| L(s) = 1 | − 2-s + 3·3-s − 4-s + 5-s − 3·6-s − 3·7-s + 8-s + 6·9-s − 10-s − 3·12-s − 7·13-s + 3·14-s + 3·15-s − 16-s − 17-s − 6·18-s − 8·19-s − 20-s − 9·21-s − 2·23-s + 3·24-s − 7·25-s + 7·26-s + 10·27-s + 3·28-s − 7·29-s − 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 1/2·4-s + 0.447·5-s − 1.22·6-s − 1.13·7-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.866·12-s − 1.94·13-s + 0.801·14-s + 0.774·15-s − 1/4·16-s − 0.242·17-s − 1.41·18-s − 1.83·19-s − 0.223·20-s − 1.96·21-s − 0.417·23-s + 0.612·24-s − 7/5·25-s + 1.37·26-s + 1.92·27-s + 0.566·28-s − 1.29·29-s − 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 8 T^{2} - 11 T^{3} + 8 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 7 T + 51 T^{2} + 184 T^{3} + 51 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + T + 44 T^{2} + 35 T^{3} + 44 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 63 T^{2} + 260 T^{3} + 63 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T - 9 T^{2} - 60 T^{3} - 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 7 T + 99 T^{2} + 408 T^{3} + 99 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 132 T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 3 T + 107 T^{2} + 214 T^{3} + 107 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 13 T + 163 T^{2} + 1098 T^{3} + 163 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 58 T^{2} + 266 T^{3} + 58 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 6 T + 110 T^{2} + 530 T^{3} + 110 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 21 T + 299 T^{2} + 2522 T^{3} + 299 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 622 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 12 T + 165 T^{2} + 1172 T^{3} + 165 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 50 T^{2} - 852 T^{3} + 50 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 179 T^{2} - 76 T^{3} + 179 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 4 T - 7 T^{2} + 1044 T^{3} - 7 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 209 T^{2} + 1636 T^{3} + 209 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 78 T^{2} + 64 T^{3} + 78 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + T + 188 T^{2} + 251 T^{3} + 188 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 25 T + 315 T^{2} + 2968 T^{3} + 315 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−8.317544415945984756744947036357, −8.016032008781402413410157594126, −7.84357476591240631627235259338, −7.70987271634921652506476453225, −7.39078694541142829927976926155, −7.00990565423151717182272202693, −6.71500925605797353984064369715, −6.71248518496429128854046956344, −6.25968231186537424322866069935, −6.16997550867019494882356806177, −5.56274765446038740054610015261, −5.43534203127225120403183062608, −5.06807042973118704688351495002, −4.49496694656401828694853456336, −4.47838945061212022466317424052, −4.47217384393441731806836454440, −3.70438088034905898208980878375, −3.55780207736382274131207699630, −3.21170702143242883469588410698, −3.20286019646154746359058285878, −2.42150673483173309650342284074, −2.35368983806938460490585653915, −1.97724606377807734527300884486, −1.77293359957941054589938912673, −1.36162796415753653709415733358, 0, 0, 0,
1.36162796415753653709415733358, 1.77293359957941054589938912673, 1.97724606377807734527300884486, 2.35368983806938460490585653915, 2.42150673483173309650342284074, 3.20286019646154746359058285878, 3.21170702143242883469588410698, 3.55780207736382274131207699630, 3.70438088034905898208980878375, 4.47217384393441731806836454440, 4.47838945061212022466317424052, 4.49496694656401828694853456336, 5.06807042973118704688351495002, 5.43534203127225120403183062608, 5.56274765446038740054610015261, 6.16997550867019494882356806177, 6.25968231186537424322866069935, 6.71248518496429128854046956344, 6.71500925605797353984064369715, 7.00990565423151717182272202693, 7.39078694541142829927976926155, 7.70987271634921652506476453225, 7.84357476591240631627235259338, 8.016032008781402413410157594126, 8.317544415945984756744947036357
