| L(s) = 1 | + 10·2-s + 5·3-s + 60·4-s + 50·6-s − 3·7-s + 280·8-s − 13·9-s − 26·11-s + 300·12-s − 61·13-s − 30·14-s + 1.12e3·16-s + 231·17-s − 130·18-s + 74·19-s − 15·21-s − 260·22-s + 115·23-s + 1.40e3·24-s − 610·26-s − 29·27-s − 180·28-s + 37·29-s + 220·31-s + 4.03e3·32-s − 130·33-s + 2.31e3·34-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 0.962·3-s + 15/2·4-s + 3.40·6-s − 0.161·7-s + 12.3·8-s − 0.481·9-s − 0.712·11-s + 7.21·12-s − 1.30·13-s − 0.572·14-s + 35/2·16-s + 3.29·17-s − 1.70·18-s + 0.893·19-s − 0.155·21-s − 2.51·22-s + 1.04·23-s + 11.9·24-s − 4.60·26-s − 0.206·27-s − 1.21·28-s + 0.236·29-s + 1.27·31-s + 22.2·32-s − 0.685·33-s + 11.6·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 5^{10} \cdot 23^{5}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(381.9906208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(381.9906208\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{5} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - p T )^{5} \) |
| good | 3 | $C_2 \wr S_5$ | \( 1 - 5 T + 38 T^{2} - 226 T^{3} + 1262 T^{4} - 4040 T^{5} + 1262 p^{3} T^{6} - 226 p^{6} T^{7} + 38 p^{9} T^{8} - 5 p^{12} T^{9} + p^{15} T^{10} \) |
| 7 | $C_2 \wr S_5$ | \( 1 + 3 T + 771 T^{2} + 3144 T^{3} + 371058 T^{4} + 2380138 T^{5} + 371058 p^{3} T^{6} + 3144 p^{6} T^{7} + 771 p^{9} T^{8} + 3 p^{12} T^{9} + p^{15} T^{10} \) |
| 11 | $C_2 \wr S_5$ | \( 1 + 26 T + 2418 T^{2} + 113386 T^{3} + 4059013 T^{4} + 212955280 T^{5} + 4059013 p^{3} T^{6} + 113386 p^{6} T^{7} + 2418 p^{9} T^{8} + 26 p^{12} T^{9} + p^{15} T^{10} \) |
| 13 | $C_2 \wr S_5$ | \( 1 + 61 T + 10540 T^{2} + 529280 T^{3} + 45358160 T^{4} + 1736952500 T^{5} + 45358160 p^{3} T^{6} + 529280 p^{6} T^{7} + 10540 p^{9} T^{8} + 61 p^{12} T^{9} + p^{15} T^{10} \) |
| 17 | $C_2 \wr S_5$ | \( 1 - 231 T + 2463 p T^{2} - 4861940 T^{3} + 483550256 T^{4} - 36207317242 T^{5} + 483550256 p^{3} T^{6} - 4861940 p^{6} T^{7} + 2463 p^{10} T^{8} - 231 p^{12} T^{9} + p^{15} T^{10} \) |
| 19 | $C_2 \wr S_5$ | \( 1 - 74 T + 12546 T^{2} - 283034 T^{3} + 11332093 T^{4} + 2967552288 T^{5} + 11332093 p^{3} T^{6} - 283034 p^{6} T^{7} + 12546 p^{9} T^{8} - 74 p^{12} T^{9} + p^{15} T^{10} \) |
| 29 | $C_2 \wr S_5$ | \( 1 - 37 T + 49794 T^{2} - 4360786 T^{3} + 1471271970 T^{4} - 188914249194 T^{5} + 1471271970 p^{3} T^{6} - 4360786 p^{6} T^{7} + 49794 p^{9} T^{8} - 37 p^{12} T^{9} + p^{15} T^{10} \) |
| 31 | $C_2 \wr S_5$ | \( 1 - 220 T + 92656 T^{2} - 17419261 T^{3} + 4814097217 T^{4} - 698419034622 T^{5} + 4814097217 p^{3} T^{6} - 17419261 p^{6} T^{7} + 92656 p^{9} T^{8} - 220 p^{12} T^{9} + p^{15} T^{10} \) |
| 37 | $C_2 \wr S_5$ | \( 1 - 220 T + 216009 T^{2} - 35499624 T^{3} + 19855535026 T^{4} - 2472537970040 T^{5} + 19855535026 p^{3} T^{6} - 35499624 p^{6} T^{7} + 216009 p^{9} T^{8} - 220 p^{12} T^{9} + p^{15} T^{10} \) |
| 41 | $C_2 \wr S_5$ | \( 1 - 112 T + 246093 T^{2} - 37957947 T^{3} + 661636213 p T^{4} - 4163193534223 T^{5} + 661636213 p^{4} T^{6} - 37957947 p^{6} T^{7} + 246093 p^{9} T^{8} - 112 p^{12} T^{9} + p^{15} T^{10} \) |
| 43 | $C_2 \wr S_5$ | \( 1 - 97 T + 247163 T^{2} - 9219484 T^{3} + 28200088918 T^{4} - 279343826070 T^{5} + 28200088918 p^{3} T^{6} - 9219484 p^{6} T^{7} + 247163 p^{9} T^{8} - 97 p^{12} T^{9} + p^{15} T^{10} \) |
| 47 | $C_2 \wr S_5$ | \( 1 - 370 T + 560732 T^{2} - 154409339 T^{3} + 120663232045 T^{4} - 24084178915350 T^{5} + 120663232045 p^{3} T^{6} - 154409339 p^{6} T^{7} + 560732 p^{9} T^{8} - 370 p^{12} T^{9} + p^{15} T^{10} \) |
| 53 | $C_2 \wr S_5$ | \( 1 - 1224 T + 849169 T^{2} - 349422432 T^{3} + 106131613898 T^{4} - 31335599759280 T^{5} + 106131613898 p^{3} T^{6} - 349422432 p^{6} T^{7} + 849169 p^{9} T^{8} - 1224 p^{12} T^{9} + p^{15} T^{10} \) |
