L(s) = 1 | + 2-s + 4-s − 2.72·5-s − 4.67·7-s + 8-s − 2.72·10-s + 11-s − 2.67·13-s − 4.67·14-s + 16-s − 0.201·17-s − 19-s − 2.72·20-s + 22-s + 1.79·23-s + 2.42·25-s − 2.67·26-s − 4.67·28-s + 4.92·29-s − 2.57·31-s + 32-s − 0.201·34-s + 12.7·35-s + 4.40·37-s − 38-s − 2.72·40-s − 4.97·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.21·5-s − 1.76·7-s + 0.353·8-s − 0.861·10-s + 0.301·11-s − 0.742·13-s − 1.25·14-s + 0.250·16-s − 0.0488·17-s − 0.229·19-s − 0.609·20-s + 0.213·22-s + 0.375·23-s + 0.485·25-s − 0.525·26-s − 0.883·28-s + 0.914·29-s − 0.461·31-s + 0.176·32-s − 0.0345·34-s + 2.15·35-s + 0.723·37-s − 0.162·38-s − 0.430·40-s − 0.776·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.363119758\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.363119758\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 2.72T + 5T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 0.201T + 17T^{2} \) |
| 23 | \( 1 - 1.79T + 23T^{2} \) |
| 29 | \( 1 - 4.92T + 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 - 4.40T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 12.4T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 - 5.24T + 59T^{2} \) |
| 61 | \( 1 - 1.59T + 61T^{2} \) |
| 67 | \( 1 - 9.08T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 2.20T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 5.45T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.436193110009829830651427723442, −7.55091294694503453353010048467, −6.88720221893020516072447734127, −6.44924272599378475146283652795, −5.47714578365147635211939259268, −4.55211498613396365470103615904, −3.76898583652084982082407561951, −3.26004313425429930875772700942, −2.38068652390596721892413141053, −0.57792624091526970490551875298,
0.57792624091526970490551875298, 2.38068652390596721892413141053, 3.26004313425429930875772700942, 3.76898583652084982082407561951, 4.55211498613396365470103615904, 5.47714578365147635211939259268, 6.44924272599378475146283652795, 6.88720221893020516072447734127, 7.55091294694503453353010048467, 8.436193110009829830651427723442