| 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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3722386528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3722386528\) |
| \(L(1)\) |
|
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$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.56347084902920474100059323560, −7.43138048122246676517233177101, −7.30561533657428930333882431848, −7.00760170215587445860040121859, −6.74817660930024033205749713351, −6.74519411853016298074262981156, −6.36246232372965394144168797952, −6.09539013505025997470713816034, −5.85882397546256010053620974521, −5.85192143840969066843830563583, −5.22201094020313486186815448491, −4.90063392364097198514084202757, −4.55728651411710535468314231644, −4.47613194258610712613781110086, −4.06029751534957778611113359193, −3.92263838188277592935385883376, −3.46684174023696029298636198034, −3.34712125717025835726075998281, −3.14514879464342040078017039586, −2.45324313751879379526484172041, −2.39952796095851266681977725277, −2.01668785135785887605871052522, −1.54385720485636178687101007795, −1.15900893831949460014946500972, −0.28824053321103833858328876944,
0.28824053321103833858328876944, 1.15900893831949460014946500972, 1.54385720485636178687101007795, 2.01668785135785887605871052522, 2.39952796095851266681977725277, 2.45324313751879379526484172041, 3.14514879464342040078017039586, 3.34712125717025835726075998281, 3.46684174023696029298636198034, 3.92263838188277592935385883376, 4.06029751534957778611113359193, 4.47613194258610712613781110086, 4.55728651411710535468314231644, 4.90063392364097198514084202757, 5.22201094020313486186815448491, 5.85192143840969066843830563583, 5.85882397546256010053620974521, 6.09539013505025997470713816034, 6.36246232372965394144168797952, 6.74519411853016298074262981156, 6.74817660930024033205749713351, 7.00760170215587445860040121859, 7.30561533657428930333882431848, 7.43138048122246676517233177101, 7.56347084902920474100059323560
