| L(s) = 1 | + 4·4-s + 13.0·7-s + 9·9-s − 3·11-s + 16·16-s − 30.5·17-s + 19·19-s − 34.8·23-s + 52.3·28-s + 36·36-s − 13.0·43-s − 12·44-s + 56.6·47-s + 122·49-s − 103·61-s + 117.·63-s + 64·64-s − 122.·68-s − 143.·73-s + 76·76-s − 39.2·77-s + 81·81-s + 139.·83-s − 139.·92-s − 27·99-s + 102·101-s + 209.·112-s + ⋯ |
| L(s) = 1 | + 4-s + 1.86·7-s + 9-s − 0.272·11-s + 16-s − 1.79·17-s + 19-s − 1.51·23-s + 1.86·28-s + 36-s − 0.304·43-s − 0.272·44-s + 1.20·47-s + 2.48·49-s − 1.68·61-s + 1.86·63-s + 64-s − 1.79·68-s − 1.97·73-s + 76-s − 0.509·77-s + 81-s + 1.68·83-s − 1.51·92-s − 0.272·99-s + 1.00·101-s + 1.86·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.767651664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.767651664\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 - 4T^{2} \) |
| 3 | \( 1 - 9T^{2} \) |
| 7 | \( 1 - 13.0T + 49T^{2} \) |
| 11 | \( 1 + 3T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 30.5T + 289T^{2} \) |
| 23 | \( 1 + 34.8T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 13.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 56.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 103T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 143.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 139.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
| 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
−10.86648999984355439565740326795, −10.20945267062976497253133349147, −8.883624479836231303768541361869, −7.81209143931222182985716760584, −7.35382058626634152076670602783, −6.19718017463719614393507599470, −5.01031584116395778221351209355, −4.10311856062080438863899358103, −2.31037341491722725181699735169, −1.48141944622674725867777868361,
1.48141944622674725867777868361, 2.31037341491722725181699735169, 4.10311856062080438863899358103, 5.01031584116395778221351209355, 6.19718017463719614393507599470, 7.35382058626634152076670602783, 7.81209143931222182985716760584, 8.883624479836231303768541361869, 10.20945267062976497253133349147, 10.86648999984355439565740326795
