| L(s) = 1 | − 2-s − 2·3-s − 5-s + 2·6-s − 4·7-s − 8-s + 9-s + 10-s − 2·11-s + 2·13-s + 4·14-s + 2·15-s − 16-s + 2·17-s − 18-s − 6·19-s + 8·21-s + 2·22-s + 2·24-s + 2·25-s − 2·26-s − 2·27-s + 6·29-s − 2·30-s − 3·31-s + 6·32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 0.447·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.554·13-s + 1.06·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s + 1.74·21-s + 0.426·22-s + 0.408·24-s + 2/5·25-s − 0.392·26-s − 0.384·27-s + 1.11·29-s − 0.365·30-s − 0.538·31-s + 1.06·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3391 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3391 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3391 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 14 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T - 23 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 113 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 103 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T - 33 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 64 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 30 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 137 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17 T + 225 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 72 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−18.2940715580, −17.9453628772, −17.1393060210, −16.7484680036, −16.5067317712, −15.8471816219, −15.3480705143, −15.0015794666, −13.9549331975, −13.2544096789, −12.9204773129, −12.2768159800, −11.7001510758, −11.2571785890, −10.3607365808, −10.1873950777, −9.41200126679, −8.63432815933, −8.25284897950, −7.09876609635, −6.50046423925, −6.02378398721, −5.14386467707, −4.01891649507, −2.91666897434, 0,
2.91666897434, 4.01891649507, 5.14386467707, 6.02378398721, 6.50046423925, 7.09876609635, 8.25284897950, 8.63432815933, 9.41200126679, 10.1873950777, 10.3607365808, 11.2571785890, 11.7001510758, 12.2768159800, 12.9204773129, 13.2544096789, 13.9549331975, 15.0015794666, 15.3480705143, 15.8471816219, 16.5067317712, 16.7484680036, 17.1393060210, 17.9453628772, 18.2940715580
