| L(s) = 1 | − 3·2-s + 3-s + 6·4-s + 4·5-s − 3·6-s + 3·7-s − 10·8-s − 9-s − 12·10-s + 6·12-s + 2·13-s − 9·14-s + 4·15-s + 15·16-s − 2·17-s + 3·18-s − 3·19-s + 24·20-s + 3·21-s − 9·23-s − 10·24-s + 15·25-s − 6·26-s − 6·27-s + 18·28-s − 5·29-s − 12·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 3·4-s + 1.78·5-s − 1.22·6-s + 1.13·7-s − 3.53·8-s − 1/3·9-s − 3.79·10-s + 1.73·12-s + 0.554·13-s − 2.40·14-s + 1.03·15-s + 15/4·16-s − 0.485·17-s + 0.707·18-s − 0.688·19-s + 5.36·20-s + 0.654·21-s − 1.87·23-s − 2.04·24-s + 3·25-s − 1.17·26-s − 1.15·27-s + 3.40·28-s − 0.928·29-s − 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.839536751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.839536751\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 2 T^{2} + p T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 T + T^{2} + 14 T^{3} + p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 16 T^{2} - 39 T^{3} + 16 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 60 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T - 5 T^{2} - 128 T^{3} - 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 64 T^{2} + 377 T^{3} + 64 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 5 T + 70 T^{2} + 287 T^{3} + 70 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 19 T + 224 T^{2} - 1607 T^{3} + 224 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 3 p T^{2} - 782 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 8 T + 81 T^{2} + 400 T^{3} + 81 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - T + 124 T^{2} - 73 T^{3} + 124 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 7 T + 42 T^{2} - 101 T^{3} + 42 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 19 T + 186 T^{2} - 1351 T^{3} + 186 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 125 T^{2} - 262 T^{3} + 125 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 93 T^{2} - 470 T^{3} + 93 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 26 T + 427 T^{2} - 4312 T^{3} + 427 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 24 T + 381 T^{2} + 3994 T^{3} + 381 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 6 T + 223 T^{2} + 1014 T^{3} + 223 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 179 T^{2} - 1202 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 2 T + 55 T^{2} + 228 T^{3} + 55 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 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
−7.39597254000487089246211992781, −7.37883874809667829032621936348, −6.77736437917999143664827973798, −6.70760128963967126730816758608, −6.40475607252675987963658635211, −6.31723901047262290583855306603, −6.10428638762583701897771879671, −5.65511702973582029872704008055, −5.52198945959374135977115680069, −5.22815803053823858551366136988, −5.15902613292321038892585479952, −4.52731114959747842396424105177, −4.28947479191086103117819479408, −3.93944802950442954376885331794, −3.75838671573245680812131231816, −3.42172148720626194645784587816, −2.62123234943556521113617980930, −2.58297207814802566209379448715, −2.57215457398253165825447970588, −2.11632631642437874484209671922, −1.96887321197883358656526089477, −1.54533962392715632385638538752, −1.25209992372634408293157581465, −0.71147990964783460043178164173, −0.57992338873482411399320397081,
0.57992338873482411399320397081, 0.71147990964783460043178164173, 1.25209992372634408293157581465, 1.54533962392715632385638538752, 1.96887321197883358656526089477, 2.11632631642437874484209671922, 2.57215457398253165825447970588, 2.58297207814802566209379448715, 2.62123234943556521113617980930, 3.42172148720626194645784587816, 3.75838671573245680812131231816, 3.93944802950442954376885331794, 4.28947479191086103117819479408, 4.52731114959747842396424105177, 5.15902613292321038892585479952, 5.22815803053823858551366136988, 5.52198945959374135977115680069, 5.65511702973582029872704008055, 6.10428638762583701897771879671, 6.31723901047262290583855306603, 6.40475607252675987963658635211, 6.70760128963967126730816758608, 6.77736437917999143664827973798, 7.37883874809667829032621936348, 7.39597254000487089246211992781
