| L(s) = 1 | − 2-s − 3·4-s + 3·5-s − 2·7-s + 4·8-s − 3·10-s − 3·11-s + 2·14-s + 3·16-s + 10·17-s + 6·19-s − 9·20-s + 3·22-s − 5·23-s + 6·25-s + 6·28-s − 29-s − 3·31-s − 6·32-s − 10·34-s − 6·35-s − 5·37-s − 6·38-s + 12·40-s − 22·41-s − 16·43-s + 9·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3/2·4-s + 1.34·5-s − 0.755·7-s + 1.41·8-s − 0.948·10-s − 0.904·11-s + 0.534·14-s + 3/4·16-s + 2.42·17-s + 1.37·19-s − 2.01·20-s + 0.639·22-s − 1.04·23-s + 6/5·25-s + 1.13·28-s − 0.185·29-s − 0.538·31-s − 1.06·32-s − 1.71·34-s − 1.01·35-s − 0.821·37-s − 0.973·38-s + 1.89·40-s − 3.43·41-s − 2.43·43-s + 1.35·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 + T + p^{2} T^{2} + 3 T^{3} + p^{3} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 2 T + 20 T^{2} + 27 T^{3} + 20 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{3} \) |
| 17 | $A_4\times C_2$ | \( 1 - 10 T + 75 T^{2} - 348 T^{3} + 75 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 6 T + 48 T^{2} - 187 T^{3} + 48 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 5 T + 75 T^{2} + 231 T^{3} + 75 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + T + 43 T^{2} - 69 T^{3} + 43 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 3 T + 68 T^{2} + 215 T^{3} + 68 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 5 T + 89 T^{2} + 273 T^{3} + 89 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 22 T + 282 T^{2} + 2181 T^{3} + 282 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 16 T + 212 T^{2} + 1515 T^{3} + 212 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 12 T + 140 T^{2} - 1087 T^{3} + 140 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 5 T + 67 T^{2} + 447 T^{3} + 67 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 89 T^{2} + 23 T^{3} + 89 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 10 T + 102 T^{2} + 787 T^{3} + 102 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 20 T + 318 T^{2} - 2849 T^{3} + 318 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 6 T + 162 T^{2} - 545 T^{3} + 162 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 2 T + 36 T^{2} - 451 T^{3} + 36 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 2 T + 82 T^{2} + 539 T^{3} + 82 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 27 T + 401 T^{2} - 4105 T^{3} + 401 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 4 T + 186 T^{2} - 291 T^{3} + 186 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 9 T + 66 T^{2} - 495 T^{3} + 66 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.57888896113266503125078178847, −7.03481789169272162563647934184, −6.90875668307938392582431820275, −6.47539254565936839592891254847, −6.35426416578736672797753276736, −6.22482485687130175816787021748, −5.97706547212588188090392545911, −5.40999574431198587675399184427, −5.30872215802324084846326312382, −5.21627087256708054364714415820, −4.96533745892528000941714853364, −4.95790247058889792920447027046, −4.79985938722232849010931628765, −3.95075310271566759764650975042, −3.76051361176831871598243034335, −3.69920616155098527062632714554, −3.38760400805042818150113847936, −3.31156340511747248736498561745, −2.79425238308548319785102952387, −2.61570958057102512126135630168, −2.07270122922327625707397822267, −1.96352906260938269054984911867, −1.30647509642092252782956777630, −1.17297888499376732639342492099, −1.17012602857637456355746939806, 0, 0, 0,
1.17012602857637456355746939806, 1.17297888499376732639342492099, 1.30647509642092252782956777630, 1.96352906260938269054984911867, 2.07270122922327625707397822267, 2.61570958057102512126135630168, 2.79425238308548319785102952387, 3.31156340511747248736498561745, 3.38760400805042818150113847936, 3.69920616155098527062632714554, 3.76051361176831871598243034335, 3.95075310271566759764650975042, 4.79985938722232849010931628765, 4.95790247058889792920447027046, 4.96533745892528000941714853364, 5.21627087256708054364714415820, 5.30872215802324084846326312382, 5.40999574431198587675399184427, 5.97706547212588188090392545911, 6.22482485687130175816787021748, 6.35426416578736672797753276736, 6.47539254565936839592891254847, 6.90875668307938392582431820275, 7.03481789169272162563647934184, 7.57888896113266503125078178847
