| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 3·6-s − 10·8-s + 11-s − 6·12-s + 15·16-s − 17-s + 19-s − 3·22-s + 10·24-s + 3·25-s − 21·32-s − 33-s + 3·34-s − 3·38-s + 41-s − 43-s + 6·44-s − 15·48-s + 3·49-s − 9·50-s + 51-s − 57-s + 59-s + 28·64-s + ⋯ |
| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 3·6-s − 10·8-s + 11-s − 6·12-s + 15·16-s − 17-s + 19-s − 3·22-s + 10·24-s + 3·25-s − 21·32-s − 33-s + 3·34-s − 3·38-s + 41-s − 43-s + 6·44-s − 15·48-s + 3·49-s − 9·50-s + 51-s − 57-s + 59-s + 28·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2250591071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2250591071\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
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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
−8.998161009629300678845992802965, −8.552374132242745331936134606213, −8.248827395580906062131398994385, −8.074440035193628999367285644952, −7.62822687003279847760861294482, −7.49620290506170427262453948143, −7.02993990228159782487727427512, −6.82103557523437074653198951012, −6.66851426204752728950055238975, −6.66702078228276421763668550772, −6.06743034957441844393410998142, −5.88887507984258659902066155046, −5.56837472357926365888576415778, −5.23071495286438696624843171875, −4.99713063808816386791871969561, −4.43102333568490005881517382781, −3.90523008275740844800289010599, −3.44407131966947999155178676608, −3.32244613703469756194265771825, −2.61049569801933838006443846531, −2.49146545336076355069241483812, −2.15602446262476721571130859222, −1.46715953389719651997465312122, −0.958699560042509244582022976604, −0.826751479743222131863633548589,
0.826751479743222131863633548589, 0.958699560042509244582022976604, 1.46715953389719651997465312122, 2.15602446262476721571130859222, 2.49146545336076355069241483812, 2.61049569801933838006443846531, 3.32244613703469756194265771825, 3.44407131966947999155178676608, 3.90523008275740844800289010599, 4.43102333568490005881517382781, 4.99713063808816386791871969561, 5.23071495286438696624843171875, 5.56837472357926365888576415778, 5.88887507984258659902066155046, 6.06743034957441844393410998142, 6.66702078228276421763668550772, 6.66851426204752728950055238975, 6.82103557523437074653198951012, 7.02993990228159782487727427512, 7.49620290506170427262453948143, 7.62822687003279847760861294482, 8.074440035193628999367285644952, 8.248827395580906062131398994385, 8.552374132242745331936134606213, 8.998161009629300678845992802965
