| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 4·11-s − 12-s + 2·13-s − 14-s − 15-s + 16-s − 2·17-s + 18-s + 4·19-s + 20-s + 21-s + 4·22-s + 8·23-s − 24-s + 25-s + 2·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.953116217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.953116217\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.16579573434351, −12.58257860606440, −12.04716803806266, −11.78836238942521, −11.22550789187475, −10.90884780325234, −10.15880499227148, −10.03651106057378, −9.183779421700905, −8.847158777961644, −8.534559577327059, −7.418130053455667, −7.104082121268586, −6.829265913048512, −6.063644668208889, −5.915588247120454, −5.297014001285487, −4.668158239904701, −4.355136496155136, −3.533661727643896, −3.236407361163928, −2.579791119859792, −1.688566332667304, −1.282309596860427, −0.6051665484002640,
0.6051665484002640, 1.282309596860427, 1.688566332667304, 2.579791119859792, 3.236407361163928, 3.533661727643896, 4.355136496155136, 4.668158239904701, 5.297014001285487, 5.915588247120454, 6.063644668208889, 6.829265913048512, 7.104082121268586, 7.418130053455667, 8.534559577327059, 8.847158777961644, 9.183779421700905, 10.03651106057378, 10.15880499227148, 10.90884780325234, 11.22550789187475, 11.78836238942521, 12.04716803806266, 12.58257860606440, 13.16579573434351
