| L(s) = 1 | + 688·7-s − 1.45e3·17-s − 1.29e3·23-s + 2.14e5·25-s + 8.92e4·31-s − 5.21e5·41-s + 1.56e6·47-s − 2.48e6·49-s − 7.59e6·71-s + 2.08e6·73-s − 1.60e7·79-s − 2.16e6·89-s − 1.08e6·97-s + 6.64e7·103-s + 4.62e6·113-s − 9.98e5·119-s + 6.46e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 8.91e5·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.758·7-s − 0.0716·17-s − 0.0222·23-s + 2.74·25-s + 0.538·31-s − 1.18·41-s + 2.20·47-s − 3.02·49-s − 2.51·71-s + 0.628·73-s − 3.65·79-s − 0.326·89-s − 0.121·97-s + 5.99·103-s + 0.301·113-s − 0.0543·119-s + 3.31·121-s − 0.0168·161-s + 3.27·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{30} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.01197607130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01197607130\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 214718 T^{2} + 1269826431 p^{2} T^{4} - 4608734769796 p^{4} T^{6} + 1269826431 p^{16} T^{8} - 214718 p^{28} T^{10} + p^{42} T^{12} \) |
| 7 | \( ( 1 - 344 T + 1422245 T^{2} - 124668368 T^{3} + 1422245 p^{7} T^{4} - 344 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 11 | \( 1 - 64629022 T^{2} + 2363172524483783 T^{4} - \)\(55\!\cdots\!04\)\( T^{6} + 2363172524483783 p^{14} T^{8} - 64629022 p^{28} T^{10} + p^{42} T^{12} \) |
| 13 | \( 1 - 205410958 T^{2} + 1687860473809811 p T^{4} - \)\(16\!\cdots\!64\)\( T^{6} + 1687860473809811 p^{15} T^{8} - 205410958 p^{28} T^{10} + p^{42} T^{12} \) |
| 17 | \( ( 1 + 726 T + 323301023 T^{2} + 9708009717300 T^{3} + 323301023 p^{7} T^{4} + 726 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 19 | \( 1 - 2002416334 T^{2} + 2872909317854295863 T^{4} - \)\(28\!\cdots\!20\)\( T^{6} + 2872909317854295863 p^{14} T^{8} - 2002416334 p^{28} T^{10} + p^{42} T^{12} \) |
| 23 | \( ( 1 + 648 T + 9708521717 T^{2} + 6547475963760 T^{3} + 9708521717 p^{7} T^{4} + 648 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 29 | \( 1 - 47836636078 T^{2} + \)\(14\!\cdots\!79\)\( T^{4} - \)\(30\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!79\)\( p^{14} T^{8} - 47836636078 p^{28} T^{10} + p^{42} T^{12} \) |
| 31 | \( ( 1 - 1440 p T + 48354349725 T^{2} - 4324137771289408 T^{3} + 48354349725 p^{7} T^{4} - 1440 p^{15} T^{5} + p^{21} T^{6} )^{2} \) |
| 37 | \( 1 - 78937168126 T^{2} + \)\(11\!\cdots\!51\)\( T^{4} - \)\(18\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!51\)\( p^{14} T^{8} - 78937168126 p^{28} T^{10} + p^{42} T^{12} \) |
| 41 | \( ( 1 + 260622 T + 351195126263 T^{2} + 83834535574878564 T^{3} + 351195126263 p^{7} T^{4} + 260622 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 43 | \( 1 - 1505999929054 T^{2} + \)\(97\!\cdots\!71\)\( T^{4} - \)\(34\!\cdots\!32\)\( T^{6} + \)\(97\!\cdots\!71\)\( p^{14} T^{8} - 1505999929054 p^{28} T^{10} + p^{42} T^{12} \) |
| 47 | \( ( 1 - 783216 T + 1470820452333 T^{2} - 778339576797138720 T^{3} + 1470820452333 p^{7} T^{4} - 783216 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 53 | \( 1 - 3131635055902 T^{2} + \)\(64\!\cdots\!63\)\( T^{4} - \)\(86\!\cdots\!56\)\( T^{6} + \)\(64\!\cdots\!63\)\( p^{14} T^{8} - 3131635055902 p^{28} T^{10} + p^{42} T^{12} \) |
| 59 | \( 1 - 8312323804862 T^{2} + \)\(38\!\cdots\!03\)\( T^{4} - \)\(11\!\cdots\!64\)\( T^{6} + \)\(38\!\cdots\!03\)\( p^{14} T^{8} - 8312323804862 p^{28} T^{10} + p^{42} T^{12} \) |
