| L(s) = 1 | + 2·2-s + 4-s − 5-s + 8-s − 2·10-s + 7·11-s − 2·13-s + 3·16-s − 5·19-s − 20-s + 14·22-s + 6·23-s − 8·25-s − 4·26-s + 13·29-s + 8·31-s + 4·32-s − 8·37-s − 10·38-s − 40-s − 2·41-s − 9·43-s + 7·44-s + 12·46-s − 9·47-s − 16·50-s − 2·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.632·10-s + 2.11·11-s − 0.554·13-s + 3/4·16-s − 1.14·19-s − 0.223·20-s + 2.98·22-s + 1.25·23-s − 8/5·25-s − 0.784·26-s + 2.41·29-s + 1.43·31-s + 0.707·32-s − 1.31·37-s − 1.62·38-s − 0.158·40-s − 0.312·41-s − 1.37·43-s + 1.05·44-s + 1.76·46-s − 1.31·47-s − 2.26·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{9} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.146941116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.146941116\) |
| \(L(\frac{3}{2})\) |
|
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 | $S_4\times C_2$ | \( 1 - p T + 3 T^{2} - 5 T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + T + 9 T^{2} + 13 T^{3} + 9 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 45 T^{2} - 157 T^{3} + 45 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 20 T^{2} + 5 T^{3} + 20 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 18 T^{2} + 9 T^{3} + 18 p T^{4} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 53 T^{2} + 161 T^{3} + 53 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 72 T^{2} - 267 T^{3} + 72 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 13 T + 117 T^{2} - 763 T^{3} + 117 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 94 T^{2} - 427 T^{3} + 94 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 106 T^{2} + 499 T^{3} + 106 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 18 T^{2} - 223 T^{3} + 18 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 117 T^{2} + 673 T^{3} + 117 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 99 T^{2} + 855 T^{3} + 99 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 24 T + 324 T^{2} - 2787 T^{3} + 324 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 15 T + 243 T^{2} - 1851 T^{3} + 243 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 134 T^{2} - T^{3} + 134 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 14 T + 254 T^{2} - 1907 T^{3} + 254 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 3 T + 105 T^{2} + 669 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 7 T + 85 T^{2} + 41 T^{3} + 85 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 168 T^{2} - 821 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 3 T + 69 T^{2} + 1227 T^{3} + 69 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 5 T + 174 T^{2} - 401 T^{3} + 174 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 307 T^{2} - 2692 T^{3} + 307 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−8.510972566094292981428204874845, −8.350549895640300295113138998835, −8.221145893715625897739793843526, −7.57957284100525440084382724710, −7.27386790827781380835338059794, −7.19618995858780359176017427665, −6.72826351805603779380876472696, −6.53388726348620445856529681002, −6.30451642111038210770496949945, −6.23206650799529291774439212010, −5.61088362510871379251986271949, −5.16386850030209061018530219915, −5.14364003819429536585102039665, −4.73894802588754734597392581667, −4.40694138984125799620544291856, −4.36256627475879219694345637827, −3.76575394111948690607787978118, −3.63177191814584300706651281048, −3.60221754580290317897134758412, −2.80627788180869429299571593262, −2.63582510607946938269152046297, −1.92347874271809749943724395961, −1.81289808073203684792573398015, −0.946329713497372231954577889354, −0.71654060367547032107368546326,
0.71654060367547032107368546326, 0.946329713497372231954577889354, 1.81289808073203684792573398015, 1.92347874271809749943724395961, 2.63582510607946938269152046297, 2.80627788180869429299571593262, 3.60221754580290317897134758412, 3.63177191814584300706651281048, 3.76575394111948690607787978118, 4.36256627475879219694345637827, 4.40694138984125799620544291856, 4.73894802588754734597392581667, 5.14364003819429536585102039665, 5.16386850030209061018530219915, 5.61088362510871379251986271949, 6.23206650799529291774439212010, 6.30451642111038210770496949945, 6.53388726348620445856529681002, 6.72826351805603779380876472696, 7.19618995858780359176017427665, 7.27386790827781380835338059794, 7.57957284100525440084382724710, 8.221145893715625897739793843526, 8.350549895640300295113138998835, 8.510972566094292981428204874845
