| L(s) = 1 | + 3·2-s + 3·4-s − 3·5-s − 9·10-s + 6·11-s + 3·13-s − 3·16-s − 6·17-s + 3·19-s − 9·20-s + 18·22-s + 12·23-s − 6·25-s + 9·26-s + 9·29-s + 3·31-s − 6·32-s − 18·34-s − 3·37-s + 9·38-s − 3·43-s + 18·44-s + 36·46-s − 3·47-s − 18·50-s + 9·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s − 1.34·5-s − 2.84·10-s + 1.80·11-s + 0.832·13-s − 3/4·16-s − 1.45·17-s + 0.688·19-s − 2.01·20-s + 3.83·22-s + 2.50·23-s − 6/5·25-s + 1.76·26-s + 1.67·29-s + 0.538·31-s − 1.06·32-s − 3.08·34-s − 0.493·37-s + 1.45·38-s − 0.457·43-s + 2.71·44-s + 5.30·46-s − 0.437·47-s − 2.54·50-s + 1.24·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.20136234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.20136234\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $A_4\times C_2$ | \( 1 - 3 T + 3 p T^{2} - 9 T^{3} + 3 p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $A_4\times C_2$ | \( 1 + 3 T + 3 p T^{2} + 27 T^{3} + 3 p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 135 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 29 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 6 T + 60 T^{2} + 207 T^{3} + 60 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 97 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 - 12 T + 96 T^{2} - 549 T^{3} + 96 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 9 T + 51 T^{2} - 189 T^{3} + 51 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 3 T + 15 T^{2} + 137 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 33 T^{2} - 101 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 114 T^{2} + 9 T^{3} + 114 p T^{4} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 259 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 87 T^{2} + 333 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 639 T^{3} + 150 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 405 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 6 T + 168 T^{2} + 713 T^{3} + 168 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 12 T + 222 T^{2} + 1591 T^{3} + 222 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - 9 T + 159 T^{2} - 1305 T^{3} + 159 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 21 T + 303 T^{2} - 2797 T^{3} + 303 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 21 T + 357 T^{2} + 3499 T^{3} + 357 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 18 T + 294 T^{2} - 2997 T^{3} + 294 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 12 T + 204 T^{2} + 1323 T^{3} + 204 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 3 T + 123 T^{2} - 259 T^{3} + 123 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.30394943554587703443441544490, −7.19494929518110329553662179102, −6.91434687199022591079729091594, −6.62139691538858663366397536556, −6.40115179086270890211068944386, −6.22121959168113238585658593585, −6.07424482951053050998266199482, −5.55812349492244493377850539477, −5.36279926417163303112258394961, −5.09698755810419350901288054824, −4.66901922864519360430678073066, −4.65605091171053168362806434722, −4.53173475166533445711632021729, −4.04114773342861518039251386124, −3.91322442388454155614814699402, −3.81767676503564471828293940856, −3.33510686307247030728745468715, −3.14663059564583488423574395184, −3.13877817271932927446041297401, −2.40193234121672161280718253935, −2.20909402578530652458598324714, −1.58602013408515275323041461300, −1.37441126382271694925582723902, −0.70689527052284079729243613953, −0.54815999522225199395145748178,
0.54815999522225199395145748178, 0.70689527052284079729243613953, 1.37441126382271694925582723902, 1.58602013408515275323041461300, 2.20909402578530652458598324714, 2.40193234121672161280718253935, 3.13877817271932927446041297401, 3.14663059564583488423574395184, 3.33510686307247030728745468715, 3.81767676503564471828293940856, 3.91322442388454155614814699402, 4.04114773342861518039251386124, 4.53173475166533445711632021729, 4.65605091171053168362806434722, 4.66901922864519360430678073066, 5.09698755810419350901288054824, 5.36279926417163303112258394961, 5.55812349492244493377850539477, 6.07424482951053050998266199482, 6.22121959168113238585658593585, 6.40115179086270890211068944386, 6.62139691538858663366397536556, 6.91434687199022591079729091594, 7.19494929518110329553662179102, 7.30394943554587703443441544490
