| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 2·5-s + 12·6-s − 2·7-s − 3·8-s + 7·9-s + 6·10-s − 8·11-s − 16·12-s − 5·13-s + 6·14-s + 8·15-s + 3·16-s − 2·17-s − 21·18-s − 4·19-s − 8·20-s + 8·21-s + 24·22-s − 2·23-s + 12·24-s + 2·25-s + 15·26-s − 4·27-s − 8·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 0.894·5-s + 4.89·6-s − 0.755·7-s − 1.06·8-s + 7/3·9-s + 1.89·10-s − 2.41·11-s − 4.61·12-s − 1.38·13-s + 1.60·14-s + 2.06·15-s + 3/4·16-s − 0.485·17-s − 4.94·18-s − 0.917·19-s − 1.78·20-s + 1.74·21-s + 5.11·22-s − 0.417·23-s + 2.44·24-s + 2/5·25-s + 2.94·26-s − 0.769·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5449 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5449 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 14 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 16 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T - 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 68 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 39 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 98 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 52 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 87 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 112 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.8174132959, −17.5225692673, −17.0389190593, −16.5940744547, −16.2501629863, −15.6383724508, −15.5038469990, −14.6459506624, −13.5042266013, −12.7036105378, −12.5472785000, −11.8310912786, −11.4055402129, −10.7109348683, −10.4029348586, −10.0448078551, −9.46628115447, −8.39315076080, −8.04613822747, −7.48970350066, −6.60831888901, −6.14323516324, −5.07700679836, −4.76401901792, −2.80558811221, 0, 0,
2.80558811221, 4.76401901792, 5.07700679836, 6.14323516324, 6.60831888901, 7.48970350066, 8.04613822747, 8.39315076080, 9.46628115447, 10.0448078551, 10.4029348586, 10.7109348683, 11.4055402129, 11.8310912786, 12.5472785000, 12.7036105378, 13.5042266013, 14.6459506624, 15.5038469990, 15.6383724508, 16.2501629863, 16.5940744547, 17.0389190593, 17.5225692673, 17.8174132959
