| L(s) = 1 | + 3·7-s − 8-s + 3·11-s + 3·13-s − 3·23-s + 27-s + 3·43-s + 6·49-s − 3·56-s + 9·77-s − 3·88-s − 6·89-s + 9·91-s − 6·101-s − 3·104-s + 6·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 9·143-s + 149-s + 151-s + 157-s − 9·161-s + 163-s + ⋯ |
| L(s) = 1 | + 3·7-s − 8-s + 3·11-s + 3·13-s − 3·23-s + 27-s + 3·43-s + 6·49-s − 3·56-s + 9·77-s − 3·88-s − 6·89-s + 9·91-s − 6·101-s − 3·104-s + 6·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 9·143-s + 149-s + 151-s + 157-s − 9·161-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 11^{3} \cdot 43^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{3} \cdot 11^{3} \cdot 43^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.560381286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.560381286\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| 43 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 3 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 5 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{3} \) |
| 17 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 29 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_6$ | \( 1 - T^{3} + T^{6} \) |
| 89 | $C_1$ | \( ( 1 + T )^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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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
−8.004882549140158634437511989735, −7.64231818270741851847047515332, −7.24547891395879307438421624710, −7.19426177888150238745669960175, −6.76638392391006655567731245148, −6.52048485066540500825120694662, −6.18624882559631998814532766966, −6.06302169485209141663847961218, −5.72726028587247438138464199581, −5.67807234338798601033004025089, −5.55814748472375137751210012237, −4.87490126767663730093975192361, −4.51534859691384440120815840279, −4.42399214841441578022954060346, −4.03433915692979508637271852040, −3.86781303800271866951312424392, −3.72912371720122813422709302562, −3.70587497738175348849265781882, −2.77207653651579140919739568288, −2.56070836490988475395305869405, −2.29975057206021329908867987123, −1.51729172701338116293685378176, −1.50771114210261946106002010694, −1.18542852239258822122214858277, −1.18326126478213211520431473515,
1.18326126478213211520431473515, 1.18542852239258822122214858277, 1.50771114210261946106002010694, 1.51729172701338116293685378176, 2.29975057206021329908867987123, 2.56070836490988475395305869405, 2.77207653651579140919739568288, 3.70587497738175348849265781882, 3.72912371720122813422709302562, 3.86781303800271866951312424392, 4.03433915692979508637271852040, 4.42399214841441578022954060346, 4.51534859691384440120815840279, 4.87490126767663730093975192361, 5.55814748472375137751210012237, 5.67807234338798601033004025089, 5.72726028587247438138464199581, 6.06302169485209141663847961218, 6.18624882559631998814532766966, 6.52048485066540500825120694662, 6.76638392391006655567731245148, 7.19426177888150238745669960175, 7.24547891395879307438421624710, 7.64231818270741851847047515332, 8.004882549140158634437511989735
