| L(s) = 1 | − 2-s + 5-s + 7-s + 3·9-s − 10-s + 11-s − 13-s − 14-s − 3·17-s − 3·18-s − 22-s − 3·23-s + 26-s + 3·34-s + 35-s + 37-s + 3·45-s + 3·46-s − 47-s + 55-s − 59-s + 61-s + 3·63-s − 65-s − 70-s − 74-s + 77-s + ⋯ |
| L(s) = 1 | − 2-s + 5-s + 7-s + 3·9-s − 10-s + 11-s − 13-s − 14-s − 3·17-s − 3·18-s − 22-s − 3·23-s + 26-s + 3·34-s + 35-s + 37-s + 3·45-s + 3·46-s − 47-s + 55-s − 59-s + 61-s + 3·63-s − 65-s − 70-s − 74-s + 77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59776471 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59776471 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4276503346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4276503346\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 5 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 7 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 11 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 13 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $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 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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
−10.29080243678577801084215712339, −9.820797783529460321261542445946, −9.723452749814873235128155562424, −9.459913253745065435387604747644, −9.202321744849756295407235622819, −9.017403375426438783937988662065, −8.520113650101730533034693135993, −7.997054632133074710551133335153, −7.991168463386503424101149563279, −7.64475662497593167334294664987, −7.06889463628623973309689148608, −6.87466912499423897226527556522, −6.60952238255392947715760871793, −6.16435018389957524038800258969, −6.13427020024295517036538102459, −5.24015446652141393155905036262, −4.98020646459742582694584103381, −4.45498838752007386513925043254, −4.33529886376829021432355802480, −4.04712605788126764901957483751, −3.70046826995720298863453249682, −2.42824587641723525380590562180, −2.21946831534787701453424689715, −1.71998419396861055596451023158, −1.49727035299391329052135149753,
1.49727035299391329052135149753, 1.71998419396861055596451023158, 2.21946831534787701453424689715, 2.42824587641723525380590562180, 3.70046826995720298863453249682, 4.04712605788126764901957483751, 4.33529886376829021432355802480, 4.45498838752007386513925043254, 4.98020646459742582694584103381, 5.24015446652141393155905036262, 6.13427020024295517036538102459, 6.16435018389957524038800258969, 6.60952238255392947715760871793, 6.87466912499423897226527556522, 7.06889463628623973309689148608, 7.64475662497593167334294664987, 7.991168463386503424101149563279, 7.997054632133074710551133335153, 8.520113650101730533034693135993, 9.017403375426438783937988662065, 9.202321744849756295407235622819, 9.459913253745065435387604747644, 9.723452749814873235128155562424, 9.820797783529460321261542445946, 10.29080243678577801084215712339
