| L(s) = 1 | − 3-s − 4-s − 2·5-s − 3·8-s − 4·9-s − 3·11-s + 12-s − 11·13-s + 2·15-s − 16-s − 3·17-s − 11·19-s + 2·20-s + 12·23-s + 3·24-s − 4·25-s + 6·27-s − 9·29-s − 3·31-s + 6·32-s + 3·33-s + 4·36-s − 4·37-s + 11·39-s + 6·40-s − 5·41-s + 2·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 4/3·9-s − 0.904·11-s + 0.288·12-s − 3.05·13-s + 0.516·15-s − 1/4·16-s − 0.727·17-s − 2.52·19-s + 0.447·20-s + 2.50·23-s + 0.612·24-s − 4/5·25-s + 1.15·27-s − 1.67·29-s − 0.538·31-s + 1.06·32-s + 0.522·33-s + 2/3·36-s − 0.657·37-s + 1.76·39-s + 0.948·40-s − 0.780·41-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T^{2} + 3 T^{3} + p T^{4} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + p T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 8 T^{2} + 3 p T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11 T + 75 T^{2} + 321 T^{3} + 75 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 49 T^{2} + 95 T^{3} + 49 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 11 T + 77 T^{2} + p^{2} T^{3} + 77 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 12 T + 112 T^{2} - 599 T^{3} + 112 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 9 T + 67 T^{2} + 469 T^{3} + 67 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 49 T^{2} + 79 T^{3} + 49 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 75 T^{2} + 144 T^{3} + 75 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 5 T + 43 T^{2} + 301 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 104 T^{2} - 131 T^{3} + 104 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 3 T + 139 T^{2} + 275 T^{3} + 139 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 17 T + 233 T^{2} - 1823 T^{3} + 233 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 20 T^{2} + 379 T^{3} + 20 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 24 T + 355 T^{2} + 3304 T^{3} + 355 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 16 T + 269 T^{2} + 2216 T^{3} + 269 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 7 T + 127 T^{2} - 575 T^{3} + 127 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 244 T^{2} + 2295 T^{3} + 244 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 3 T + 199 T^{2} - 333 T^{3} + 199 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 11 T + 265 T^{2} + 1823 T^{3} + 265 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - T + 259 T^{2} - 175 T^{3} + 259 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 279 T^{2} - 1699 T^{3} + 279 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−10.26658329915847082761187630619, −9.728274928714342853397712350932, −9.370719277056865651776781881010, −9.190855776298553568879598828402, −8.859074389249418760599333235781, −8.622332400215452707349614317324, −8.551807951980438392660205628556, −7.88847781648112056309951457175, −7.74828989340668861098029346259, −7.23390944318713128044117633117, −7.14115808936953489511206754834, −6.88639247848090395940145970180, −6.28043340198195456598623150479, −6.07931724988479289282338665983, −5.64101511325646641247097671437, −5.31710467468675038827123040827, −5.06192667716581458177367709659, −4.71898823530409726432234475291, −4.45016504790278300255613480830, −3.99848172981647169995591228537, −3.55903607239581580550562159406, −2.82082233898832548055628883683, −2.71504829467129263430489083099, −2.47840816480128217762166572978, −1.78477482011343864942782577083, 0, 0, 0,
1.78477482011343864942782577083, 2.47840816480128217762166572978, 2.71504829467129263430489083099, 2.82082233898832548055628883683, 3.55903607239581580550562159406, 3.99848172981647169995591228537, 4.45016504790278300255613480830, 4.71898823530409726432234475291, 5.06192667716581458177367709659, 5.31710467468675038827123040827, 5.64101511325646641247097671437, 6.07931724988479289282338665983, 6.28043340198195456598623150479, 6.88639247848090395940145970180, 7.14115808936953489511206754834, 7.23390944318713128044117633117, 7.74828989340668861098029346259, 7.88847781648112056309951457175, 8.551807951980438392660205628556, 8.622332400215452707349614317324, 8.859074389249418760599333235781, 9.190855776298553568879598828402, 9.370719277056865651776781881010, 9.728274928714342853397712350932, 10.26658329915847082761187630619
