| L(s) = 1 | − 2·2-s − 3-s − 5-s + 2·6-s − 4·7-s + 4·8-s − 3·9-s + 2·10-s − 11-s + 2·13-s + 8·14-s + 15-s − 4·16-s − 3·17-s + 6·18-s − 2·19-s + 4·21-s + 2·22-s + 23-s − 4·24-s − 4·26-s + 4·27-s + 8·29-s − 2·30-s − 8·31-s + 33-s + 6·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s − 0.447·5-s + 0.816·6-s − 1.51·7-s + 1.41·8-s − 9-s + 0.632·10-s − 0.301·11-s + 0.554·13-s + 2.13·14-s + 0.258·15-s − 16-s − 0.727·17-s + 1.41·18-s − 0.458·19-s + 0.872·21-s + 0.426·22-s + 0.208·23-s − 0.816·24-s − 0.784·26-s + 0.769·27-s + 1.48·29-s − 0.365·30-s − 1.43·31-s + 0.174·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2059 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2059 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 9 T + p T^{2} ) \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 9 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 49 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_4$ | \( 1 + 6 T + 106 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 154 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 20 T + 232 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−19.0990465480, −18.3855396893, −18.0384025584, −17.6273899031, −16.9042794007, −16.6733137515, −16.1180000920, −15.6093557259, −14.8705576560, −14.0882470176, −13.3136900402, −13.1312058945, −12.2330087092, −11.6465847000, −10.7930841746, −10.4759483637, −9.70013483660, −9.10419218973, −8.60655478909, −8.11172877045, −7.03690977138, −6.36033287239, −5.49400412471, −4.34100873259, −3.12326742958, 0,
3.12326742958, 4.34100873259, 5.49400412471, 6.36033287239, 7.03690977138, 8.11172877045, 8.60655478909, 9.10419218973, 9.70013483660, 10.4759483637, 10.7930841746, 11.6465847000, 12.2330087092, 13.1312058945, 13.3136900402, 14.0882470176, 14.8705576560, 15.6093557259, 16.1180000920, 16.6733137515, 16.9042794007, 17.6273899031, 18.0384025584, 18.3855396893, 19.0990465480
