L(s) = 1 | − 3·2-s + 3-s + 6·4-s + 5-s − 3·6-s − 10·8-s − 3·10-s + 11-s + 6·12-s + 13-s + 15-s + 15·16-s − 17-s + 6·20-s − 3·22-s − 23-s − 10·24-s − 3·26-s − 3·30-s − 21·32-s + 33-s + 3·34-s + 39-s − 10·40-s + 43-s + 6·44-s + 3·46-s + ⋯ |
L(s) = 1 | − 3·2-s + 3-s + 6·4-s + 5-s − 3·6-s − 10·8-s − 3·10-s + 11-s + 6·12-s + 13-s + 15-s + 15·16-s − 17-s + 6·20-s − 3·22-s − 23-s − 10·24-s − 3·26-s − 3·30-s − 21·32-s + 33-s + 3·34-s + 39-s − 10·40-s + 43-s + 6·44-s + 3·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 67^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 67^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3306896563\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3306896563\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 67 | $C_1$ | \( ( 1 + T )^{3} \) |
good | 3 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 5 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 13 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 53 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 71 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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
−10.01004310592409514211731123294, −9.254848184169428343740878709725, −9.134732906217286538180687485268, −8.981041046042858229249609130945, −8.941668702027069244549406045497, −8.460983270798979522927504731056, −8.387050183722462468831537672236, −7.80772800895855874971029050887, −7.73067948880539300272265776007, −7.16143587950239904710736941497, −7.09852289495463052425338075446, −6.69541207727282465800285058801, −6.29751677501735064867847053492, −5.96562901123342482248701724320, −5.85747932399892133889911789951, −5.63226423934626143401753205157, −4.81108574171467223934865707331, −3.99809265331282871294376617322, −3.83257673977440739533329411334, −3.32176610777350327064755448938, −2.62428330104468251834559643497, −2.53382066237305735866409952275, −2.18569641276333263378569885026, −1.37045074564157809475076615507, −1.36077569376897373138759855754,
1.36077569376897373138759855754, 1.37045074564157809475076615507, 2.18569641276333263378569885026, 2.53382066237305735866409952275, 2.62428330104468251834559643497, 3.32176610777350327064755448938, 3.83257673977440739533329411334, 3.99809265331282871294376617322, 4.81108574171467223934865707331, 5.63226423934626143401753205157, 5.85747932399892133889911789951, 5.96562901123342482248701724320, 6.29751677501735064867847053492, 6.69541207727282465800285058801, 7.09852289495463052425338075446, 7.16143587950239904710736941497, 7.73067948880539300272265776007, 7.80772800895855874971029050887, 8.387050183722462468831537672236, 8.460983270798979522927504731056, 8.941668702027069244549406045497, 8.981041046042858229249609130945, 9.134732906217286538180687485268, 9.254848184169428343740878709725, 10.01004310592409514211731123294