| L(s) = 1 | + 2·5-s − 4·7-s − 2·11-s − 6·17-s − 10·19-s − 2·23-s + 3·25-s − 14·31-s − 8·35-s − 6·37-s + 16·41-s + 16·43-s + 18·47-s − 2·49-s − 2·53-s − 4·55-s + 2·59-s − 6·61-s + 14·67-s − 32·71-s + 18·73-s + 8·77-s + 2·79-s − 16·83-s − 12·85-s − 18·89-s − 20·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 0.603·11-s − 1.45·17-s − 2.29·19-s − 0.417·23-s + 3/5·25-s − 2.51·31-s − 1.35·35-s − 0.986·37-s + 2.49·41-s + 2.43·43-s + 2.62·47-s − 2/7·49-s − 0.274·53-s − 0.539·55-s + 0.260·59-s − 0.768·61-s + 1.71·67-s − 3.79·71-s + 2.10·73-s + 0.911·77-s + 0.225·79-s − 1.75·83-s − 1.30·85-s − 1.90·89-s − 2.05·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376952955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376952955\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 14 T^{3} - 36 T^{4} + 14 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 17 T^{2} - 2 T^{3} + 276 T^{4} - 2 p T^{5} - 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + T^{2} + 6 T^{3} + 324 T^{4} + 6 p T^{5} + p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 10 T + 39 T^{2} + 230 T^{3} + 1460 T^{4} + 230 p T^{5} + 39 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 25 T^{2} - 34 T^{3} + 220 T^{4} - 34 p T^{5} - 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 14 T + 87 T^{2} + 658 T^{3} + 4844 T^{4} + 658 p T^{5} + 87 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T - 15 T^{2} - 138 T^{3} - 100 T^{4} - 138 p T^{5} - 15 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 151 T^{2} - 1422 T^{3} + 12492 T^{4} - 1422 p T^{5} + 151 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2 T - 17 T^{2} + 194 T^{3} - 3276 T^{4} + 194 p T^{5} - 17 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 63 T^{2} - 138 T^{3} + 3884 T^{4} - 138 p T^{5} - 63 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14 T + 31 T^{2} - 434 T^{3} + 9604 T^{4} - 434 p T^{5} + 31 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 18 T + 129 T^{2} - 882 T^{3} + 9044 T^{4} - 882 p T^{5} + 129 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 2 T - 105 T^{2} + 98 T^{3} + 5324 T^{4} + 98 p T^{5} - 105 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 202 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.46705538265789120194214515982, −6.29407696114662383008771876202, −6.19314583743978987909365963628, −5.80782968375471052110239540502, −5.67769649781624278022477795787, −5.54257868408583480109740703238, −5.50920288530660914532469599065, −5.00116569502349454267245570883, −4.80585981380250097377097045291, −4.49449887936160267971747222005, −4.21362855853902025081543124817, −4.08858520995211163311843274738, −4.01642009766734062773107335376, −3.73894549615166940802697413485, −3.49653973916179157300724984606, −2.94131897065558634758799790675, −2.70615467982877915783299600136, −2.64997831285060103552805870645, −2.56522560073079030027790656502, −2.03021763563797252334440830100, −1.89828625428670760791592705503, −1.65783364686828943782753065995, −1.15674434684834749741577129889, −0.42822537039830879917595440216, −0.35560772121347049397480245998,
0.35560772121347049397480245998, 0.42822537039830879917595440216, 1.15674434684834749741577129889, 1.65783364686828943782753065995, 1.89828625428670760791592705503, 2.03021763563797252334440830100, 2.56522560073079030027790656502, 2.64997831285060103552805870645, 2.70615467982877915783299600136, 2.94131897065558634758799790675, 3.49653973916179157300724984606, 3.73894549615166940802697413485, 4.01642009766734062773107335376, 4.08858520995211163311843274738, 4.21362855853902025081543124817, 4.49449887936160267971747222005, 4.80585981380250097377097045291, 5.00116569502349454267245570883, 5.50920288530660914532469599065, 5.54257868408583480109740703238, 5.67769649781624278022477795787, 5.80782968375471052110239540502, 6.19314583743978987909365963628, 6.29407696114662383008771876202, 6.46705538265789120194214515982
