| L(s) = 1 | + 4·2-s − 4·4-s + 20·5-s − 10·7-s − 32·8-s + 80·10-s + 46·11-s + 54·13-s − 40·14-s − 39·16-s + 250·17-s − 28·19-s − 80·20-s + 184·22-s − 115·23-s + 7·25-s + 216·26-s + 40·28-s + 460·29-s − 360·31-s − 64·32-s + 1.00e3·34-s − 200·35-s − 92·37-s − 112·38-s − 640·40-s + 788·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 1.78·5-s − 0.539·7-s − 1.41·8-s + 2.52·10-s + 1.26·11-s + 1.15·13-s − 0.763·14-s − 0.609·16-s + 3.56·17-s − 0.338·19-s − 0.894·20-s + 1.78·22-s − 1.04·23-s + 0.0559·25-s + 1.62·26-s + 0.269·28-s + 2.94·29-s − 2.08·31-s − 0.353·32-s + 5.04·34-s − 0.965·35-s − 0.408·37-s − 0.478·38-s − 2.52·40-s + 3.00·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(14.24475095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.24475095\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + p T )^{5} \) |
| good | 2 | $C_2 \wr S_5$ | \( 1 - p^{2} T + 5 p^{2} T^{2} - p^{6} T^{3} + 247 T^{4} - 87 p^{3} T^{5} + 247 p^{3} T^{6} - p^{12} T^{7} + 5 p^{11} T^{8} - p^{14} T^{9} + p^{15} T^{10} \) |
| 5 | $C_2 \wr S_5$ | \( 1 - 4 p T + 393 T^{2} - 6476 T^{3} + 85694 T^{4} - 1043128 T^{5} + 85694 p^{3} T^{6} - 6476 p^{6} T^{7} + 393 p^{9} T^{8} - 4 p^{13} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 + 10 T + 17 p^{2} T^{2} + 15996 T^{3} + 38296 p T^{4} + 8619972 T^{5} + 38296 p^{4} T^{6} + 15996 p^{6} T^{7} + 17 p^{11} T^{8} + 10 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 - 46 T + 323 p T^{2} - 90896 T^{3} + 6145308 T^{4} - 146750948 T^{5} + 6145308 p^{3} T^{6} - 90896 p^{6} T^{7} + 323 p^{10} T^{8} - 46 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 - 54 T + 7437 T^{2} - 32592 p T^{3} + 25810582 T^{4} - 1347737748 T^{5} + 25810582 p^{3} T^{6} - 32592 p^{7} T^{7} + 7437 p^{9} T^{8} - 54 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 250 T + 38615 T^{2} - 4232500 T^{3} + 390449104 T^{4} - 29571265044 T^{5} + 390449104 p^{3} T^{6} - 4232500 p^{6} T^{7} + 38615 p^{9} T^{8} - 250 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 + 28 T + 29623 T^{2} + 532804 T^{3} + 371915302 T^{4} + 246635720 p T^{5} + 371915302 p^{3} T^{6} + 532804 p^{6} T^{7} + 29623 p^{9} T^{8} + 28 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 460 T + 174629 T^{2} - 43592544 T^{3} + 9543545038 T^{4} - 1583081788264 T^{5} + 9543545038 p^{3} T^{6} - 43592544 p^{6} T^{7} + 174629 p^{9} T^{8} - 460 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 + 360 T + 182823 T^{2} + 43574560 T^{3} + 11874350854 T^{4} + 1956422302448 T^{5} + 11874350854 p^{3} T^{6} + 43574560 p^{6} T^{7} + 182823 p^{9} T^{8} + 360 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 + 92 T + 91299 T^{2} + 4925604 T^{3} + 4159790608 T^{4} - 71738308544 T^{5} + 4159790608 p^{3} T^{6} + 4925604 p^{6} T^{7} + 91299 p^{9} T^{8} + 92 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 788 T + 486385 T^{2} - 211173488 T^{3} + 75960521542 T^{4} - 21605505269816 T^{5} + 75960521542 p^{3} T^{6} - 211173488 p^{6} T^{7} + 486385 p^{9} T^{8} - 788 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 108 T + 81119 T^{2} - 17804716 T^{3} + 11842738534 T^{4} - 910617386936 T^{5} + 11842738534 p^{3} T^{6} - 17804716 p^{6} T^{7} + 81119 p^{9} T^{8} - 108 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 724 T + 479523 T^{2} - 171240720 T^{3} + 64228476626 T^{4} - 17720375620216 T^{5} + 64228476626 p^{3} T^{6} - 171240720 p^{6} T^{7} + 479523 p^{9} T^{8} - 724 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 580 T + 417865 T^{2} - 60694692 T^{3} + 21127620942 T^{4} + 9696243070280 T^{5} + 21127620942 p^{3} T^{6} - 60694692 p^{6} T^{7} + 417865 p^{9} T^{8} - 580 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 + 408 T + 294211 T^{2} + 90303888 T^{3} + 