| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.301250417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.301250417\) |
| \(L(1)\) |
|
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
−7.72144353607170784294955099800, −7.54946426416013762055219631359, −7.45960847911963929506956738637, −7.37780448373464619927710713203, −6.86050452637603901286454166791, −6.40499822003154484400151102937, −6.34953698832712705674546156232, −6.09350897342253287192453393859, −5.67593993862809497656307995780, −5.67507716069513400494748451826, −5.20673246037266709569777226224, −4.91803072298919354356013789907, −4.64678473405522221930922235303, −4.41140542420386437908004265698, −4.30240189844941248715088429277, −3.85916138656805277008664074495, −3.84868404567995083274629970542, −3.30674401722641486837972125062, −2.95629712335063861276273404495, −2.13273765115692338867722965639, −2.10622945594196746899664446430, −2.07383442989205793459930963050, −1.91468534582930596214457340518, −0.975203408124054156661525296154, −0.844880053643547088479242091099,
0.844880053643547088479242091099, 0.975203408124054156661525296154, 1.91468534582930596214457340518, 2.07383442989205793459930963050, 2.10622945594196746899664446430, 2.13273765115692338867722965639, 2.95629712335063861276273404495, 3.30674401722641486837972125062, 3.84868404567995083274629970542, 3.85916138656805277008664074495, 4.30240189844941248715088429277, 4.41140542420386437908004265698, 4.64678473405522221930922235303, 4.91803072298919354356013789907, 5.20673246037266709569777226224, 5.67507716069513400494748451826, 5.67593993862809497656307995780, 6.09350897342253287192453393859, 6.34953698832712705674546156232, 6.40499822003154484400151102937, 6.86050452637603901286454166791, 7.37780448373464619927710713203, 7.45960847911963929506956738637, 7.54946426416013762055219631359, 7.72144353607170784294955099800
