| L(s) = 1 | − 2·4-s − 12·5-s − 10·7-s + 12·11-s + 48·17-s − 42·19-s + 24·20-s − 24·23-s + 58·25-s + 20·28-s + 102·31-s + 120·35-s + 22·37-s + 28·43-s − 24·44-s + 132·47-s + 49·49-s − 120·53-s − 144·55-s + 24·59-s − 72·61-s + 8·64-s + 110·67-s − 96·68-s − 312·71-s − 66·73-s + 84·76-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.39·5-s − 1.42·7-s + 1.09·11-s + 2.82·17-s − 2.21·19-s + 6/5·20-s − 1.04·23-s + 2.31·25-s + 5/7·28-s + 3.29·31-s + 24/7·35-s + 0.594·37-s + 0.651·43-s − 0.545·44-s + 2.80·47-s + 49-s − 2.26·53-s − 2.61·55-s + 0.406·59-s − 1.18·61-s + 1/8·64-s + 1.64·67-s − 1.41·68-s − 4.39·71-s − 0.904·73-s + 1.10·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7359842147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7359842147\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 10 T + 51 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T + 86 T^{2} + 456 T^{3} + 2019 T^{4} + 456 p^{2} T^{5} + 86 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 6 T - 85 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 98 T^{2} + 54915 T^{4} + 98 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 48 T + 1514 T^{2} - 35808 T^{3} + 694947 T^{4} - 35808 p^{2} T^{5} + 1514 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 42 T + 1433 T^{2} + 35490 T^{3} + 795972 T^{4} + 35490 p^{2} T^{5} + 1433 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 24 T + 22 T^{2} - 12096 T^{3} - 277629 T^{4} - 12096 p^{2} T^{5} + 22 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 530 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 102 T + 6041 T^{2} - 8466 p T^{3} + 9396 p^{2} T^{4} - 8466 p^{3} T^{5} + 6041 p^{4} T^{6} - 102 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 3674 p^{2} T^{5} - 2087 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2476 T^{2} + 6405414 T^{4} - 2476 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 14 T + 3675 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T + 11654 T^{2} - 771672 T^{3} + 42125907 T^{4} - 771672 p^{2} T^{5} + 11654 p^{4} T^{6} - 132 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 120 T + 110 p T^{2} + 354240 T^{3} + 25104819 T^{4} + 354240 p^{2} T^{5} + 110 p^{5} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 6026 T^{2} - 140016 T^{3} + 22586547 T^{4} - 140016 p^{2} T^{5} + 6026 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 539280 p^{2} T^{5} + 9218 p^{4} T^{6} + 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 110 T + 3625 T^{2} + 55330 T^{3} - 2642396 T^{4} + 55330 p^{2} T^{5} + 3625 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 156 T + 178 p T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 66 T + 2873 T^{2} + 93786 T^{3} - 18641292 T^{4} + 93786 p^{2} T^{5} + 2873 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 10 T - 3695 T^{2} - 86870 T^{3} - 25172156 T^{4} - 86870 p^{2} T^{5} - 3695 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 116 p T^{2} + 107694438 T^{4} - 116 p^{5} T^{6} + p^{8} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 36 T + 8353 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 36580 T^{2} + 511416774 T^{4} - 36580 p^{4} T^{6} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.544254091697781155379272964989, −9.268959572161725507693014105160, −9.198451460664972552486179676912, −8.539120188488608359804971261453, −8.462407142832589518286540870139, −8.097725127011049536803350414833, −7.967205827201395179097672270862, −7.76652422910699565943890142564, −7.27856737918960901944340489852, −7.03938743408257599033833084683, −6.79895035347936507592712923639, −6.16979816722841391061974085761, −6.15613721402608089145109930584, −5.79354444252777014471318938547, −5.56361385139454075428257393573, −4.55473651830101178572905839220, −4.38339910994110619223548300583, −4.26517285066874729248914854187, −4.13748045058694122429008457519, −3.37087450268952251697580910419, −3.10300510715676986415934547553, −3.08558533999374937300566535101, −2.06485288354938203844627194635, −0.984959416037264205730751098369, −0.45728706469460177563151992409,
0.45728706469460177563151992409, 0.984959416037264205730751098369, 2.06485288354938203844627194635, 3.08558533999374937300566535101, 3.10300510715676986415934547553, 3.37087450268952251697580910419, 4.13748045058694122429008457519, 4.26517285066874729248914854187, 4.38339910994110619223548300583, 4.55473651830101178572905839220, 5.56361385139454075428257393573, 5.79354444252777014471318938547, 6.15613721402608089145109930584, 6.16979816722841391061974085761, 6.79895035347936507592712923639, 7.03938743408257599033833084683, 7.27856737918960901944340489852, 7.76652422910699565943890142564, 7.967205827201395179097672270862, 8.097725127011049536803350414833, 8.462407142832589518286540870139, 8.539120188488608359804971261453, 9.198451460664972552486179676912, 9.268959572161725507693014105160, 9.544254091697781155379272964989
