| L(s) = 1 | − 2·3-s − 3·5-s + 4·7-s − 9-s − 2·11-s + 4·13-s + 6·15-s + 6·17-s + 3·19-s − 8·21-s + 8·23-s + 6·25-s + 2·27-s + 2·29-s + 2·31-s + 4·33-s − 12·35-s + 2·37-s − 8·39-s + 16·41-s − 8·43-s + 3·45-s + 4·47-s − 49-s − 12·51-s + 10·53-s + 6·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 1.51·7-s − 1/3·9-s − 0.603·11-s + 1.10·13-s + 1.54·15-s + 1.45·17-s + 0.688·19-s − 1.74·21-s + 1.66·23-s + 6/5·25-s + 0.384·27-s + 0.371·29-s + 0.359·31-s + 0.696·33-s − 2.02·35-s + 0.328·37-s − 1.28·39-s + 2.49·41-s − 1.21·43-s + 0.447·45-s + 0.583·47-s − 1/7·49-s − 1.68·51-s + 1.37·53-s + 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.103906546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.103906546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 10 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 52 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 25 T^{2} + 32 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 33 T^{2} - 86 T^{3} + 33 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 - 8 T + 81 T^{2} - 356 T^{3} + 81 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 67 T^{2} - 92 T^{3} + 67 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 2 T + 73 T^{2} - 100 T^{3} + 73 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 101 T^{2} - 146 T^{3} + 101 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 16 T + 187 T^{2} - 1360 T^{3} + 187 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 113 T^{2} + 524 T^{3} + 113 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 137 T^{2} - 372 T^{3} + 137 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 181 T^{2} - 1054 T^{3} + 181 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 260 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 59 T^{2} - 8 p T^{3} + 59 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 69 T^{2} - 238 T^{3} + 69 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 26 T + 417 T^{2} - 4148 T^{3} + 417 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 199 T^{2} - 268 T^{3} + 199 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 273 T^{2} + 1996 T^{3} + 273 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 26 T + 471 T^{2} - 5084 T^{3} + 471 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 381 T^{2} - 3574 T^{3} + 381 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.74576552227558984792912644017, −7.47380689408166588316036958665, −7.37764584577594519548853006544, −7.12676371782241235200987184536, −6.56763656430343476918737195647, −6.39328922219487964270446671202, −6.29693301520996344656435156335, −5.64415951115799450045803361617, −5.62331217967078586357642492289, −5.54903913786008379698450055072, −5.05143929559786029579676372545, −4.84481773471542965328947219505, −4.82984003847554994514218816328, −4.20062744463933678767525474755, −4.19325285995919837500300503544, −3.72357075164891452634586260296, −3.32428631479572379754329648161, −3.06802362097335032790066460448, −3.06005996811577949814079734347, −2.39983098248294358194396420678, −1.97747545382210122743039473936, −1.63804827902822657248179707060, −0.867849798057032037100840160828, −0.808138683263191966920336607615, −0.64709512870536653825115727659,
0.64709512870536653825115727659, 0.808138683263191966920336607615, 0.867849798057032037100840160828, 1.63804827902822657248179707060, 1.97747545382210122743039473936, 2.39983098248294358194396420678, 3.06005996811577949814079734347, 3.06802362097335032790066460448, 3.32428631479572379754329648161, 3.72357075164891452634586260296, 4.19325285995919837500300503544, 4.20062744463933678767525474755, 4.82984003847554994514218816328, 4.84481773471542965328947219505, 5.05143929559786029579676372545, 5.54903913786008379698450055072, 5.62331217967078586357642492289, 5.64415951115799450045803361617, 6.29693301520996344656435156335, 6.39328922219487964270446671202, 6.56763656430343476918737195647, 7.12676371782241235200987184536, 7.37764584577594519548853006544, 7.47380689408166588316036958665, 7.74576552227558984792912644017
