| L(s) = 1 | − 2-s + 4-s − 5-s − 1.81·7-s − 8-s + 10-s − 3.81·11-s − 2.73·13-s + 1.81·14-s + 16-s − 0.483·17-s + 2.92·19-s − 20-s + 3.81·22-s + 1.40·23-s + 25-s + 2.73·26-s − 1.81·28-s − 9.47·29-s + 31-s − 32-s + 0.483·34-s + 1.81·35-s − 3.70·37-s − 2.92·38-s + 40-s − 9.47·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.685·7-s − 0.353·8-s + 0.316·10-s − 1.15·11-s − 0.759·13-s + 0.485·14-s + 0.250·16-s − 0.117·17-s + 0.670·19-s − 0.223·20-s + 0.813·22-s + 0.293·23-s + 0.200·25-s + 0.536·26-s − 0.342·28-s − 1.75·29-s + 0.179·31-s − 0.176·32-s + 0.0829·34-s + 0.306·35-s − 0.609·37-s − 0.474·38-s + 0.158·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6429522021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6429522021\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 0.483T + 17T^{2} \) |
| 19 | \( 1 - 2.92T + 19T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 + 9.47T + 29T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 + 9.47T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 8.55T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 8.70T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 + 2.81T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 - 0.154T + 97T^{2} \) |
| 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
−8.887477339303603230377946109135, −7.950126766684677249561687570173, −7.40941630924432473733523177004, −6.83104986364759299673095950441, −5.71147320309174252185903230588, −5.09128807120580688592606078859, −3.85807640909542944040320047866, −2.97934026203930142165974950223, −2.11803709982847350975517260360, −0.52296432586472041904313330071,
0.52296432586472041904313330071, 2.11803709982847350975517260360, 2.97934026203930142165974950223, 3.85807640909542944040320047866, 5.09128807120580688592606078859, 5.71147320309174252185903230588, 6.83104986364759299673095950441, 7.40941630924432473733523177004, 7.950126766684677249561687570173, 8.887477339303603230377946109135
