| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s + 2·7-s − 8-s − 2·9-s − 3·11-s + 2·12-s − 3·13-s + 2·14-s + 3·16-s + 6·17-s − 2·18-s − 4·21-s − 3·22-s + 2·24-s + 2·25-s − 3·26-s + 10·27-s − 2·28-s + 7·29-s − 3·31-s + 3·32-s + 6·33-s + 6·34-s + 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s + 0.755·7-s − 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.577·12-s − 0.832·13-s + 0.534·14-s + 3/4·16-s + 1.45·17-s − 0.471·18-s − 0.872·21-s − 0.639·22-s + 0.408·24-s + 2/5·25-s − 0.588·26-s + 1.92·27-s − 0.377·28-s + 1.29·29-s − 0.538·31-s + 0.530·32-s + 1.04·33-s + 1.02·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1142 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4932578385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4932578385\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 571 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 94 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + T - 110 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 154 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 116 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−19.8751859837, −19.1425885463, −18.5178556846, −17.9983866379, −17.5092790046, −16.9166385811, −16.7239225879, −15.8960248054, −15.0400359948, −14.4910346178, −14.1067522878, −13.5254942900, −12.5564790581, −12.2242510520, −11.7120809661, −10.8683668257, −10.3953025470, −9.63082345343, −8.40432661376, −8.04311525002, −6.90881083430, −5.73048022359, −5.22711295545, −4.79376174186, −3.18351320342,
3.18351320342, 4.79376174186, 5.22711295545, 5.73048022359, 6.90881083430, 8.04311525002, 8.40432661376, 9.63082345343, 10.3953025470, 10.8683668257, 11.7120809661, 12.2242510520, 12.5564790581, 13.5254942900, 14.1067522878, 14.4910346178, 15.0400359948, 15.8960248054, 16.7239225879, 16.9166385811, 17.5092790046, 17.9983866379, 18.5178556846, 19.1425885463, 19.8751859837
