| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 5·5-s + 8·6-s − 4·7-s − 4·8-s + 8·9-s + 10·10-s − 11-s − 8·12-s + 8·14-s + 20·15-s + 8·16-s + 4·17-s − 16·18-s − 3·19-s − 10·20-s + 16·21-s + 2·22-s − 9·23-s + 16·24-s + 10·25-s − 12·27-s − 8·28-s − 10·29-s − 40·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 2.23·5-s + 3.26·6-s − 1.51·7-s − 1.41·8-s + 8/3·9-s + 3.16·10-s − 0.301·11-s − 2.30·12-s + 2.13·14-s + 5.16·15-s + 2·16-s + 0.970·17-s − 3.77·18-s − 0.688·19-s − 2.23·20-s + 3.49·21-s + 0.426·22-s − 1.87·23-s + 3.26·24-s + 2·25-s − 2.30·27-s − 1.51·28-s − 1.85·29-s − 7.30·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5641 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5641 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5641 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 70 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 9 T + 53 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 81 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 105 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 19 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 30 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 62 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 20 T + 244 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T - 101 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−17.8716348883, −17.2408286174, −16.7155510117, −16.5208175795, −16.1581989797, −15.6468185365, −15.2114122239, −14.7918784345, −13.4108734249, −12.6366758466, −12.2716210805, −11.9292597246, −11.5334049858, −11.0790179390, −10.4234379986, −9.83438859301, −9.46707132659, −8.34400562979, −7.95561643441, −7.27771121984, −6.55985046799, −5.97259022683, −5.44115532946, −4.04092722488, −3.45992068722, 0, 0,
3.45992068722, 4.04092722488, 5.44115532946, 5.97259022683, 6.55985046799, 7.27771121984, 7.95561643441, 8.34400562979, 9.46707132659, 9.83438859301, 10.4234379986, 11.0790179390, 11.5334049858, 11.9292597246, 12.2716210805, 12.6366758466, 13.4108734249, 14.7918784345, 15.2114122239, 15.6468185365, 16.1581989797, 16.5208175795, 16.7155510117, 17.2408286174, 17.8716348883
