| L(s) = 1 | + 3-s + 3·5-s − 7-s − 4·9-s + 11·13-s + 3·15-s − 3·17-s − 3·19-s − 21-s − 9·23-s + 6·25-s − 5·27-s + 7·29-s + 6·31-s − 3·35-s + 20·37-s + 11·39-s − 22·41-s + 10·43-s − 12·45-s − 4·49-s − 3·51-s + 7·53-s − 3·57-s − 11·59-s + 16·61-s + 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s − 4/3·9-s + 3.05·13-s + 0.774·15-s − 0.727·17-s − 0.688·19-s − 0.218·21-s − 1.87·23-s + 6/5·25-s − 0.962·27-s + 1.29·29-s + 1.07·31-s − 0.507·35-s + 3.28·37-s + 1.76·39-s − 3.43·41-s + 1.52·43-s − 1.78·45-s − 4/7·49-s − 0.420·51-s + 0.961·53-s − 0.397·57-s − 1.43·59-s + 2.04·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.439376122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.439376122\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - T + 5 T^{2} - 4 T^{3} + 5 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + 30 T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 11 T + 55 T^{2} - 200 T^{3} + 55 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 47 T^{2} + 98 T^{3} + 47 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 89 T^{2} + 422 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 43 T^{2} - 114 T^{3} + 43 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 340 T^{3} + 77 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 20 T + 237 T^{2} - 1724 T^{3} + 237 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $D_{6}$ | \( 1 + 22 T + 223 T^{2} + 1572 T^{3} + 223 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 97 T^{2} - 508 T^{3} + 97 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 29 T^{2} - 128 T^{3} + 29 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 7 T - 9 T^{2} + 600 T^{3} - 9 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 11 T + 37 T^{2} - 246 T^{3} + 37 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 16 T + 207 T^{2} - 1600 T^{3} + 207 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - T + 101 T^{2} - 396 T^{3} + 101 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 5 T + 47 T^{2} - 498 T^{3} + 47 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 26 T + 445 T^{2} - 4604 T^{3} + 445 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 14 T + 297 T^{2} - 2340 T^{3} + 297 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} - 188 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 233 T^{2} - 1260 T^{3} + 233 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.01372998176844115583948611385, −6.70888284435640576972737198364, −6.46876268819083560984684182330, −6.32795093286366676878012494361, −6.13509348844305428213235381982, −6.11451256409621543085184820596, −5.90921971615937092889217270074, −5.53761207580578084478902855521, −5.22629499736552280282287453021, −5.11617777935953789516519151917, −4.48230197682382186633679597051, −4.43807133307377021649043973952, −4.27157168323402117886246749196, −3.74430535266617879847806274755, −3.52660438925227212222630476628, −3.46918107391822402028254002480, −2.99389721360434769960444070035, −2.73972352232129884699292131021, −2.54640257611730506235942687583, −2.07320124755585031596456274824, −1.99050770353959184166733177060, −1.68188868295884402923333227636, −1.02798591366807118343159379613, −0.874157850629667606829669262013, −0.45088149046645619948501968141,
0.45088149046645619948501968141, 0.874157850629667606829669262013, 1.02798591366807118343159379613, 1.68188868295884402923333227636, 1.99050770353959184166733177060, 2.07320124755585031596456274824, 2.54640257611730506235942687583, 2.73972352232129884699292131021, 2.99389721360434769960444070035, 3.46918107391822402028254002480, 3.52660438925227212222630476628, 3.74430535266617879847806274755, 4.27157168323402117886246749196, 4.43807133307377021649043973952, 4.48230197682382186633679597051, 5.11617777935953789516519151917, 5.22629499736552280282287453021, 5.53761207580578084478902855521, 5.90921971615937092889217270074, 6.11451256409621543085184820596, 6.13509348844305428213235381982, 6.32795093286366676878012494361, 6.46876268819083560984684182330, 6.70888284435640576972737198364, 7.01372998176844115583948611385
