| L(s) = 1 | − 3·2-s + 6·4-s − 10·8-s + 15·16-s + 3·29-s − 21·32-s − 9·58-s + 3·59-s + 28·64-s − 3·79-s − 3·103-s + 18·116-s − 9·118-s + 125-s + 127-s − 36·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 9·158-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
| L(s) = 1 | − 3·2-s + 6·4-s − 10·8-s + 15·16-s + 3·29-s − 21·32-s − 9·58-s + 3·59-s + 28·64-s − 3·79-s − 3·103-s + 18·116-s − 9·118-s + 125-s + 127-s − 36·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 9·158-s + 163-s + 167-s + 3·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{15}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3155501534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3155501534\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{3} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 7 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 11 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{3} \) |
| 31 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 83 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 97 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.567829111662892568415076272962, −8.093605303178597732526453541498, −7.941498288918585018109255080033, −7.88036236676164309306504612920, −7.23260275003179863655042046979, −7.08280621041487644546692938243, −7.01629530945912672160229337612, −6.61663812960613489011247666488, −6.45779747300551218651224450547, −6.28154377639610931461103043614, −5.63999987653063032774817052066, −5.55825936805979295729179718378, −5.50643999088193473666130993833, −4.84061598291919795815450810712, −4.37993309650858139770483304520, −4.16858317580071414449040969850, −3.52290927241259923646516597637, −3.33178975813547499083786461403, −2.87859354304462526185304447061, −2.64685575154973394836888047024, −2.36727086647577685208815237210, −2.01224006221501520517532287844, −1.29431574017196265416246133313, −1.26880964580096207533413469356, −0.62263320687505932692759918555,
0.62263320687505932692759918555, 1.26880964580096207533413469356, 1.29431574017196265416246133313, 2.01224006221501520517532287844, 2.36727086647577685208815237210, 2.64685575154973394836888047024, 2.87859354304462526185304447061, 3.33178975813547499083786461403, 3.52290927241259923646516597637, 4.16858317580071414449040969850, 4.37993309650858139770483304520, 4.84061598291919795815450810712, 5.50643999088193473666130993833, 5.55825936805979295729179718378, 5.63999987653063032774817052066, 6.28154377639610931461103043614, 6.45779747300551218651224450547, 6.61663812960613489011247666488, 7.01629530945912672160229337612, 7.08280621041487644546692938243, 7.23260275003179863655042046979, 7.88036236676164309306504612920, 7.941498288918585018109255080033, 8.093605303178597732526453541498, 8.567829111662892568415076272962
