| L(s) = 1 | − 2-s + 3·3-s − 2·4-s − 2·5-s − 3·6-s + 3·7-s + 2·8-s + 6·9-s + 2·10-s + 10·11-s − 6·12-s − 3·14-s − 6·15-s + 16-s − 6·18-s + 8·19-s + 4·20-s + 9·21-s − 10·22-s − 2·23-s + 6·24-s − 3·25-s + 10·27-s − 6·28-s − 2·29-s + 6·30-s + 12·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s − 4-s − 0.894·5-s − 1.22·6-s + 1.13·7-s + 0.707·8-s + 2·9-s + 0.632·10-s + 3.01·11-s − 1.73·12-s − 0.801·14-s − 1.54·15-s + 1/4·16-s − 1.41·18-s + 1.83·19-s + 0.894·20-s + 1.96·21-s − 2.13·22-s − 0.417·23-s + 1.22·24-s − 3/5·25-s + 1.92·27-s − 1.13·28-s − 0.371·29-s + 1.09·30-s + 2.15·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.706102119\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.706102119\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T + 3 T^{2} + 3 T^{3} + 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 24 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 10 T + 61 T^{2} - 240 T^{3} + 61 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 272 T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 49 T^{2} + 84 T^{3} + 49 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 76 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 12 T + 77 T^{2} - 424 T^{3} + 77 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 155 T^{2} - 1168 T^{3} + 155 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 16 T + 161 T^{2} - 1248 T^{3} + 161 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 113 T^{2} + 52 T^{3} + 113 p T^{4} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 189 T^{2} - 948 T^{3} + 189 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 11 T^{2} + 684 T^{3} + 11 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 16 T + 265 T^{2} - 2176 T^{3} + 265 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 6 T + 65 T^{2} + 112 T^{3} + 65 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 211 T^{2} - 2128 T^{3} + 211 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 309 T^{2} + 2608 T^{3} + 309 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 77 T^{2} - 76 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 179 T^{2} - 1144 T^{3} + 179 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 275 T^{2} - 16 T^{3} + 275 p T^{4} + p^{3} T^{6} \) |
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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
−7.77413074694215663076686136348, −7.49652599203086958816560430416, −7.33110691521464644351737354070, −7.02897502083393427200363509827, −6.66247257452931973074961644938, −6.42491749936537339322016856389, −6.17159030371173806038187978916, −5.88370425449482791150964214864, −5.57049528485269429409755456389, −5.19067340222787098051356772817, −4.66364959222611504138175330058, −4.60312520983527540253723642406, −4.49348312716926288007413831436, −3.96411502051196030562193003822, −3.89233764457289730215758755295, −3.80466786114123250100870740959, −3.47875472106902916868512218885, −3.02814892907035946544790614141, −2.63449112596429107480113724234, −2.42518786887449764774251610784, −1.89420834345118193446086860771, −1.65121472023420837354246588912, −0.964000324785569416308808834783, −0.932209497665472400080976210716, −0.76364839070747289946839244006,
0.76364839070747289946839244006, 0.932209497665472400080976210716, 0.964000324785569416308808834783, 1.65121472023420837354246588912, 1.89420834345118193446086860771, 2.42518786887449764774251610784, 2.63449112596429107480113724234, 3.02814892907035946544790614141, 3.47875472106902916868512218885, 3.80466786114123250100870740959, 3.89233764457289730215758755295, 3.96411502051196030562193003822, 4.49348312716926288007413831436, 4.60312520983527540253723642406, 4.66364959222611504138175330058, 5.19067340222787098051356772817, 5.57049528485269429409755456389, 5.88370425449482791150964214864, 6.17159030371173806038187978916, 6.42491749936537339322016856389, 6.66247257452931973074961644938, 7.02897502083393427200363509827, 7.33110691521464644351737354070, 7.49652599203086958816560430416, 7.77413074694215663076686136348
