| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.488276775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.488276775\) |
| \(L(1)\) |
|
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
−7.88468811283647148399358716241, −7.60363197106569923307441230596, −7.17595549488162523124884531950, −6.87272458937408017483545626413, −6.65731425008005358110448736166, −6.49230580357821437198496591912, −6.45951029978828735781132316588, −6.16265574451822560086975420663, −5.95542350216486602368027016045, −5.75770806525735628612151250926, −5.44481564690281149166402784249, −5.08154688768842691812903680625, −4.34834128436904513283669993090, −4.26266384120556633154248937036, −3.98859776502791976150448711589, −3.79424708824988800139256930321, −3.71078107345942374440007683711, −3.45084751274367364628853951817, −3.02918022749115229452536232868, −2.94784093858683753910129529635, −1.95033934355911737673492020969, −1.78328791965744335308024802379, −1.76573051085906256068597758079, −1.08091573858374526827062071023, −0.64184401535554343450833392252,
0.64184401535554343450833392252, 1.08091573858374526827062071023, 1.76573051085906256068597758079, 1.78328791965744335308024802379, 1.95033934355911737673492020969, 2.94784093858683753910129529635, 3.02918022749115229452536232868, 3.45084751274367364628853951817, 3.71078107345942374440007683711, 3.79424708824988800139256930321, 3.98859776502791976150448711589, 4.26266384120556633154248937036, 4.34834128436904513283669993090, 5.08154688768842691812903680625, 5.44481564690281149166402784249, 5.75770806525735628612151250926, 5.95542350216486602368027016045, 6.16265574451822560086975420663, 6.45951029978828735781132316588, 6.49230580357821437198496591912, 6.65731425008005358110448736166, 6.87272458937408017483545626413, 7.17595549488162523124884531950, 7.60363197106569923307441230596, 7.88468811283647148399358716241
