| L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s + 7-s − 8-s − 10-s − 2·12-s + 4·13-s + 14-s − 2·15-s + 3·16-s − 5·19-s + 20-s + 2·21-s − 2·24-s − 3·25-s + 4·26-s − 2·27-s − 28-s − 12·29-s − 2·30-s + 3·32-s − 35-s − 5·38-s + 8·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.577·12-s + 1.10·13-s + 0.267·14-s − 0.516·15-s + 3/4·16-s − 1.14·19-s + 0.223·20-s + 0.436·21-s − 0.408·24-s − 3/5·25-s + 0.784·26-s − 0.384·27-s − 0.188·28-s − 2.22·29-s − 0.365·30-s + 0.530·32-s − 0.169·35-s − 0.811·38-s + 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10114 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10114 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.590002896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.590002896\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 5 T + p T^{2} ) \) |
| 389 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 24 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 76 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 16 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 88 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 15 T + 130 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - T + 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 134 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 106 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−16.6978661327, −15.8700771846, −15.1827832860, −15.0812109664, −14.5610262032, −14.0878872347, −13.6244097466, −13.1861001488, −12.9449318691, −12.1093187304, −11.5864508568, −11.0434203614, −10.4142348698, −9.78376953422, −8.97923446282, −8.56405759652, −8.31775623373, −7.58310327153, −6.90200699627, −5.81714205144, −5.45173625502, −4.32214003625, −3.87669171935, −3.26334035199, −2.01685848405,
2.01685848405, 3.26334035199, 3.87669171935, 4.32214003625, 5.45173625502, 5.81714205144, 6.90200699627, 7.58310327153, 8.31775623373, 8.56405759652, 8.97923446282, 9.78376953422, 10.4142348698, 11.0434203614, 11.5864508568, 12.1093187304, 12.9449318691, 13.1861001488, 13.6244097466, 14.0878872347, 14.5610262032, 15.0812109664, 15.1827832860, 15.8700771846, 16.6978661327
