| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 5·5-s + 12·6-s − 2·7-s − 3·8-s + 7·9-s + 15·10-s − 4·11-s − 16·12-s − 2·13-s + 6·14-s + 20·15-s + 3·16-s − 4·17-s − 21·18-s + 2·19-s − 20·20-s + 8·21-s + 12·22-s + 12·24-s + 10·25-s + 6·26-s − 4·27-s − 8·28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 2.23·5-s + 4.89·6-s − 0.755·7-s − 1.06·8-s + 7/3·9-s + 4.74·10-s − 1.20·11-s − 4.61·12-s − 0.554·13-s + 1.60·14-s + 5.16·15-s + 3/4·16-s − 0.970·17-s − 4.94·18-s + 0.458·19-s − 4.47·20-s + 1.74·21-s + 2.55·22-s + 2.44·24-s + 2·25-s + 1.17·26-s − 0.769·27-s − 1.51·28-s − 0.371·29-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$ | \( ( 1 + T )( 1 + 80 T + p T^{2} ) \) |
| 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 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + 4 T + p T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 52 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 15 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 106 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 94 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 83 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 96 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−18.4591181114, −17.9636956494, −17.6360205721, −17.1089016924, −16.5986459687, −16.2677920166, −15.8858719243, −15.3939816337, −14.9662294560, −13.5512906784, −12.7583925178, −12.1772653521, −11.8078640393, −11.3709692996, −10.8618791166, −10.3158196872, −9.93867452538, −8.86453502441, −8.43021839996, −7.60176719027, −7.24964560736, −6.51129535267, −5.47447461279, −4.79816559228, −3.50770790118, 0, 0,
3.50770790118, 4.79816559228, 5.47447461279, 6.51129535267, 7.24964560736, 7.60176719027, 8.43021839996, 8.86453502441, 9.93867452538, 10.3158196872, 10.8618791166, 11.3709692996, 11.8078640393, 12.1772653521, 12.7583925178, 13.5512906784, 14.9662294560, 15.3939816337, 15.8858719243, 16.2677920166, 16.5986459687, 17.1089016924, 17.6360205721, 17.9636956494, 18.4591181114
