| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 2·5-s + 12·6-s − 3·8-s + 7·9-s + 6·10-s − 11-s − 16·12-s − 2·13-s + 8·15-s + 3·16-s − 21·18-s − 7·19-s − 8·20-s + 3·22-s − 5·23-s + 12·24-s + 3·25-s + 6·26-s − 4·27-s + 5·29-s − 24·30-s − 2·31-s − 6·32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 0.894·5-s + 4.89·6-s − 1.06·8-s + 7/3·9-s + 1.89·10-s − 0.301·11-s − 4.61·12-s − 0.554·13-s + 2.06·15-s + 3/4·16-s − 4.94·18-s − 1.60·19-s − 1.78·20-s + 0.639·22-s − 1.04·23-s + 2.44·24-s + 3/5·25-s + 1.17·26-s − 0.769·27-s + 0.928·29-s − 4.38·30-s − 0.359·31-s − 1.06·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 587 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 587 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 587 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 36 T + p T^{2} ) \) |
| 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 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 33 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 20 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 65 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 65 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 60 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 76 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 174 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−19.6352703335, −19.3600266755, −18.5731061875, −18.2805247654, −17.6746848723, −17.2422789540, −16.8856096887, −16.4714245243, −15.7932934253, −15.2149613183, −14.2103430439, −12.7956512685, −12.2195340955, −11.7460692853, −11.0669067939, −10.3891754836, −10.2108998470, −8.97587542682, −8.32048345866, −7.49749287390, −6.57180236549, −5.72995305341, −4.54506016608, 0,
4.54506016608, 5.72995305341, 6.57180236549, 7.49749287390, 8.32048345866, 8.97587542682, 10.2108998470, 10.3891754836, 11.0669067939, 11.7460692853, 12.2195340955, 12.7956512685, 14.2103430439, 15.2149613183, 15.7932934253, 16.4714245243, 16.8856096887, 17.2422789540, 17.6746848723, 18.2805247654, 18.5731061875, 19.3600266755, 19.6352703335