| 61 | \( 1 - 14350504934382 T^{2} + \)\(96\!\cdots\!03\)\( T^{4} - \)\(38\!\cdots\!24\)\( T^{6} + \)\(96\!\cdots\!03\)\( p^{14} T^{8} - 14350504934382 p^{28} T^{10} + p^{42} T^{12} \) |
| 67 | \( 1 - 35072892237678 T^{2} + \)\(51\!\cdots\!43\)\( T^{4} - \)\(41\!\cdots\!64\)\( T^{6} + \)\(51\!\cdots\!43\)\( p^{14} T^{8} - 35072892237678 p^{28} T^{10} + p^{42} T^{12} \) |
| 71 | \( ( 1 + 3798552 T + 17820666583269 T^{2} + 72937737977373055056 T^{3} + 17820666583269 p^{7} T^{4} + 3798552 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 73 | \( ( 1 - 1044782 T + 23207418111255 T^{2} - 22868192285089705636 T^{3} + 23207418111255 p^{7} T^{4} - 1044782 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 79 | \( ( 1 + 8007952 T + 49828330384013 T^{2} + \)\(25\!\cdots\!96\)\( T^{3} + 49828330384013 p^{7} T^{4} + 8007952 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 83 | \( 1 - 124932236904014 T^{2} + \)\(70\!\cdots\!51\)\( T^{4} - \)\(23\!\cdots\!92\)\( T^{6} + \)\(70\!\cdots\!51\)\( p^{14} T^{8} - 124932236904014 p^{28} T^{10} + p^{42} T^{12} \) |
| 89 | \( ( 1 + 1084542 T + 130289056617383 T^{2} + 94766843887733093316 T^{3} + 130289056617383 p^{7} T^{4} + 1084542 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
| 97 | \( ( 1 + 544154 T + 184934786492783 T^{2} + 86271378317799707180 T^{3} + 184934786492783 p^{7} T^{4} + 544154 p^{14} T^{5} + p^{21} T^{6} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.14771507170683438146214607249, −4.79282825915702747501864497014, −4.76576137186453328773351532869, −4.71300457931473001014492138410, −4.47053645442175440718556339731, −4.25350040481324500070445983841, −4.08128650783753630447401107988, −3.96058232578594535571834582775, −3.37186127622600639590274552563, −3.28084848783510750438977194328, −3.20498114826299436883931411724, −3.20202829979098983500915116186, −2.74905175919918062335940748359, −2.68322236086268883882329053214, −2.27413555065958061017605826389, −2.27285804578250109880161395284, −1.72827778681345480184168019995, −1.66084392326879090507353123228, −1.62045457744826818029361575659, −1.35559526038004661783258325906, −0.873617584121314528631454761597, −0.871403355405954173297565186615, −0.70824623523083055574386937919, −0.39864735936895246310173420275, −0.00693295924804094371925221661,
0.00693295924804094371925221661, 0.39864735936895246310173420275, 0.70824623523083055574386937919, 0.871403355405954173297565186615, 0.873617584121314528631454761597, 1.35559526038004661783258325906, 1.62045457744826818029361575659, 1.66084392326879090507353123228, 1.72827778681345480184168019995, 2.27285804578250109880161395284, 2.27413555065958061017605826389, 2.68322236086268883882329053214, 2.74905175919918062335940748359, 3.20202829979098983500915116186, 3.20498114826299436883931411724, 3.28084848783510750438977194328, 3.37186127622600639590274552563, 3.96058232578594535571834582775, 4.08128650783753630447401107988, 4.25350040481324500070445983841, 4.47053645442175440718556339731, 4.71300457931473001014492138410, 4.76576137186453328773351532869, 4.79282825915702747501864497014, 5.14771507170683438146214607249
Plot not available for L-functions of degree greater than 10.
