| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·13-s + 2·14-s + 16-s − 6·17-s + 6·23-s + 2·26-s − 2·28-s + 4·29-s − 32-s + 6·34-s − 10·37-s + 8·41-s − 2·43-s − 6·46-s − 2·47-s − 3·49-s − 2·52-s − 2·53-s + 2·56-s − 4·58-s + 14·61-s + 64-s − 4·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 1.25·23-s + 0.392·26-s − 0.377·28-s + 0.742·29-s − 0.176·32-s + 1.02·34-s − 1.64·37-s + 1.24·41-s − 0.304·43-s − 0.884·46-s − 0.291·47-s − 3/7·49-s − 0.277·52-s − 0.274·53-s + 0.267·56-s − 0.525·58-s + 1.79·61-s + 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5822940479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5822940479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.22920054681147, −12.71399347026878, −12.41360493748835, −11.60304481098924, −11.44876206994834, −10.77691899515348, −10.29642366135921, −10.02352260074546, −9.260680185262224, −9.026817317968174, −8.619035279948606, −7.999029124810822, −7.371666856324210, −6.870638757712823, −6.665243000209890, −6.045365516140104, −5.393858502355934, −4.801469293463051, −4.310509299600308, −3.533875078311316, −2.969791780497106, −2.500119493718730, −1.851706124870551, −1.091032707431199, −0.2684379637223729,
0.2684379637223729, 1.091032707431199, 1.851706124870551, 2.500119493718730, 2.969791780497106, 3.533875078311316, 4.310509299600308, 4.801469293463051, 5.393858502355934, 6.045365516140104, 6.665243000209890, 6.870638757712823, 7.371666856324210, 7.999029124810822, 8.619035279948606, 9.026817317968174, 9.260680185262224, 10.02352260074546, 10.29642366135921, 10.77691899515348, 11.44876206994834, 11.60304481098924, 12.41360493748835, 12.71399347026878, 13.22920054681147
