| L(s) = 1 | − 2·2-s + 4-s − 3·5-s + 2·8-s + 6·10-s + 3·11-s + 4·13-s − 4·16-s − 6·17-s − 20·19-s − 3·20-s − 6·22-s − 9·23-s + 4·25-s − 8·26-s − 6·29-s + 4·31-s + 2·32-s + 12·34-s + 8·37-s + 40·38-s − 6·40-s − 15·41-s − 43-s + 3·44-s + 18·46-s − 8·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 1.34·5-s + 0.707·8-s + 1.89·10-s + 0.904·11-s + 1.10·13-s − 16-s − 1.45·17-s − 4.58·19-s − 0.670·20-s − 1.27·22-s − 1.87·23-s + 4/5·25-s − 1.56·26-s − 1.11·29-s + 0.718·31-s + 0.353·32-s + 2.05·34-s + 1.31·37-s + 6.48·38-s − 0.948·40-s − 2.34·41-s − 0.152·43-s + 0.452·44-s + 2.65·46-s − 1.13·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2682383648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2682383648\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + 3 T + p T^{2} )^{2}( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 7 T^{2} + 18 T^{3} + 36 T^{4} + 18 p T^{5} - 7 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + 9 T + p T^{2} )^{2}( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 144 T^{3} - 729 T^{4} - 144 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 + 15 T + 95 T^{2} + 720 T^{3} + 5994 T^{4} + 720 p T^{5} + 95 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + T - 11 T^{2} - 74 T^{3} - 1748 T^{4} - 74 p T^{5} - 11 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3 T - 37 T^{2} + 216 T^{3} - 1896 T^{4} + 216 p T^{5} - 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 11 T + 43 T^{2} - 484 T^{3} - 5018 T^{4} - 484 p T^{5} + 43 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 13 T + p T^{2} )^{2}( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_{4}$ | \( ( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T - 47 T^{2} - 434 T^{3} - 896 T^{4} - 434 p T^{5} - 47 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 74 T^{2} - 1152 T^{3} - 13941 T^{4} - 1152 p T^{5} + 74 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - T - 119 T^{2} + 74 T^{3} + 4894 T^{4} + 74 p T^{5} - 119 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.27136147018680210970690925443, −6.14196212710416342330354075622, −5.98665839000364842234863639619, −5.85226172527283788495105302770, −5.77288356695673606082050465152, −5.12116957149806071557854535891, −4.88391883568771403707519164296, −4.70509043611836095136459756239, −4.36750722272787624607755259834, −4.34433536655879307228137982098, −4.28942089495553533015171873728, −4.05919071321066348581436952970, −3.73353549886869086356468251002, −3.70362205891450015831833619285, −3.31763545629517522209181401279, −2.95384740747348867281078248298, −2.83333661503940653261254196275, −2.18086377070435055712507938903, −2.07356995158326963442944146276, −1.94072142819859218115040320542, −1.78480311807014878988048656063, −1.41446694998930329502487465103, −0.837306395347892643112177775106, −0.37569472229557115620044006060, −0.25260059690760270253988934246,
0.25260059690760270253988934246, 0.37569472229557115620044006060, 0.837306395347892643112177775106, 1.41446694998930329502487465103, 1.78480311807014878988048656063, 1.94072142819859218115040320542, 2.07356995158326963442944146276, 2.18086377070435055712507938903, 2.83333661503940653261254196275, 2.95384740747348867281078248298, 3.31763545629517522209181401279, 3.70362205891450015831833619285, 3.73353549886869086356468251002, 4.05919071321066348581436952970, 4.28942089495553533015171873728, 4.34433536655879307228137982098, 4.36750722272787624607755259834, 4.70509043611836095136459756239, 4.88391883568771403707519164296, 5.12116957149806071557854535891, 5.77288356695673606082050465152, 5.85226172527283788495105302770, 5.98665839000364842234863639619, 6.14196212710416342330354075622, 6.27136147018680210970690925443
