| L(s) = 1 | − 2·5-s − 4·7-s + 3·11-s + 4·13-s − 3·17-s − 4·19-s + 8·23-s − 25-s − 4·29-s + 9·31-s + 8·35-s + 3·37-s − 6·41-s − 8·43-s − 15·47-s + 10·49-s − 13·53-s − 6·55-s − 8·59-s + 9·61-s − 8·65-s + 8·71-s + 32·73-s − 12·77-s − 79-s − 13·83-s + 6·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 0.904·11-s + 1.10·13-s − 0.727·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.742·29-s + 1.61·31-s + 1.35·35-s + 0.493·37-s − 0.937·41-s − 1.21·43-s − 2.18·47-s + 10/7·49-s − 1.78·53-s − 0.809·55-s − 1.04·59-s + 1.15·61-s − 0.992·65-s + 0.949·71-s + 3.74·73-s − 1.36·77-s − 0.112·79-s − 1.42·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.492351975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.492351975\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 2 T + p T^{2} + 6 T^{3} + 4 T^{4} + 6 p T^{5} + p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 3 T + 10 T^{2} + 13 T^{3} + 26 T^{4} + 13 p T^{5} + 10 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 3 T + 32 T^{2} + 133 T^{3} + 526 T^{4} + 133 p T^{5} + 32 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 4 T + 31 T^{2} + 68 T^{3} + 440 T^{4} + 68 p T^{5} + 31 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 51 T^{2} - 320 T^{3} + 2040 T^{4} - 320 p T^{5} + 51 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 4 T + 71 T^{2} + 312 T^{3} + 2544 T^{4} + 312 p T^{5} + 71 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 9 T + 92 T^{2} - 613 T^{3} + 3526 T^{4} - 613 p T^{5} + 92 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 3 T + 112 T^{2} - 313 T^{3} + 5566 T^{4} - 313 p T^{5} + 112 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 6 T + 136 T^{2} + 602 T^{3} + 7854 T^{4} + 602 p T^{5} + 136 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 8 T + 131 T^{2} + 800 T^{3} + 8320 T^{4} + 800 p T^{5} + 131 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 15 T + 232 T^{2} + 1931 T^{3} + 16622 T^{4} + 1931 p T^{5} + 232 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 13 T + 70 T^{2} + 463 T^{3} + 4946 T^{4} + 463 p T^{5} + 70 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 8 T + 148 T^{2} + 1064 T^{3} + 12806 T^{4} + 1064 p T^{5} + 148 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 9 T + 14 T^{2} + 197 T^{3} + 322 T^{4} + 197 p T^{5} + 14 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 8 T^{2} - 32 T^{3} + 8158 T^{4} - 32 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 8 T + 196 T^{2} - 1352 T^{3} + 20054 T^{4} - 1352 p T^{5} + 196 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 32 T + 611 T^{2} - 8012 T^{3} + 79456 T^{4} - 8012 p T^{5} + 611 p^{2} T^{6} - 32 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + T + 198 T^{2} - 387 T^{3} + 17970 T^{4} - 387 p T^{5} + 198 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 13 T + 254 T^{2} + 2165 T^{3} + 28530 T^{4} + 2165 p T^{5} + 254 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 7 T + 334 T^{2} + 1657 T^{3} + 43282 T^{4} + 1657 p T^{5} + 334 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 19 T + 318 T^{2} - 4413 T^{3} + 47202 T^{4} - 4413 p T^{5} + 318 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.60370689816635661332419381733, −5.32561875902785677235372442791, −5.24200838789580628420007805784, −5.11338553377693740943539442895, −4.88617392295493595034071342370, −4.53730567782966131173655109930, −4.41007977799196912567937913972, −4.26067725285184497186197222997, −4.24240239874643498232540058496, −3.82939220350716653495954430040, −3.61309332568759226894914227778, −3.40912013770260579253065967071, −3.38306226246001116569496452875, −3.14230627319726528636977767536, −3.02171528193303923173612529326, −2.78728019686838880609735016116, −2.49361350297825209490912722151, −2.09112107651948929183086261254, −1.85344667876622628203397727141, −1.67003482802193018964180309918, −1.65724978154692170193808555028, −1.07612740598429450872268836750, −0.56540492065813242325640668957, −0.53862356822311387795333880186, −0.48470515312216258116657243421,
0.48470515312216258116657243421, 0.53862356822311387795333880186, 0.56540492065813242325640668957, 1.07612740598429450872268836750, 1.65724978154692170193808555028, 1.67003482802193018964180309918, 1.85344667876622628203397727141, 2.09112107651948929183086261254, 2.49361350297825209490912722151, 2.78728019686838880609735016116, 3.02171528193303923173612529326, 3.14230627319726528636977767536, 3.38306226246001116569496452875, 3.40912013770260579253065967071, 3.61309332568759226894914227778, 3.82939220350716653495954430040, 4.24240239874643498232540058496, 4.26067725285184497186197222997, 4.41007977799196912567937913972, 4.53730567782966131173655109930, 4.88617392295493595034071342370, 5.11338553377693740943539442895, 5.24200838789580628420007805784, 5.32561875902785677235372442791, 5.60370689816635661332419381733
