| L(s) = 1 | − 2·3-s − 9-s + 10·11-s − 8·13-s − 6·17-s − 10·19-s + 12·23-s − 9·25-s + 2·27-s + 8·29-s − 4·31-s − 20·33-s + 8·37-s + 16·39-s − 12·43-s − 12·47-s − 2·49-s + 12·51-s − 20·53-s + 20·57-s + 26·59-s − 4·61-s + 6·67-s − 24·69-s + 18·73-s + 18·75-s + 16·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/3·9-s + 3.01·11-s − 2.21·13-s − 1.45·17-s − 2.29·19-s + 2.50·23-s − 9/5·25-s + 0.384·27-s + 1.48·29-s − 0.718·31-s − 3.48·33-s + 1.31·37-s + 2.56·39-s − 1.82·43-s − 1.75·47-s − 2/7·49-s + 1.68·51-s − 2.74·53-s + 2.64·57-s + 3.38·59-s − 0.512·61-s + 0.733·67-s − 2.88·69-s + 2.10·73-s + 2.07·75-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.204812711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204812711\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 16 T^{4} + 10 p T^{5} + 5 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 82 T^{3} + 136 T^{4} - 82 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 446 T^{4} + 120 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 5 T^{2} - 18 T^{3} + 60 T^{4} - 18 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 10 T + 41 T^{2} + 66 T^{3} - 40 T^{4} + 66 p T^{5} + 41 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 105 T^{2} - 684 T^{3} + 3824 T^{4} - 684 p T^{5} + 105 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 248 T^{3} + 1918 T^{4} - 248 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T - 47 T^{2} + 4 T^{3} + 2512 T^{4} + 4 p T^{5} - 47 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 3494 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2702 T^{4} + 444 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 17 T^{2} + 396 T^{3} + 7152 T^{4} + 396 p T^{5} + 17 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 20 T + 101 T^{2} - 1072 T^{3} - 16076 T^{4} - 1072 p T^{5} + 101 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 26 T + 365 T^{2} - 3506 T^{3} + 28624 T^{4} - 3506 p T^{5} + 365 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 624 T^{3} + 1892 T^{4} + 624 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 45 T^{2} + 738 T^{3} - 4576 T^{4} + 738 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 18 T + 277 T^{2} - 3042 T^{3} + 31116 T^{4} - 3042 p T^{5} + 277 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 109 T^{2} + 176 T^{3} - 4856 T^{4} + 176 p T^{5} + 109 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 2180 T^{3} + 23086 T^{4} + 2180 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T + 116 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.03350547330118243044908154345, −6.92871559610792842261685219032, −6.76548197420866309833746218330, −6.65433685474631928332619089069, −6.52847607697598907245280415435, −6.06160001861033763480814449133, −5.86790296144874867529036350531, −5.85465556458220317751978513155, −5.31419106516289071823255936286, −5.26141276203254230786236656837, −4.77404170311886084711592576332, −4.66553081791054139510031579165, −4.38943077000012282917496375662, −4.32911033952160309642897355390, −4.19198380934839356077926241098, −3.42446630115724765894595525836, −3.37878498740333461157152868869, −3.26982626237346963005827087738, −2.73371198431788596126168676553, −2.17275247422227873408778555074, −2.08706131395674319153982679995, −1.87645502493920070182894141666, −1.41736484623952063919892029951, −0.56570053331061323733911949344, −0.48520801386260515329982921837,
0.48520801386260515329982921837, 0.56570053331061323733911949344, 1.41736484623952063919892029951, 1.87645502493920070182894141666, 2.08706131395674319153982679995, 2.17275247422227873408778555074, 2.73371198431788596126168676553, 3.26982626237346963005827087738, 3.37878498740333461157152868869, 3.42446630115724765894595525836, 4.19198380934839356077926241098, 4.32911033952160309642897355390, 4.38943077000012282917496375662, 4.66553081791054139510031579165, 4.77404170311886084711592576332, 5.26141276203254230786236656837, 5.31419106516289071823255936286, 5.85465556458220317751978513155, 5.86790296144874867529036350531, 6.06160001861033763480814449133, 6.52847607697598907245280415435, 6.65433685474631928332619089069, 6.76548197420866309833746218330, 6.92871559610792842261685219032, 7.03350547330118243044908154345
