| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s + 8·10-s − 2·11-s − 3·16-s − 8·20-s + 4·22-s + 8·25-s + 4·32-s + 2·41-s − 4·44-s + 4·47-s − 16·50-s + 8·55-s + 2·59-s − 2·64-s + 2·71-s + 12·80-s − 4·82-s + 4·83-s + 2·89-s − 8·94-s + 16·100-s − 16·110-s − 4·118-s + 2·121-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s + 8·10-s − 2·11-s − 3·16-s − 8·20-s + 4·22-s + 8·25-s + 4·32-s + 2·41-s − 4·44-s + 4·47-s − 16·50-s + 8·55-s + 2·59-s − 2·64-s + 2·71-s + 12·80-s − 4·82-s + 4·83-s + 2·89-s − 8·94-s + 16·100-s − 16·110-s − 4·118-s + 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1330557044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1330557044\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−7.03367002451891164136444409618, −6.99422507541652918191920193203, −6.86514578702649269646091878782, −6.71747595334367771324026584177, −6.20659325994628558139793537003, −5.84994662107991887093620573936, −5.65189500314977577782628096744, −5.35234724367793009639864270967, −5.29686357523040008227032431144, −4.83321405960523162610136511806, −4.49944076069812045372961427709, −4.47262492310976724208332708394, −4.39267900832395775358801435636, −3.96832653364552590580840849863, −3.89602411932736342681797380301, −3.49437219657911600204011765774, −3.35961610706580435650370450950, −3.13700057190234228982555463008, −2.54243219611971349729838532073, −2.39672333993709651934274194279, −2.10151424966125618268324803474, −2.00310205356506553350699916122, −0.910976812824809821657976240477, −0.824396872911944168976828545014, −0.56916806197531090722533623677,
0.56916806197531090722533623677, 0.824396872911944168976828545014, 0.910976812824809821657976240477, 2.00310205356506553350699916122, 2.10151424966125618268324803474, 2.39672333993709651934274194279, 2.54243219611971349729838532073, 3.13700057190234228982555463008, 3.35961610706580435650370450950, 3.49437219657911600204011765774, 3.89602411932736342681797380301, 3.96832653364552590580840849863, 4.39267900832395775358801435636, 4.47262492310976724208332708394, 4.49944076069812045372961427709, 4.83321405960523162610136511806, 5.29686357523040008227032431144, 5.35234724367793009639864270967, 5.65189500314977577782628096744, 5.84994662107991887093620573936, 6.20659325994628558139793537003, 6.71747595334367771324026584177, 6.86514578702649269646091878782, 6.99422507541652918191920193203, 7.03367002451891164136444409618
