680*alpha^5*Gamma[3+s]*p4*s^2+30720*alpha^5*Gamma[3+s]*p4*s^4+1320960*alpha^5*Gamma[3+s]*p4*s^3+
514560*alpha^5*Gamma[3+s]*p1*s^3+1052160*alpha^5*Gamma[3+s]*p3*s^3+5599680*alpha^5*Gamma[3+s]*p2*s^2+
30720*alpha^5*Gamma[3+s]*p1*s^4+55746240*alpha^5*Gamma[3+s]*p4*s+9430080*alpha^5*Gamma[3+s]*p3*s^2+
32150640*alpha^5*Gamma[3+s]*p3*s+30720*alpha^5*Gamma[3+s]*p2*s^4
)/Gamma[9/2+s]==0

eq6:=
1935/8*alpha^6*p2+alpha^6*p0*s^3+23/4*alpha^6*p0*s+alpha^6*p1*s^3+33/2*alpha^6*p1*s^2+9/2*alpha^6*p0*s^2+
81/2*alpha^6*p3*s^2+215/4*alpha^6*p1*s+15/8*alpha^6*p0+alpha^6*p3*s^3+57/2*alpha^6*p2*s^2+5655/8*alpha^6*p3+
12495/8*alpha^6*p4+alpha^6*p2*s^3+alpha^6*p4*s^3+1319/4*alpha^6*p3*s+647/4*alpha^6*p2*s+375/8*alpha^6*p1+
2231/4*alpha^6*p4*s+105/2*alpha^6*p4*s^2-33808341042557/640==0

eq7:=
1/30720*(
-135419269305231752*Gamma[9/2+s]+11244480*alpha^7*Gamma[4+s]*p2*s^2+37810560*alpha^7*Gamma[4+s]*p2*s+
729600*alpha^7*Gamma[4+s]*p1*s^3+30720*alpha^7*Gamma[4+s]*p1*s^4+35134080*alpha^7*Gamma[4+s]*p4*s^2+
30720*alpha^7*Gamma[4+s]*p3*s^4+144408600*alpha^7*Gamma[4+s]*p3+361725525*alpha^7*Gamma[4+s]*p4+
1697280*alpha^7*Gamma[4+s]*p3*s^3+660480*alpha^7*Gamma[4+s]*p0*s^2+2181120*alpha^7*Gamma[4+s]*p4*s^3+
198022080*alpha^7*Gamma[4+s]*p4*s+675840*alpha^7*Gamma[4+s]*p0*s+21526080*alpha^7*Gamma[4+s]*p3*s^2+
245760*alpha^7*Gamma[4+s]*p0*s^3+9264000*alpha^7*Gamma[4+s]*p1*s+4289280*alpha^7*Gamma[4+s]*p1*s^2+
30720*alpha^7*Gamma[4+s]*p0*s^4+6552000*alpha^7*Gamma[4+s]*p1+42008400*alpha^7*Gamma[4+s]*p2+
1213440*alpha^7*Gamma[4+s]*p2*s^3+30720*alpha^7*Gamma[4+s]*p2*s^4+97126320*alpha^7*Gamma[4+s]*p3*s+
201600*alpha^7*Gamma[4+s]*p0+30720*alpha^7*Gamma[4+s]*p4*s^4
)/Gamma[9/2+s]==0


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