| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 6·5-s + 12·6-s − 6·7-s − 3·8-s + 7·9-s + 18·10-s − 4·11-s − 16·12-s − 5·13-s + 18·14-s + 24·15-s + 3·16-s − 6·17-s − 21·18-s − 5·19-s − 24·20-s + 24·21-s + 12·22-s − 9·23-s + 12·24-s + 18·25-s + 15·26-s − 4·27-s − 24·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 2.68·5-s + 4.89·6-s − 2.26·7-s − 1.06·8-s + 7/3·9-s + 5.69·10-s − 1.20·11-s − 4.61·12-s − 1.38·13-s + 4.81·14-s + 6.19·15-s + 3/4·16-s − 1.45·17-s − 4.94·18-s − 1.14·19-s − 5.36·20-s + 5.23·21-s + 2.55·22-s − 1.87·23-s + 2.44·24-s + 18/5·25-s + 2.94·26-s − 0.769·27-s − 4.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 563011 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 563011 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 563011 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 120 T + p T^{2} ) \) |
| 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 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 11 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 66 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 3 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 79 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 38 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 69 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 83 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T - 29 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 97 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T + 7 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 103 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 165 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 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
−12.9203399885, −12.6820204497, −12.3729611774, −12.0370500305, −11.4847859818, −11.2498694341, −10.9774160576, −10.5368935130, −10.2234915834, −9.74748231154, −9.36526617540, −8.99039043029, −8.35258900020, −7.92032052427, −7.62834656928, −7.22726163633, −6.81095164597, −6.31348391929, −5.96326926962, −5.31994711780, −4.71923759766, −4.14643946300, −3.64222489146, −3.03497440596, −2.00563625231, 0, 0, 0, 0,
2.00563625231, 3.03497440596, 3.64222489146, 4.14643946300, 4.71923759766, 5.31994711780, 5.96326926962, 6.31348391929, 6.81095164597, 7.22726163633, 7.62834656928, 7.92032052427, 8.35258900020, 8.99039043029, 9.36526617540, 9.74748231154, 10.2234915834, 10.5368935130, 10.9774160576, 11.2498694341, 11.4847859818, 12.0370500305, 12.3729611774, 12.6820204497, 12.9203399885
