| L(s) = 1 | − 2·2-s − 2·3-s + 4-s − 2·5-s + 4·6-s − 4·7-s + 3·9-s + 4·10-s − 2·11-s − 2·12-s + 8·14-s + 4·15-s + 16-s − 8·17-s − 6·18-s − 8·19-s − 2·20-s + 8·21-s + 4·22-s − 8·23-s + 3·25-s − 4·27-s − 4·28-s − 4·29-s − 8·30-s + 2·32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s + 1.63·6-s − 1.51·7-s + 9-s + 1.26·10-s − 0.603·11-s − 0.577·12-s + 2.13·14-s + 1.03·15-s + 1/4·16-s − 1.94·17-s − 1.41·18-s − 1.83·19-s − 0.447·20-s + 1.74·21-s + 0.852·22-s − 1.66·23-s + 3/5·25-s − 0.769·27-s − 0.755·28-s − 0.742·29-s − 1.46·30-s + 0.353·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.71356479536445702086775698581, −11.89236308217832953012385481959, −11.40122504862480372315304683882, −11.08616747244285173223687672393, −10.31600728048091969811308172255, −10.24270735616098881995544842593, −9.366705082037274225159796771471, −9.307790568978610657685973125159, −8.380360614730459104025703531425, −8.145492917234143663137211812727, −7.44551346438669193859360856384, −6.64325833235590056459070713525, −6.31227099398064812462775450942, −5.95319055097928357696185137670, −4.60397357310367788518780208858, −4.37694419147338609265442814529, −3.38533677853718646521597180185, −2.19032666796141313339072907284, 0, 0,
2.19032666796141313339072907284, 3.38533677853718646521597180185, 4.37694419147338609265442814529, 4.60397357310367788518780208858, 5.95319055097928357696185137670, 6.31227099398064812462775450942, 6.64325833235590056459070713525, 7.44551346438669193859360856384, 8.145492917234143663137211812727, 8.380360614730459104025703531425, 9.307790568978610657685973125159, 9.366705082037274225159796771471, 10.24270735616098881995544842593, 10.31600728048091969811308172255, 11.08616747244285173223687672393, 11.40122504862480372315304683882, 11.89236308217832953012385481959, 12.71356479536445702086775698581