46389350126 T^{4} + 23464212250800 T^{5} + 46389350126 p^{3} T^{6} + 90303888 p^{6} T^{7} + 294211 p^{9} T^{8} + 408 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 + 620 T + 829499 T^{2} + 442674196 T^{3} + 324762956688 T^{4} + 138937628024128 T^{5} + 324762956688 p^{3} T^{6} + 442674196 p^{6} T^{7} + 829499 p^{9} T^{8} + 620 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 - 100 T + 980247 T^{2} - 64921236 T^{3} + 487248244054 T^{4} - 22131765740840 T^{5} + 487248244054 p^{3} T^{6} - 64921236 p^{6} T^{7} + 980247 p^{9} T^{8} - 100 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 124 T + 643575 T^{2} + 132379984 T^{3} + 365080849854 T^{4} + 36192869200872 T^{5} + 365080849854 p^{3} T^{6} + 132379984 p^{6} T^{7} + 643575 p^{9} T^{8} + 124 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 + 1946 T + 2370617 T^{2} + 2143489904 T^{3} + 1736042995838 T^{4} + 1174456533829676 T^{5} + 1736042995838 p^{3} T^{6} + 2143489904 p^{6} T^{7} + 2370617 p^{9} T^{8} + 1946 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 + 198 T + 1431801 T^{2} + 468601876 T^{3} + 1064856569560 T^{4} + 359643416452220 T^{5} + 1064856569560 p^{3} T^{6} + 468601876 p^{6} T^{7} + 1431801 p^{9} T^{8} + 198 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 + 578 T + 1798769 T^{2} + 931952608 T^{3} + 1510477399620 T^{4} + 688304793775644 T^{5} + 1510477399620 p^{3} T^{6} + 931952608 p^{6} T^{7} + 1798769 p^{9} T^{8} + 578 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 - 930 T + 2287407 T^{2} - 1655809268 T^{3} + 2465301302144 T^{4} - 1519191014987812 T^{5} + 2465301302144 p^{3} T^{6} - 1655809268 p^{6} T^{7} + 2287407 p^{9} T^{8} - 930 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 - 2618 T + 5523829 T^{2} - 8467467016 T^{3} + 10949175375914 T^{4} - 11138124085391228 T^{5} + 10949175375914 p^{3} T^{6} - 8467467016 p^{6} T^{7} + 5523829 p^{9} T^{8} - 2618 p^{12} T^{9} + p^{15} T^{10} \) |
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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
−7.28887574292134761758444983104, −6.95250112815833210794678515158, −6.56224063344379584728012998064, −6.43566669833059959505477431725, −6.19423741824248611622139055674, −5.95917566277379626665297029651, −5.73206156287880404139187362270, −5.60936316718092703437329676708, −5.50319483670083435157510524295, −5.42574125613765367688579355251, −4.83775401640788457386440679477, −4.45290685670348296308001579275, −4.36834867687477614999862163444, −4.17231258515693335381530269346, −4.08313067219659822444257639644, −3.55488846972681680891631769316, −3.35754746972964286531860743478, −3.05203697766192185391647555891, −2.95287341380340514233224861128, −2.32512030423992637989540071260, −1.85991163886242491369445042400, −1.72306380748019374005552700602, −1.05289156654566722191420736601, −1.04286640923213544177094269159, −0.45350742923805704074649034012,
0.45350742923805704074649034012, 1.04286640923213544177094269159, 1.05289156654566722191420736601, 1.72306380748019374005552700602, 1.85991163886242491369445042400, 2.32512030423992637989540071260, 2.95287341380340514233224861128, 3.05203697766192185391647555891, 3.35754746972964286531860743478, 3.55488846972681680891631769316, 4.08313067219659822444257639644, 4.17231258515693335381530269346, 4.36834867687477614999862163444, 4.45290685670348296308001579275, 4.83775401640788457386440679477, 5.42574125613765367688579355251, 5.50319483670083435157510524295, 5.60936316718092703437329676708, 5.73206156287880404139187362270, 5.95917566277379626665297029651, 6.19423741824248611622139055674, 6.43566669833059959505477431725, 6.56224063344379584728012998064, 6.95250112815833210794678515158, 7.28887574292134761758444983104
