| L(s) = 1 | + 3·2-s − 3·3-s + 6·4-s + 5-s − 9·6-s + 9·7-s + 10·8-s + 6·9-s + 3·10-s + 5·11-s − 18·12-s + 27·14-s − 3·15-s + 15·16-s − 8·17-s + 18·18-s − 4·19-s + 6·20-s − 27·21-s + 15·22-s − 30·24-s + 2·25-s − 10·27-s + 54·28-s + 11·29-s − 9·30-s + 5·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.73·3-s + 3·4-s + 0.447·5-s − 3.67·6-s + 3.40·7-s + 3.53·8-s + 2·9-s + 0.948·10-s + 1.50·11-s − 5.19·12-s + 7.21·14-s − 0.774·15-s + 15/4·16-s − 1.94·17-s + 4.24·18-s − 0.917·19-s + 1.34·20-s − 5.89·21-s + 3.19·22-s − 6.12·24-s + 2/5·25-s − 1.92·27-s + 10.2·28-s + 2.04·29-s − 1.64·30-s + 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.00797794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.00797794\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 - T - T^{2} + 19 T^{3} - p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 - 9 T + 41 T^{2} - 125 T^{3} + 41 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 5 T + 25 T^{2} - 111 T^{3} + 25 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 8 T + 63 T^{2} + 264 T^{3} + 63 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 88 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 41 T^{2} + 56 T^{3} + 41 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 11 T + 111 T^{2} - 21 p T^{3} + 111 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 5 T + 43 T^{2} - 185 T^{3} + 43 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 8 T + 67 T^{2} + 584 T^{3} + 67 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 2 T + 59 T^{2} + 68 T^{3} + 59 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 12 T + 149 T^{2} - 928 T^{3} + 149 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 - 4 T + 109 T^{2} - 312 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 5 T + 123 T^{2} - 573 T^{3} + 123 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 5 T + 141 T^{2} + 423 T^{3} + 141 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 - 22 T + 335 T^{2} - 3012 T^{3} + 335 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 6 T + 185 T^{2} - 700 T^{3} + 185 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 18 T + 293 T^{2} + 2548 T^{3} + 293 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 + 13 T + 189 T^{2} + 1885 T^{3} + 189 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 - 31 T + 513 T^{2} - 5431 T^{3} + 513 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 13 T + 121 T^{2} - 591 T^{3} + 121 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 14 T + 211 T^{2} + 2436 T^{3} + 211 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 23 T + 381 T^{2} + 47 p T^{3} + 381 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.825392026924166280406903100099, −8.426917069270007905553237765455, −8.223431750201238654576153016522, −7.941379992296977425843908756712, −7.60187006403622546574433194017, −7.23108407478501189576114712665, −6.81378051354124590363795940283, −6.61029116663737738141223926140, −6.52324786766301180099130336963, −6.33295178599944839729793915694, −5.79597003784574618625920826059, −5.34080870550437940206385632237, −5.31684280651627511901195199074, −5.10726397891155128952839450307, −4.63013126634936077953335818758, −4.56041111266473658755362544040, −4.11821172036526101207763772406, −3.99184081576424569928344067595, −3.95263311549670629431967014927, −2.75691521100841403669945068711, −2.47826276801074121661477756091, −2.20576365014078648635754951857, −1.60127985088511496131363056257, −1.34806012139143938612331236626, −1.00170322081288417021166830514,
1.00170322081288417021166830514, 1.34806012139143938612331236626, 1.60127985088511496131363056257, 2.20576365014078648635754951857, 2.47826276801074121661477756091, 2.75691521100841403669945068711, 3.95263311549670629431967014927, 3.99184081576424569928344067595, 4.11821172036526101207763772406, 4.56041111266473658755362544040, 4.63013126634936077953335818758, 5.10726397891155128952839450307, 5.31684280651627511901195199074, 5.34080870550437940206385632237, 5.79597003784574618625920826059, 6.33295178599944839729793915694, 6.52324786766301180099130336963, 6.61029116663737738141223926140, 6.81378051354124590363795940283, 7.23108407478501189576114712665, 7.60187006403622546574433194017, 7.941379992296977425843908756712, 8.223431750201238654576153016522, 8.426917069270007905553237765455, 8.825392026924166280406903100099
