| L(s) = 1 | + 2·2-s + 3-s − 4-s − 25·5-s + 2·6-s − 10·7-s − 82·9-s − 50·10-s − 126·11-s − 12-s + 83·13-s − 20·14-s − 25·15-s + 21·16-s − 164·18-s + 55·19-s + 25·20-s − 10·21-s − 252·22-s + 2·23-s + 375·25-s + 166·26-s − 37·27-s + 10·28-s − 195·29-s − 50·30-s − 97·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.192·3-s − 1/8·4-s − 2.23·5-s + 0.136·6-s − 0.539·7-s − 3.03·9-s − 1.58·10-s − 3.45·11-s − 0.0240·12-s + 1.77·13-s − 0.381·14-s − 0.430·15-s + 0.328·16-s − 2.14·18-s + 0.664·19-s + 0.279·20-s − 0.103·21-s − 2.44·22-s + 0.0181·23-s + 3·25-s + 1.25·26-s − 0.263·27-s + 0.0674·28-s − 1.24·29-s − 0.304·30-s − 0.561·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 17^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{5} \cdot 17^{10}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.090242557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.090242557\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + p T )^{5} \) |
| 17 | | \( 1 \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 - p T + 5 T^{2} - 3 p^{2} T^{3} + p^{3} T^{4} + p^{6} T^{6} - 3 p^{8} T^{7} + 5 p^{9} T^{8} - p^{13} T^{9} + p^{15} T^{10} \) |
| 3 | $C_2 \wr S_5$ | \( 1 - T + 83 T^{2} - 128 T^{3} + 3266 T^{4} - 5566 T^{5} + 3266 p^{3} T^{6} - 128 p^{6} T^{7} + 83 p^{9} T^{8} - p^{12} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 + 10 T + 11 p^{2} T^{2} + 1404 T^{3} + 302202 T^{4} + 315948 p T^{5} + 302202 p^{3} T^{6} + 1404 p^{6} T^{7} + 11 p^{11} T^{8} + 10 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 126 T + 11551 T^{2} + 736580 T^{3} + 37561402 T^{4} + 1514510076 T^{5} + 37561402 p^{3} T^{6} + 736580 p^{6} T^{7} + 11551 p^{9} T^{8} + 126 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 83 T + 11705 T^{2} - 664236 T^{3} + 52251210 T^{4} - 2125046386 T^{5} + 52251210 p^{3} T^{6} - 664236 p^{6} T^{7} + 11705 p^{9} T^{8} - 83 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 55 T + 12923 T^{2} + 84108 T^{3} + 2926314 p T^{4} + 4054411654 T^{5} + 2926314 p^{4} T^{6} + 84108 p^{6} T^{7} + 12923 p^{9} T^{8} - 55 p^{12} T^{9} + p^{15} T^{10} \) |
| 23 | $C_2 \wr S_5$ | \( 1 - 2 T + 837 p T^{2} + 806556 T^{3} + 258240306 T^{4} + 17940344204 T^{5} + 258240306 p^{3} T^{6} + 806556 p^{6} T^{7} + 837 p^{10} T^{8} - 2 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 + 195 T + 82317 T^{2} + 8538924 T^{3} + 2545246150 T^{4} + 187375518322 T^{5} + 2545246150 p^{3} T^{6} + 8538924 p^{6} T^{7} + 82317 p^{9} T^{8} + 195 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 97 T + 44029 T^{2} + 4241504 T^{3} + 1797375928 T^{4} + 154051126334 T^{5} + 1797375928 p^{3} T^{6} + 4241504 p^{6} T^{7} + 44029 p^{9} T^{8} + 97 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 476 T + 153305 T^{2} - 12622704 T^{3} - 3543517406 T^{4} + 1858372578392 T^{5} - 3543517406 p^{3} T^{6} - 12622704 p^{6} T^{7} + 153305 p^{9} T^{8} - 476 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 + 298 T + 186853 T^{2} + 35611608 T^{3} + 16629118802 T^{4} + 2556337753020 T^{5} + 16629118802 p^{3} T^{6} + 35611608 p^{6} T^{7} + 186853 p^{9} T^{8} + 298 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 168 T + 131991 T^{2} - 36322976 T^{3} + 17312851778 T^{4} - 2154677969328 T^{5} + 17312851778 p^{3} T^{6} - 36322976 p^{6} T^{7} + 131991 p^{9} T^{8} - 168 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 1095 T + 728137 T^{2} - 355701832 T^{3} + 143827477240 T^{4} - 49413065471010 T^{5} + 143827477240 p^{3} T^{6} - 355701832 p^{6} T^{7} + 728137 p^{9} T^{8} - 1095 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 + 1035 T + 1103701 T^{2} + 12380476 p T^{3} + 380002580550 T^{4} + 149021215754242 T^{5} + 380002580550 p^{3} T^{6} + 12380476 p^{7} T^{7} + 1103701 p^{9} T^{8} + 1035 