| L(s) = 1 | − 3·5-s − 3·8-s + 6·11-s − 3·13-s − 3·17-s − 6·23-s + 3·29-s − 6·31-s + 15·37-s + 9·40-s + 12·41-s + 12·43-s + 12·53-s − 18·55-s + 18·59-s + 3·61-s + 64-s + 9·65-s + 6·67-s − 9·73-s + 6·79-s − 12·83-s + 9·85-s − 18·88-s − 15·89-s − 12·97-s + 30·101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.06·8-s + 1.80·11-s − 0.832·13-s − 0.727·17-s − 1.25·23-s + 0.557·29-s − 1.07·31-s + 2.46·37-s + 1.42·40-s + 1.87·41-s + 1.82·43-s + 1.64·53-s − 2.42·55-s + 2.34·59-s + 0.384·61-s + 1/8·64-s + 1.11·65-s + 0.733·67-s − 1.05·73-s + 0.675·79-s − 1.31·83-s + 0.976·85-s − 1.91·88-s − 1.58·89-s − 1.21·97-s + 2.98·101-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
| \(L(1)\) |
\(\approx\) |
\(2.300671870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.300671870\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{6}$ | \( 1 + 3 T^{3} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 9 T^{2} + 18 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 126 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 3 T^{2} - 34 T^{3} + 3 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 27 T^{2} + 114 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 9 T^{2} + 56 T^{3} + 9 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 6 T + 45 T^{2} + 180 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} - 210 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 912 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 12 T + 84 T^{2} - 380 T^{3} + 84 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 117 T^{2} + 24 T^{3} + 117 p T^{4} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 12 T + 180 T^{2} - 1266 T^{3} + 180 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 18 T + 189 T^{2} - 1572 T^{3} + 189 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 105 T^{2} - 178 T^{3} + 105 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 132 T^{2} - 542 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 132 T^{2} - 108 T^{3} + 132 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 9 T + 207 T^{2} + 1298 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 168 T^{2} - 686 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 1920 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 15 T + 315 T^{2} + 2622 T^{3} + 315 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 12 T + 3 p T^{2} + 2144 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.63791989372428470898650732794, −7.20119612660951083291530085392, −6.83887476135519576889787086632, −6.81660538940869216590485653531, −6.79120366243504666352524856966, −6.06852965854962000716034617599, −5.97888654730344914591318884979, −5.78064480065519281009912952176, −5.64900883202222696796549854087, −5.41400233776457405776372850896, −4.69862633264689876250437642122, −4.48570810638000671095062912820, −4.28435093628036278204316464483, −4.14179415229719923192844129860, −4.05203560258153162797986392410, −3.57185673553874596429523635695, −3.37754718510814837474554083001, −3.01412449285328134068091185504, −2.51348762964771214451820889214, −2.46740812017681228401052255691, −2.00509032390237138761513550216, −1.77554554952848420857948929480, −0.849352283350967378862293237889, −0.819087146485478635884274530385, −0.41081771851745334484665875597,
0.41081771851745334484665875597, 0.819087146485478635884274530385, 0.849352283350967378862293237889, 1.77554554952848420857948929480, 2.00509032390237138761513550216, 2.46740812017681228401052255691, 2.51348762964771214451820889214, 3.01412449285328134068091185504, 3.37754718510814837474554083001, 3.57185673553874596429523635695, 4.05203560258153162797986392410, 4.14179415229719923192844129860, 4.28435093628036278204316464483, 4.48570810638000671095062912820, 4.69862633264689876250437642122, 5.41400233776457405776372850896, 5.64900883202222696796549854087, 5.78064480065519281009912952176, 5.97888654730344914591318884979, 6.06852965854962000716034617599, 6.79120366243504666352524856966, 6.81660538940869216590485653531, 6.83887476135519576889787086632, 7.20119612660951083291530085392, 7.63791989372428470898650732794
