| L(s) = 1 | − 3·7-s − 8-s + 3·9-s − 3·29-s + 3·49-s + 3·56-s − 9·63-s − 3·72-s + 6·81-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 3·7-s − 8-s + 3·9-s − 3·29-s + 3·49-s + 3·56-s − 9·63-s − 3·72-s + 6·81-s + 3·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7880599 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7880599 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2902650820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2902650820\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 199 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 5 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{3} \) |
| 31 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 47 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 53 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 + T^{3} + T^{6} \) |
| 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
−11.34028791533882735587077167103, −11.14982263747386014025163400753, −10.59589372629902088118790119156, −10.18804188692444243521995129630, −10.00481852592935297828629036968, −9.807306358418560895532580669778, −9.431019404134876275197746852966, −9.205267545952824517272331552822, −9.145348451341343578806795880984, −8.480747175720938371004958115412, −7.76642926199218644397142939619, −7.50153090777636544655316884401, −7.21814306904324494687421207433, −6.85805154745965212337520943726, −6.51107288328753164848987666443, −6.24782618844703468086202733803, −5.96651931317747355755135466100, −5.33939534768575301260241671293, −4.87985667123173284497666272079, −4.10644534216606761505425777664, −3.93541225788366069163597494256, −3.42332742910128185750047059179, −3.22277008544016644974729723724, −2.36866656123741058642027506006, −1.60444496340080251867544954960,
1.60444496340080251867544954960, 2.36866656123741058642027506006, 3.22277008544016644974729723724, 3.42332742910128185750047059179, 3.93541225788366069163597494256, 4.10644534216606761505425777664, 4.87985667123173284497666272079, 5.33939534768575301260241671293, 5.96651931317747355755135466100, 6.24782618844703468086202733803, 6.51107288328753164848987666443, 6.85805154745965212337520943726, 7.21814306904324494687421207433, 7.50153090777636544655316884401, 7.76642926199218644397142939619, 8.480747175720938371004958115412, 9.145348451341343578806795880984, 9.205267545952824517272331552822, 9.431019404134876275197746852966, 9.807306358418560895532580669778, 10.00481852592935297828629036968, 10.18804188692444243521995129630, 10.59589372629902088118790119156, 11.14982263747386014025163400753, 11.34028791533882735587077167103
