| L(s) = 1 | + 8·2-s − 6·3-s + 16·4-s + 50·5-s − 48·6-s + 126·7-s − 128·8-s + 179·9-s + 400·10-s + 76·11-s − 96·12-s + 1.46e3·13-s + 1.00e3·14-s − 300·15-s − 1.02e3·16-s + 3.01e3·17-s + 1.43e3·18-s − 1.16e3·19-s + 800·20-s − 756·21-s + 608·22-s + 3.51e3·23-s + 768·24-s + 625·25-s + 1.17e4·26-s − 90·27-s + 2.01e3·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.384·3-s + 1/2·4-s + 0.894·5-s − 0.544·6-s + 0.971·7-s − 0.707·8-s + 0.736·9-s + 1.26·10-s + 0.189·11-s − 0.192·12-s + 2.40·13-s + 1.37·14-s − 0.344·15-s − 16-s + 2.52·17-s + 1.04·18-s − 0.737·19-s + 0.447·20-s − 0.374·21-s + 0.267·22-s + 1.38·23-s + 0.272·24-s + 1/5·25-s + 3.39·26-s − 0.0237·27-s + 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.847772810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.847772810\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 18 p T + 451 p^{2} T^{2} - 18 p^{6} T^{3} + p^{10} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T - 143 T^{2} - 614 p T^{3} - 35756 T^{4} - 614 p^{6} T^{5} - 143 p^{10} T^{6} + 2 p^{16} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 76 T - 316506 T^{2} - 13680 T^{3} + 76998032347 T^{4} - 13680 p^{5} T^{5} - 316506 p^{10} T^{6} - 76 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 732 T + 723598 T^{2} - 732 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3012 T + 3975770 T^{2} - 6797059920 T^{3} + 11075886884019 T^{4} - 6797059920 p^{5} T^{5} + 3975770 p^{10} T^{6} - 3012 p^{15} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 1160 T - 1818214 T^{2} - 2074525440 T^{3} + 546813954395 T^{4} - 2074525440 p^{5} T^{5} - 1818214 p^{10} T^{6} + 1160 p^{15} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3518 T - 117513 p T^{2} - 7762245366 T^{3} + 106295172892348 T^{4} - 7762245366 p^{5} T^{5} - 117513 p^{11} T^{6} - 3518 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 3286 T + 42427411 T^{2} - 3286 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 588 p T + 192017582 T^{2} + 48794650800 p T^{3} + 9335981778555219 T^{4} + 48794650800 p^{6} T^{5} + 192017582 p^{10} T^{6} + 588 p^{16} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 2896 T - 85801706 T^{2} - 128870239232 T^{3} + 3481318296055723 T^{4} - 128870239232 p^{5} T^{5} - 85801706 p^{10} T^{6} + 2896 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 26174 T + 371825635 T^{2} + 26174 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 13298 T + 238774251 T^{2} + 13298 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 16472 T + 8790210 T^{2} + 3231039463680 T^{3} - 45901122723940061 T^{4} + 3231039463680 p^{5} T^{5} + 8790210 p^{10} T^{6} - 16472 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 14960 T - 664989210 T^{2} - 783901367040 T^{3} + 521468813128754251 T^{4} - 783901367040 p^{5} T^{5} - 664989210 p^{10} T^{6} - 14960 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 37816 T - 57529590 T^{2} + 2183149747968 T^{3} + 679398464025174571 T^{4} + 2183149747968 p^{5} T^{5} - 57529590 p^{10} T^{6} + 37816 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 3274 T + 105790049 T^{2} + 5841678944550 T^{3} - 713357713695453748 T^{4} + 5841678944550 p^{5} T^{5} + 105790049 p^{10} T^{6} - 3274 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 41438 T - 1412115655 T^{2} + 17775794983830 T^{3} + 5650891490531002244 T^{4} + 17775794983830 p^{5} T^{5} - 1412115655 p^{10} T^{6} + 41438 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 120 p T + 3293960686 T^{2} - 120 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 28676 T + 708228650 T^{2} - 115623319869360 T^{3} - 5989324267365574141 T^{4} - 115623319869360 p^{5} T^{5} + 708228650 p^{10} T^{6} + 28676 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 3552 T - 6061164334 T^{2} - 285338411520 T^{3} + 27383717705164021347 T^{4} - 285338411520 p^{5} T^{5} - 6061164334 p^{10} T^{6} + 3552 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 54982 T + 8517406483 T^{2} + 54982 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 81578 T - 3981166599 T^{2} - 43398045135270 T^{3} + 44783935029119535172 T^{4} - 43398045135270 p^{5} T^{5} - 3981166599 p^{10} T^{6} + 81578 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 106964 T + 17911970214 T^{2} + 106964 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−9.964939859568514582731253109257, −9.815746017895298954335035418802, −9.186416352845853146395672912052, −8.846944860897845513708397156979, −8.695535210868280194659667733684, −8.328970573637217940357872460535, −8.185485262677693052406184521428, −7.63655820261202009915129419067, −7.00056913039236719153416082999, −6.80890954471647216272924306720, −6.79030393292377663855614382410, −6.00208997179980332003997708016, −5.70354615811551382494183994450, −5.65457217622781984654804379145, −5.22167225423194396844327377832, −4.80574462778597569817907610849, −4.65884286751678209312854012816, −3.96797287657288055322084682993, −3.41934308421966191022514494302, −3.41183882553498088911179472511, −3.07028374200881843376430881109, −1.75359745765404033333138680719, −1.55908395566301133171561440303, −1.43332643779807274805766144305, −0.49468292077592452121839965236,
0.49468292077592452121839965236, 1.43332643779807274805766144305, 1.55908395566301133171561440303, 1.75359745765404033333138680719, 3.07028374200881843376430881109, 3.41183882553498088911179472511, 3.41934308421966191022514494302, 3.96797287657288055322084682993, 4.65884286751678209312854012816, 4.80574462778597569817907610849, 5.22167225423194396844327377832, 5.65457217622781984654804379145, 5.70354615811551382494183994450, 6.00208997179980332003997708016, 6.79030393292377663855614382410, 6.80890954471647216272924306720, 7.00056913039236719153416082999, 7.63655820261202009915129419067, 8.185485262677693052406184521428, 8.328970573637217940357872460535, 8.695535210868280194659667733684, 8.846944860897845513708397156979, 9.186416352845853146395672912052, 9.815746017895298954335035418802, 9.964939859568514582731253109257
