| L(s) = 1 | − 308·7-s − 4.46e4·13-s + 5.65e4·19-s + 1.21e6·25-s + 2.37e6·31-s + 5.19e6·37-s + 2.75e6·43-s − 2.07e7·49-s + 3.68e7·61-s + 5.21e7·67-s + 2.24e7·73-s + 5.11e7·79-s + 1.37e7·91-s − 2.69e8·97-s − 5.70e8·103-s − 2.39e8·109-s + 5.65e8·121-s + 127-s + 131-s − 1.74e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.128·7-s − 1.56·13-s + 0.433·19-s + 3.12·25-s + 2.57·31-s + 2.77·37-s + 0.805·43-s − 3.59·49-s + 2.66·61-s + 2.58·67-s + 0.790·73-s + 1.31·79-s + 0.200·91-s − 3.04·97-s − 5.06·103-s − 1.69·109-s + 2.63·121-s − 0.0556·133-s − 1.82·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(9.115758050\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.115758050\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 1219618 T^{2} + 647675551779 T^{4} - 1219618 p^{16} T^{6} + p^{32} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 22 p T + 212091 p^{2} T^{2} + 22 p^{9} T^{3} + p^{16} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 51372566 p T^{2} + 159918763139994291 T^{4} - 51372566 p^{17} T^{6} + p^{32} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 22340 T + 1494116934 T^{2} + 22340 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 817823804 p T^{2} + \)\(14\!\cdots\!70\)\( T^{4} - 817823804 p^{17} T^{6} + p^{32} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 28256 T + 13290126018 T^{2} - 28256 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 263647467748 T^{2} + \)\(29\!\cdots\!98\)\( T^{4} - 263647467748 p^{16} T^{6} + p^{32} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1529313383356 T^{2} + \)\(10\!\cdots\!34\)\( T^{4} - 1529313383356 p^{16} T^{6} + p^{32} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 1187714 T + 813595564131 T^{2} - 1187714 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 2598664 T + 3728357318994 T^{2} - 2598664 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 30599693396452 T^{2} + \)\(36\!\cdots\!58\)\( T^{4} - 30599693396452 p^{16} T^{6} + p^{32} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 32036 p T + 5511982508646 T^{2} - 32036 p^{9} T^{3} + p^{16} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 67952666914108 T^{2} + \)\(22\!\cdots\!58\)\( T^{4} - 67952666914108 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 31157940225730 T^{2} + \)\(64\!\cdots\!39\)\( T^{4} - 31157940225730 p^{16} T^{6} + p^{32} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 218168248603708 T^{2} + \)\(34\!\cdots\!50\)\( T^{4} - 218168248603708 p^{16} T^{6} + p^{32} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 18421192 T + 256791572847186 T^{2} - 18421192 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 26085188 T + 894751023309126 T^{2} - 26085188 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 221755868821948 T^{2} + \)\(84\!\cdots\!70\)\( T^{4} - 221755868821948 p^{16} T^{6} + p^{32} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 11220466 T + 1147639193782323 T^{2} - 11220466 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 25573220 T + 2748523931252934 T^{2} - 25573220 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 695092323341698 T^{2} + \)\(77\!\cdots\!15\)\( T^{4} - 695092323341698 p^{16} T^{6} + p^{32} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 4619397677711684 T^{2} + \)\(35\!\cdots\!54\)\( T^{4} + 4619397677711684 p^{16} T^{6} + p^{32} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 134677610 T + 14669066367352059 T^{2} + 134677610 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.69056592517640309257163181057, −6.63044578211046761469007793325, −6.41573039451198539948820850696, −5.94114145734581018462238206806, −5.70901059437002304761782532977, −5.28896174731493124549619511705, −5.21411462620066484092095245936, −4.95084944478962374017500196438, −4.75605462274250991421561293511, −4.43081020259346558505751263066, −4.34115278185972574639838365339, −3.85968776789276408453353956841, −3.75587312417829172448421883249, −3.18748670305424338924685728985, −2.95849685719241465465017775645, −2.74787599779755130217656504368, −2.61030327284324211417832844677, −2.41982571452337638478101020589, −2.06789456463284250729738060149, −1.51509463980263496257708849510, −1.29497955425419556181763361177, −0.960914268696811485905392520264, −0.823195879386526408441139908838, −0.45909837133092762552007705618, −0.33984909297093714988227617548,
0.33984909297093714988227617548, 0.45909837133092762552007705618, 0.823195879386526408441139908838, 0.960914268696811485905392520264, 1.29497955425419556181763361177, 1.51509463980263496257708849510, 2.06789456463284250729738060149, 2.41982571452337638478101020589, 2.61030327284324211417832844677, 2.74787599779755130217656504368, 2.95849685719241465465017775645, 3.18748670305424338924685728985, 3.75587312417829172448421883249, 3.85968776789276408453353956841, 4.34115278185972574639838365339, 4.43081020259346558505751263066, 4.75605462274250991421561293511, 4.95084944478962374017500196438, 5.21411462620066484092095245936, 5.28896174731493124549619511705, 5.70901059437002304761782532977, 5.94114145734581018462238206806, 6.41573039451198539948820850696, 6.63044578211046761469007793325, 6.69056592517640309257163181057
