| L(s) = 1 | − 3·3-s − 3·5-s − 5·7-s + 6·9-s − 3·11-s + 3·13-s + 9·15-s + 17-s − 4·19-s + 15·21-s − 5·23-s + 6·25-s − 10·27-s − 4·29-s − 10·31-s + 9·33-s + 15·35-s + 5·37-s − 9·39-s + 5·41-s − 12·43-s − 18·45-s − 2·47-s + 10·49-s − 3·51-s − 13·53-s + 9·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.34·5-s − 1.88·7-s + 2·9-s − 0.904·11-s + 0.832·13-s + 2.32·15-s + 0.242·17-s − 0.917·19-s + 3.27·21-s − 1.04·23-s + 6/5·25-s − 1.92·27-s − 0.742·29-s − 1.79·31-s + 1.56·33-s + 2.53·35-s + 0.821·37-s − 1.44·39-s + 0.780·41-s − 1.82·43-s − 2.68·45-s − 0.291·47-s + 10/7·49-s − 0.420·51-s − 1.78·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4646469549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4646469549\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $A_4\times C_2$ | \( 1 + 5 T + 15 T^{2} + 38 T^{3} + 15 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_6$ | \( 1 + 3 T - 7 T^{2} - 62 T^{3} - 7 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 - T + 37 T^{2} - 42 T^{3} + 37 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 4 T + 5 T^{2} - 24 T^{3} + 5 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 5 T + 63 T^{2} + 198 T^{3} + 63 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 4 T + 35 T^{2} + 56 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 10 T + 69 T^{2} + 364 T^{3} + 69 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 5 T + 19 T^{2} + 178 T^{3} + 19 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 5 T + 31 T^{2} + 138 T^{3} + 31 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 47 | $A_4\times C_2$ | \( 1 + 2 T + 85 T^{2} + 252 T^{3} + 85 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 13 T + 115 T^{2} + 894 T^{3} + 115 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 61 | $A_4\times C_2$ | \( 1 - 21 T + 287 T^{2} - 2518 T^{3} + 287 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 8 T - 7 T^{2} - 336 T^{3} - 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - T + 113 T^{2} - 494 T^{3} + 113 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 28 T + 423 T^{2} - 4264 T^{3} + 423 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 21 T + 341 T^{2} - 3446 T^{3} + 341 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 1176 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 11 T + 35 T^{2} + 638 T^{3} + 35 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 25 T + 5 p T^{2} - 5322 T^{3} + 5 p^{2} T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.76050180586840933946459683473, −7.39322449623155933998259162736, −7.14983201578287301506150767879, −6.88001119839502555977272393857, −6.48544803677123905548403630156, −6.39370396633059569680846969807, −6.33967162250658748177003662781, −6.02591709384105128700720491627, −5.55073170065137399884792760037, −5.51582617237542203992457431357, −5.15222154151079267290819129672, −4.89884771740758099560448155591, −4.59018588183955064901716931725, −4.15539280589063798942538824902, −4.07202580031588541264983123107, −3.70431914122531738924342712761, −3.30465206974145275501544298790, −3.28655186559322266382989149359, −3.10178363812040514331324532066, −2.14713130507221737127643673172, −1.94602952100095407184738762248, −1.93083545985139733897075638198, −0.74887351469396721889278120839, −0.66172664170434032537120859125, −0.31174588601431505970251412859,
0.31174588601431505970251412859, 0.66172664170434032537120859125, 0.74887351469396721889278120839, 1.93083545985139733897075638198, 1.94602952100095407184738762248, 2.14713130507221737127643673172, 3.10178363812040514331324532066, 3.28655186559322266382989149359, 3.30465206974145275501544298790, 3.70431914122531738924342712761, 4.07202580031588541264983123107, 4.15539280589063798942538824902, 4.59018588183955064901716931725, 4.89884771740758099560448155591, 5.15222154151079267290819129672, 5.51582617237542203992457431357, 5.55073170065137399884792760037, 6.02591709384105128700720491627, 6.33967162250658748177003662781, 6.39370396633059569680846969807, 6.48544803677123905548403630156, 6.88001119839502555977272393857, 7.14983201578287301506150767879, 7.39322449623155933998259162736, 7.76050180586840933946459683473