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 - 1163 T + 1277579 T^{2} - 863941780 T^{3} + 545962835734 T^{4} - 255565025905714 T^{5} + 545962835734 p^{3} T^{6} - 863941780 p^{6} T^{7} + 1277579 p^{9} T^{8} - 1163 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 351 T + 295913 T^{2} - 288347764 T^{3} + 135746325562 T^{4} - 57075386654698 T^{5} + 135746325562 p^{3} T^{6} - 288347764 p^{6} T^{7} + 295913 p^{9} T^{8} - 351 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 1064 T + 1674935 T^{2} - 1177384608 T^{3} + 1042937516346 T^{4} - 519679411593904 T^{5} + 1042937516346 p^{3} T^{6} - 1177384608 p^{6} T^{7} + 1674935 p^{9} T^{8} - 1064 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 1393 T + 2052017 T^{2} + 1618499656 T^{3} + 1367812188052 T^{4} + 775057659646862 T^{5} + 1367812188052 p^{3} T^{6} + 1618499656 p^{6} T^{7} + 2052017 p^{9} T^{8} + 1393 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 163 T + 1672693 T^{2} - 176482324 T^{3} + 1189817299330 T^{4} - 87912331507314 T^{5} + 1189817299330 p^{3} T^{6} - 176482324 p^{6} T^{7} + 1672693 p^{9} T^{8} - 163 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 750 T + 1496875 T^{2} + 1338020988 T^{3} + 1106336975146 T^{4} + 944690547284524 T^{5} + 1106336975146 p^{3} T^{6} + 1338020988 p^{6} T^{7} + 1496875 p^{9} T^{8} + 750 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 270 T + 160131 T^{2} + 509860560 T^{3} + 451464815014 T^{4} - 209011733767396 T^{5} + 451464815014 p^{3} T^{6} + 509860560 p^{6} T^{7} + 160131 p^{9} T^{8} - 270 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 1179 T + 3358201 T^{2} - 2906599564 T^{3} + 4621028822686 T^{4} - 2956952379632754 T^{5} + 4621028822686 p^{3} T^{6} - 2906599564 p^{6} T^{7} + 3358201 p^{9} T^{8} - 1179 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 2745 T + 6087149 T^{2} - 9524427020 T^{3} + 12396954393586 T^{4} - 12829411416270710 T^{5} + 12396954393586 p^{3} T^{6} - 9524427020 p^{6} T^{7} + 6087149 p^{9} T^{8} - 2745 p^{12} T^{9} + p^{15} T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.28079708484366986062078897466, −5.06078503781473533336726842686, −4.97246494384260492802414360834, −4.93325397823829031224880771974, −4.82070830489102424800574812567, −4.24236157537745830208763125421, −4.17991747171116872897776538920, −4.06590008590434078193968329378, −3.78696587022333988761968590695, −3.59841979386443671424210383065, −3.34988782536451011487015753922, −3.27646961113365497991468832767, −3.13842779228542787404779102128, −3.08016261173319637389805720579, −2.55934510045959658059634138742, −2.48107709806424456970775624935, −2.44130502303954934113749804523, −2.29972919242337682542744604188, −1.82856799579203534916551958982, −1.29905251359393406751300336094, −1.21114939918402042655725969459, −0.55809494999769922405909332706, −0.47214166205124654929816788079, −0.42189989955654717481352653970, −0.36826532369273353141008972422,
0.36826532369273353141008972422, 0.42189989955654717481352653970, 0.47214166205124654929816788079, 0.55809494999769922405909332706, 1.21114939918402042655725969459, 1.29905251359393406751300336094, 1.82856799579203534916551958982, 2.29972919242337682542744604188, 2.44130502303954934113749804523, 2.48107709806424456970775624935, 2.55934510045959658059634138742, 3.08016261173319637389805720579, 3.13842779228542787404779102128, 3.27646961113365497991468832767, 3.34988782536451011487015753922, 3.59841979386443671424210383065, 3.78696587022333988761968590695, 4.06590008590434078193968329378, 4.17991747171116872897776538920, 4.24236157537745830208763125421, 4.82070830489102424800574812567, 4.93325397823829031224880771974, 4.97246494384260492802414360834, 5.06078503781473533336726842686, 5.28079708484366986062078897466
