| L(s) = 1 | + 3-s + 5·7-s − 2·9-s − 4·11-s − 5·13-s + 11·17-s − 3·19-s + 5·21-s + 9·23-s + 3·27-s + 3·29-s − 14·31-s − 4·33-s − 14·37-s − 5·39-s − 10·41-s + 10·43-s + 4·49-s + 11·51-s + 7·53-s − 3·57-s − 7·59-s + 20·61-s − 10·63-s + 67-s + 9·69-s − 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·7-s − 2/3·9-s − 1.20·11-s − 1.38·13-s + 2.66·17-s − 0.688·19-s + 1.09·21-s + 1.87·23-s + 0.577·27-s + 0.557·29-s − 2.51·31-s − 0.696·33-s − 2.30·37-s − 0.800·39-s − 1.56·41-s + 1.52·43-s + 4/7·49-s + 1.54·51-s + 0.961·53-s − 0.397·57-s − 0.911·59-s + 2.56·61-s − 1.25·63-s + 0.122·67-s + 1.08·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.237790759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.237790759\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + p T^{2} - 8 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 5 T + 3 p T^{2} - 62 T^{3} + 3 p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 13 T^{2} + 24 T^{3} + 13 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 5 T + 17 T^{2} + 24 T^{3} + 17 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 11 T + 83 T^{2} - 394 T^{3} + 83 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 9 T + 53 T^{2} - 254 T^{3} + 53 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 66 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 14 T + 125 T^{2} + 804 T^{3} + 125 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 14 T + 157 T^{2} + 1016 T^{3} + 157 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 79 T^{2} + 348 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 41 T^{2} - 12 T^{3} + 41 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 101 T^{2} - 64 T^{3} + 101 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 7 T + 145 T^{2} - 744 T^{3} + 145 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 7 T + 185 T^{2} + 818 T^{3} + 185 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 20 T + 215 T^{2} - 1800 T^{3} + 215 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + p T^{2} - 664 T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 133 T^{2} + 624 T^{3} + 133 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 13 T + 155 T^{2} - 1398 T^{3} + 155 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 181 T^{2} - 2228 T^{3} + 181 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 26 T + 441 T^{2} - 4668 T^{3} + 441 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 335 T^{2} - 3244 T^{3} + 335 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 6 T + 17 T^{2} + 1144 T^{3} + 17 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.71927072508492483065090036461, −7.33124903593944992314985079074, −7.16321268439504405463626874098, −6.85441607374448321700348043734, −6.70127381272525871875159664769, −6.33231692482588315660601259633, −5.88994648411276469391238509900, −5.42666737260096140132659562064, −5.41264075647374756507503211278, −5.23850370033610385845931371571, −5.01497484647096146026870370963, −4.98134014214632521359761689607, −4.56893420444493442136270014630, −4.20473219905867318606850214774, −3.61607731741278097093239322442, −3.61407624687667048868614550962, −3.15260892390985625741248744371, −3.07093843808771447452979352784, −2.75643834604038183501187529542, −2.04414497352865857730549195119, −2.02723540091350256373654228729, −1.97546520479520767758810578305, −1.29002084812376151275765842727, −0.64829723857835012303590239369, −0.62174933975054835731372941632,
0.62174933975054835731372941632, 0.64829723857835012303590239369, 1.29002084812376151275765842727, 1.97546520479520767758810578305, 2.02723540091350256373654228729, 2.04414497352865857730549195119, 2.75643834604038183501187529542, 3.07093843808771447452979352784, 3.15260892390985625741248744371, 3.61407624687667048868614550962, 3.61607731741278097093239322442, 4.20473219905867318606850214774, 4.56893420444493442136270014630, 4.98134014214632521359761689607, 5.01497484647096146026870370963, 5.23850370033610385845931371571, 5.41264075647374756507503211278, 5.42666737260096140132659562064, 5.88994648411276469391238509900, 6.33231692482588315660601259633, 6.70127381272525871875159664769, 6.85441607374448321700348043734, 7.16321268439504405463626874098, 7.33124903593944992314985079074, 7.71927072508492483065090036461