| 59 | $C_2 \wr S_5$ | \( 1 + 181 T + 819180 T^{2} + 125833479 T^{3} + 302586825665 T^{4} + 37541024009080 T^{5} + 302586825665 p^{3} T^{6} + 125833479 p^{6} T^{7} + 819180 p^{9} T^{8} + 181 p^{12} T^{9} + p^{15} T^{10} \) |
| 61 | $C_2 \wr S_5$ | \( 1 - 1574 T + 1975365 T^{2} - 1581962520 T^{3} + 1083354883294 T^{4} - 553631174759044 T^{5} + 1083354883294 p^{3} T^{6} - 1581962520 p^{6} T^{7} + 1975365 p^{9} T^{8} - 1574 p^{12} T^{9} + p^{15} T^{10} \) |
| 67 | $C_2 \wr S_5$ | \( 1 + 435 T + 241931 T^{2} + 192411872 T^{3} + 102130758294 T^{4} + 1095541382282 T^{5} + 102130758294 p^{3} T^{6} + 192411872 p^{6} T^{7} + 241931 p^{9} T^{8} + 435 p^{12} T^{9} + p^{15} T^{10} \) |
| 71 | $C_2 \wr S_5$ | \( 1 + 30 T + 1197216 T^{2} + 145823515 T^{3} + 687466453625 T^{4} + 91054818733334 T^{5} + 687466453625 p^{3} T^{6} + 145823515 p^{6} T^{7} + 1197216 p^{9} T^{8} + 30 p^{12} T^{9} + p^{15} T^{10} \) |
| 73 | $C_2 \wr S_5$ | \( 1 - 2180 T + 2734129 T^{2} - 2362943775 T^{3} + 1695229971401 T^{4} - 1082670505912667 T^{5} + 1695229971401 p^{3} T^{6} - 2362943775 p^{6} T^{7} + 2734129 p^{9} T^{8} - 2180 p^{12} T^{9} + p^{15} T^{10} \) |
| 79 | $C_2 \wr S_5$ | \( 1 - 609 T + 1833841 T^{2} - 567237892 T^{3} + 1325523479672 T^{4} - 250311566358774 T^{5} + 1325523479672 p^{3} T^{6} - 567237892 p^{6} T^{7} + 1833841 p^{9} T^{8} - 609 p^{12} T^{9} + p^{15} T^{10} \) |
| 83 | $C_2 \wr S_5$ | \( 1 - 2018 T + 3746964 T^{2} - 4288910864 T^{3} + 4548945561579 T^{4} - 3553998955652964 T^{5} + 4548945561579 p^{3} T^{6} - 4288910864 p^{6} T^{7} + 3746964 p^{9} T^{8} - 2018 p^{12} T^{9} + p^{15} T^{10} \) |
| 89 | $C_2 \wr S_5$ | \( 1 + 1663 T + 4502861 T^{2} + 4941279904 T^{3} + 7067203139890 T^{4} + 5315152786232562 T^{5} + 7067203139890 p^{3} T^{6} + 4941279904 p^{6} T^{7} + 4502861 p^{9} T^{8} + 1663 p^{12} T^{9} + p^{15} T^{10} \) |
| 97 | $C_2 \wr S_5$ | \( 1 + 1274 T + 4204265 T^{2} + 3940293832 T^{3} + 7336467257286 T^{4} + 5149129381002076 T^{5} + 7336467257286 p^{3} T^{6} + 3940293832 p^{6} T^{7} + 4204265 p^{9} T^{8} + 1274 p^{12} T^{9} + p^{15} T^{10} \) |
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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
−5.39476747899534584611149966026, −5.28780854767617493973328524920, −5.09863161325096334668699848153, −5.07769183511187301159499352370, −4.75574462872200624103723735585, −4.58602833351075500895889568851, −4.56431139843586896767003405770, −4.02862620553316432377634337731, −3.92692273002016973451843063560, −3.88053683526264648627695816616, −3.43558220590874235892924173526, −3.41257225760460408810673035941, −3.25467901375067775855177097125, −3.13584298641386818457977846116, −2.80281516551701884565744269564, −2.63596842338367123260334931880, −2.37020768795677841908477545496, −2.34952604884822204348857263062, −2.34436346585886945945887703901, −1.80814195157938397723932525605, −1.30941774467702956410574907441, −1.25791896730636064497914942336, −0.822113403014396558207343522542, −0.70477841886170280937968777818, −0.52507325521308915428296060630,
0.52507325521308915428296060630, 0.70477841886170280937968777818, 0.822113403014396558207343522542, 1.25791896730636064497914942336, 1.30941774467702956410574907441, 1.80814195157938397723932525605, 2.34436346585886945945887703901, 2.34952604884822204348857263062, 2.37020768795677841908477545496, 2.63596842338367123260334931880, 2.80281516551701884565744269564, 3.13584298641386818457977846116, 3.25467901375067775855177097125, 3.41257225760460408810673035941, 3.43558220590874235892924173526, 3.88053683526264648627695816616, 3.92692273002016973451843063560, 4.02862620553316432377634337731, 4.56431139843586896767003405770, 4.58602833351075500895889568851, 4.75574462872200624103723735585, 5.07769183511187301159499352370, 5.09863161325096334668699848153, 5.28780854767617493973328524920, 5.39476747899534584611149966026
