| L(s) = 1 | + 2-s + 4-s − 5-s + 3·7-s + 8-s − 10-s − 3·11-s + 6·13-s + 3·14-s + 16-s − 2·17-s + 8·19-s − 20-s − 3·22-s + 6·23-s + 25-s + 6·26-s + 3·28-s − 8·31-s + 32-s − 2·34-s − 3·35-s + 10·37-s + 8·38-s − 40-s − 2·41-s − 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s + 1.66·13-s + 0.801·14-s + 1/4·16-s − 0.485·17-s + 1.83·19-s − 0.223·20-s − 0.639·22-s + 1.25·23-s + 1/5·25-s + 1.17·26-s + 0.566·28-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.507·35-s + 1.64·37-s + 1.29·38-s − 0.158·40-s − 0.312·41-s − 0.452·44-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
| \(L(1)\) |
\(\approx\) |
\(6.281026790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.281026790\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.21425080628012, −12.92798505485716, −12.32229095894643, −11.69294160515916, −11.33448733233979, −11.00105343238348, −10.73322046671620, −10.06834889352864, −9.235314811812940, −8.937766056451577, −8.309239670821928, −7.857102280577530, −7.346450740795361, −7.117832811485511, −6.226064757984566, −5.697208251521842, −5.338714340314561, −4.827723937798702, −4.261855044689052, −3.754754733544397, −3.130515809860301, −2.695797954396727, −1.857051735318164, −1.225363572142015, −0.7042552453650757,
0.7042552453650757, 1.225363572142015, 1.857051735318164, 2.695797954396727, 3.130515809860301, 3.754754733544397, 4.261855044689052, 4.827723937798702, 5.338714340314561, 5.697208251521842, 6.226064757984566, 7.117832811485511, 7.346450740795361, 7.857102280577530, 8.309239670821928, 8.937766056451577, 9.235314811812940, 10.06834889352864, 10.73322046671620, 11.00105343238348, 11.33448733233979, 11.69294160515916, 12.32229095894643, 12.92798505485716, 13.21425080628012
