| L(s) = 1 | + 2-s − 3-s − 4-s + 4·5-s − 6-s − 8-s + 4·10-s − 11-s + 12-s − 2·13-s − 4·15-s + 3·16-s + 7·17-s − 8·19-s − 4·20-s − 22-s + 24-s + 6·25-s − 2·26-s + 4·27-s − 29-s − 4·30-s − 5·31-s + 3·32-s + 33-s + 7·34-s − 7·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.353·8-s + 1.26·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s − 1.03·15-s + 3/4·16-s + 1.69·17-s − 1.83·19-s − 0.894·20-s − 0.213·22-s + 0.204·24-s + 6/5·25-s − 0.392·26-s + 0.769·27-s − 0.185·29-s − 0.730·30-s − 0.898·31-s + 0.530·32-s + 0.174·33-s + 1.20·34-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10086 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10086 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281450889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281450889\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 28 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 7 T + 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 98 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−16.7408679427, −16.3415382783, −15.4198919756, −14.9026851057, −14.3949324443, −13.9519132262, −13.8517088646, −12.9039524074, −12.7037105207, −12.4327609010, −11.6394008650, −10.7198567673, −10.3718649863, −10.0591314349, −9.28667992422, −8.92555853520, −8.01003263683, −7.37161745717, −6.36769582856, −5.93651001827, −5.42321709445, −4.93452160796, −4.07214686236, −2.95163711256, −1.78129448386,
1.78129448386, 2.95163711256, 4.07214686236, 4.93452160796, 5.42321709445, 5.93651001827, 6.36769582856, 7.37161745717, 8.01003263683, 8.92555853520, 9.28667992422, 10.0591314349, 10.3718649863, 10.7198567673, 11.6394008650, 12.4327609010, 12.7037105207, 12.9039524074, 13.8517088646, 13.9519132262, 14.3949324443, 14.9026851057, 15.4198919756, 16.3415382783, 16.7408679427
