L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s + 2·11-s − 6·13-s + 16-s + 2·17-s + 6·19-s − 20-s + 2·22-s + 4·23-s + 25-s − 6·26-s + 6·29-s + 32-s + 2·34-s − 6·37-s + 6·38-s − 40-s + 4·41-s + 2·44-s + 4·46-s − 4·47-s − 7·49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s + 0.603·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 1.37·19-s − 0.223·20-s + 0.426·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s + 1.11·29-s + 0.176·32-s + 0.342·34-s − 0.986·37-s + 0.973·38-s − 0.158·40-s + 0.624·41-s + 0.301·44-s + 0.589·46-s − 0.583·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.589095480\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.589095480\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.11604707973867, −12.75457600749308, −12.22902011754118, −11.93286171439266, −11.43605020497865, −11.18435744143374, −10.18277439145401, −10.07501520329148, −9.543041505490293, −8.932149890718918, −8.316441558166412, −7.851780227054871, −7.256019004002643, −6.826850913114711, −6.669756139523621, −5.520633687938513, −5.346033302920773, −4.905232995948320, −4.205534859787665, −3.755729995444339, −3.098343247304404, −2.679443845286963, −2.033328825182872, −1.129743096875638, −0.6083480599761229,
0.6083480599761229, 1.129743096875638, 2.033328825182872, 2.679443845286963, 3.098343247304404, 3.755729995444339, 4.205534859787665, 4.905232995948320, 5.346033302920773, 5.520633687938513, 6.669756139523621, 6.826850913114711, 7.256019004002643, 7.851780227054871, 8.316441558166412, 8.932149890718918, 9.543041505490293, 10.07501520329148, 10.18277439145401, 11.18435744143374, 11.43605020497865, 11.93286171439266, 12.22902011754118, 12.75457600749308, 13.11604707973867