| L(s) = 1 | − 5-s − 12·7-s − 27·11-s + 26·13-s − 20·17-s + 9·19-s + 12·23-s + 36·25-s + 2·29-s + 90·31-s + 12·35-s − 19·37-s + 4·41-s + 84·47-s + 67·49-s − 169·53-s + 27·55-s + 27·59-s + 96·61-s − 26·65-s − 9·67-s − 191·73-s + 324·77-s + 168·79-s + 20·85-s − 122·89-s − 312·91-s + ⋯ |
| L(s) = 1 | − 1/5·5-s − 1.71·7-s − 2.45·11-s + 2·13-s − 1.17·17-s + 9/19·19-s + 0.521·23-s + 1.43·25-s + 2/29·29-s + 2.90·31-s + 0.342·35-s − 0.513·37-s + 4/41·41-s + 1.78·47-s + 1.36·49-s − 3.18·53-s + 0.490·55-s + 0.457·59-s + 1.57·61-s − 2/5·65-s − 0.134·67-s − 2.61·73-s + 4.20·77-s + 2.12·79-s + 4/17·85-s − 1.37·89-s − 3.42·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.671510737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671510737\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 12 T + 11 p T^{2} + 12 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + T - 7 p T^{2} - 14 T^{3} + 646 T^{4} - 14 p^{2} T^{5} - 7 p^{5} T^{6} + p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 27 T + 427 T^{2} + 4968 T^{3} + 48618 T^{4} + 4968 p^{2} T^{5} + 427 p^{4} T^{6} + 27 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - p T + 366 T^{2} - p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 20 T - 50 T^{2} - 2560 T^{3} + 3379 T^{4} - 2560 p^{2} T^{5} - 50 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 9 T - 47 T^{2} + 666 T^{3} - 115098 T^{4} + 666 p^{2} T^{5} - 47 p^{4} T^{6} - 9 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 1042 T^{2} - 11928 T^{3} + 733587 T^{4} - 11928 p^{2} T^{5} + 1042 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - T - 42 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 90 T + 4613 T^{2} - 172170 T^{3} + 5330748 T^{4} - 172170 p^{2} T^{5} + 4613 p^{4} T^{6} - 90 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 19 T - 1769 T^{2} - 11552 T^{3} + 2168530 T^{4} - 11552 p^{2} T^{5} - 1769 p^{4} T^{6} + 19 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 1938 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 3557 T^{2} + 9145308 T^{4} - 3557 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 42 T + 2797 T^{2} - 42 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 169 T + 16159 T^{2} + 21632 p T^{3} + 23386 p^{2} T^{4} + 21632 p^{3} T^{5} + 16159 p^{4} T^{6} + 169 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 27 T + 5893 T^{2} - 152550 T^{3} + 20651022 T^{4} - 152550 p^{2} T^{5} + 5893 p^{4} T^{6} - 27 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 96 T - 302 T^{2} - 199296 T^{3} + 42453747 T^{4} - 199296 p^{2} T^{5} - 302 p^{4} T^{6} - 96 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 9 T + 8627 T^{2} + 77400 T^{3} + 53930082 T^{4} + 77400 p^{2} T^{5} + 8627 p^{4} T^{6} + 9 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4748 T^{2} + 53176038 T^{4} - 4748 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 191 T + 229 p T^{2} + 1739246 T^{3} + 167810206 T^{4} + 1739246 p^{2} T^{5} + 229 p^{5} T^{6} + 191 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 168 T + 22703 T^{2} - 2233560 T^{3} + 196522272 T^{4} - 2233560 p^{2} T^{5} + 22703 p^{4} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1997 T^{2} - 7277460 T^{4} - 1997 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 122 T - 4622 T^{2} + 447008 T^{3} + 199845631 T^{4} + 447008 p^{2} T^{5} - 4622 p^{4} T^{6} + 122 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 25 T + 5280 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.02274979173870723588255494725, −6.56997319626101362057041811617, −6.31088692685024123764862580692, −6.29725517136893115735045843209, −6.28376292714440206306255172660, −5.74345770757908660510683731876, −5.72625933002332810102751692463, −5.34762776285285720877153622908, −5.03088973886365797937512035688, −4.81039814959306023414259282314, −4.66220047854902103990631823827, −4.48035617097246365417654399784, −3.95386778719397762241858413277, −3.94009589128434047444189579023, −3.29643618104553436662377544526, −3.24082905952287138484006197963, −3.19812203989345358908016818307, −2.65801113038504244521776244325, −2.63888156590024498564335513079, −2.36146271910130510294750024157, −1.82398091332492710099866451333, −1.40229817049541041228572021831, −0.806542841082341620147623708508, −0.75950781542192586117831180760, −0.25094212220752666850694508973,
0.25094212220752666850694508973, 0.75950781542192586117831180760, 0.806542841082341620147623708508, 1.40229817049541041228572021831, 1.82398091332492710099866451333, 2.36146271910130510294750024157, 2.63888156590024498564335513079, 2.65801113038504244521776244325, 3.19812203989345358908016818307, 3.24082905952287138484006197963, 3.29643618104553436662377544526, 3.94009589128434047444189579023, 3.95386778719397762241858413277, 4.48035617097246365417654399784, 4.66220047854902103990631823827, 4.81039814959306023414259282314, 5.03088973886365797937512035688, 5.34762776285285720877153622908, 5.72625933002332810102751692463, 5.74345770757908660510683731876, 6.28376292714440206306255172660, 6.29725517136893115735045843209, 6.31088692685024123764862580692, 6.56997319626101362057041811617, 7.02274979173870723588255494725
