| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 2·13-s − 14-s + 16-s − 6·17-s − 4·19-s + 20-s + 23-s + 25-s − 2·26-s + 28-s − 6·29-s − 4·31-s − 32-s + 6·34-s + 35-s + 2·37-s + 4·38-s − 40-s + 6·41-s + 8·43-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.208·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 1.02·34-s + 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s + 1.21·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.59562811619855, −15.78479388150913, −15.43973926130773, −14.64421069078389, −14.39528797565163, −13.45206668289667, −12.97952083150712, −12.62670409921712, −11.53869076975415, −11.17762539117993, −10.81414819301900, −10.08375853087786, −9.414770799118966, −8.897647772735755, −8.458278718265369, −7.756407918977336, −7.026421064435071, −6.500826264717945, −5.854413720116718, −5.173975268977952, −4.275305689513048, −3.682457348381860, −2.480864505261472, −2.078641504430146, −1.145730288354950, 0,
1.145730288354950, 2.078641504430146, 2.480864505261472, 3.682457348381860, 4.275305689513048, 5.173975268977952, 5.854413720116718, 6.500826264717945, 7.026421064435071, 7.756407918977336, 8.458278718265369, 8.897647772735755, 9.414770799118966, 10.08375853087786, 10.81414819301900, 11.17762539117993, 11.53869076975415, 12.62670409921712, 12.97952083150712, 13.45206668289667, 14.39528797565163, 14.64421069078389, 15.43973926130773, 15.78479388150913, 16.59562811619855
