| L(s) = 1 | − 2-s − 4·3-s + 4-s − 5·5-s + 4·6-s − 4·7-s − 8-s + 8·9-s + 5·10-s − 2·11-s − 4·12-s − 5·13-s + 4·14-s + 20·15-s + 16-s − 7·17-s − 8·18-s + 2·19-s − 5·20-s + 16·21-s + 2·22-s + 6·23-s + 4·24-s + 11·25-s + 5·26-s − 12·27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s − 2.23·5-s + 1.63·6-s − 1.51·7-s − 0.353·8-s + 8/3·9-s + 1.58·10-s − 0.603·11-s − 1.15·12-s − 1.38·13-s + 1.06·14-s + 5.16·15-s + 1/4·16-s − 1.69·17-s − 1.88·18-s + 0.458·19-s − 1.11·20-s + 3.49·21-s + 0.426·22-s + 1.25·23-s + 0.816·24-s + 11/5·25-s + 0.980·26-s − 2.30·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 1493 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 2 T - 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 71 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 76 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T - 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 21 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 153 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−16.8742939000, −16.2909875054, −16.0939423190, −15.6623256122, −15.1669749922, −15.0699980177, −13.7036228245, −13.0069209029, −12.4755650927, −12.3102051461, −11.6719239207, −11.3721371091, −10.9598818717, −10.5386058242, −9.81191625983, −9.28636344834, −8.49141999340, −7.65486263538, −7.14399403125, −6.85044312218, −6.20690252400, −5.30242783001, −4.77019325995, −3.92387836743, −2.93883113263, 0, 0,
2.93883113263, 3.92387836743, 4.77019325995, 5.30242783001, 6.20690252400, 6.85044312218, 7.14399403125, 7.65486263538, 8.49141999340, 9.28636344834, 9.81191625983, 10.5386058242, 10.9598818717, 11.3721371091, 11.6719239207, 12.3102051461, 12.4755650927, 13.0069209029, 13.7036228245, 15.0699980177, 15.1669749922, 15.6623256122, 16.0939423190, 16.2909875054, 16.8742939000
