| L(s) = 1 | + 3-s − 3·5-s + 2·7-s + 3·9-s − 3·15-s − 3·17-s − 16·19-s + 2·21-s + 3·23-s − 2·25-s + 8·27-s − 5·29-s − 6·35-s − 7·41-s − 12·43-s − 9·45-s − 9·47-s − 9·49-s − 3·51-s + 17·53-s − 16·57-s − 9·59-s + 2·61-s + 6·63-s + 18·67-s + 3·69-s − 19·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 0.755·7-s + 9-s − 0.774·15-s − 0.727·17-s − 3.67·19-s + 0.436·21-s + 0.625·23-s − 2/5·25-s + 1.53·27-s − 0.928·29-s − 1.01·35-s − 1.09·41-s − 1.82·43-s − 1.34·45-s − 1.31·47-s − 9/7·49-s − 0.420·51-s + 2.33·53-s − 2.11·57-s − 1.17·59-s + 0.256·61-s + 0.755·63-s + 2.19·67-s + 0.361·69-s − 2.25·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5163018292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5163018292\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 3 T + 11 T^{2} + 24 T^{3} + 54 T^{4} + 24 p T^{5} + 11 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 15 T^{2} + 56 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3 T + 35 T^{2} + 96 T^{3} + 786 T^{4} + 96 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - 19 T^{2} + 66 T^{3} + 24 T^{4} + 66 p T^{5} - 19 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 5 T - 31 T^{2} - 10 T^{3} + 1570 T^{4} - 10 p T^{5} - 31 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 45 T^{2} + 2024 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 39 T^{2} + 2120 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 7 T - 37 T^{2} + 28 T^{3} + 3706 T^{4} + 28 p T^{5} - 37 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 12 T + 55 T^{2} + 36 T^{3} - 120 T^{4} + 36 p T^{5} + 55 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 9 T + 125 T^{2} + 882 T^{3} + 8664 T^{4} + 882 p T^{5} + 125 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 17 T + 119 T^{2} - 1088 T^{3} + 10774 T^{4} - 1088 p T^{5} + 119 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 9 T - 49 T^{2} + 108 T^{3} + 8640 T^{4} + 108 p T^{5} - 49 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 225 T^{2} - 2106 T^{3} + 16436 T^{4} - 2106 p T^{5} + 225 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 19 T + 137 T^{2} + 1558 T^{3} + 19504 T^{4} + 1558 p T^{5} + 137 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^3$ | \( 1 - 113 T^{2} + 7440 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 3 T^{2} - 90 T^{3} - 5068 T^{4} - 90 p T^{5} - 3 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 9 T - 109 T^{2} - 108 T^{3} + 20970 T^{4} - 108 p T^{5} - 109 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 27 T + 473 T^{2} - 6210 T^{3} + 67062 T^{4} - 6210 p T^{5} + 473 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79961272524286964433345407049, −7.77676167283076151248547662878, −7.64475544101979242707318398130, −7.60214248610931069808981348516, −6.83862966856864334618717282795, −6.73280321544057648036614571663, −6.61809709182501694191577846238, −6.60651387671210076274568049346, −6.00181919454474874307956555416, −5.97161491155794860244535191658, −5.25028038751468802472410700363, −5.16493751875721731644196150026, −4.72533244877282792710544312305, −4.66840920298270849449294116441, −4.39522139934633295859352812344, −4.02442529010585267171412503994, −3.87750940412423696688896881007, −3.54493410165920295391710061150, −3.34369411082870113551504688849, −2.79473975621977905721167040647, −2.31551640330033601826994633281, −2.09225977665157019408925212517, −1.75603747263669751300752846849, −1.34631190596867538276927062502, −0.23585980539511422680049855115,
0.23585980539511422680049855115, 1.34631190596867538276927062502, 1.75603747263669751300752846849, 2.09225977665157019408925212517, 2.31551640330033601826994633281, 2.79473975621977905721167040647, 3.34369411082870113551504688849, 3.54493410165920295391710061150, 3.87750940412423696688896881007, 4.02442529010585267171412503994, 4.39522139934633295859352812344, 4.66840920298270849449294116441, 4.72533244877282792710544312305, 5.16493751875721731644196150026, 5.25028038751468802472410700363, 5.97161491155794860244535191658, 6.00181919454474874307956555416, 6.60651387671210076274568049346, 6.61809709182501694191577846238, 6.73280321544057648036614571663, 6.83862966856864334618717282795, 7.60214248610931069808981348516, 7.64475544101979242707318398130, 7.77676167283076151248547662878, 7.79961272524286964433345407049
