L(s) = 1 | + 2-s + 1.52·3-s + 4-s + 4.16·5-s + 1.52·6-s + 7-s + 8-s − 0.688·9-s + 4.16·10-s + 4.64·11-s + 1.52·12-s + 14-s + 6.33·15-s + 16-s − 6.33·17-s − 0.688·18-s − 2.16·19-s + 4.16·20-s + 1.52·21-s + 4.64·22-s − 3.68·23-s + 1.52·24-s + 12.3·25-s − 5.60·27-s + 28-s + 5.04·29-s + 6.33·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.877·3-s + 0.5·4-s + 1.86·5-s + 0.620·6-s + 0.377·7-s + 0.353·8-s − 0.229·9-s + 1.31·10-s + 1.40·11-s + 0.438·12-s + 0.267·14-s + 1.63·15-s + 0.250·16-s − 1.53·17-s − 0.162·18-s − 0.497·19-s + 0.932·20-s + 0.331·21-s + 0.991·22-s − 0.769·23-s + 0.310·24-s + 2.47·25-s − 1.07·27-s + 0.188·28-s + 0.935·29-s + 1.15·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.416965040\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.416965040\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 1.52T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 17 | \( 1 + 6.33T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 + 3.68T + 23T^{2} \) |
| 29 | \( 1 - 5.04T + 29T^{2} \) |
| 31 | \( 1 + 4.64T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 + 9.68T + 41T^{2} \) |
| 43 | \( 1 - 4.33T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 - 0.479T + 61T^{2} \) |
| 67 | \( 1 - 0.311T + 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 + 1.35T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 1.66T + 89T^{2} \) |
| 97 | \( 1 + 2.31T + 97T^{2} \) |
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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
−8.906123619163605899284352327526, −8.552259333490708427860288525877, −7.22818613385843860098231981997, −6.35898262174812305638618170521, −6.03616163689913395473623472538, −4.97576235086074684566455379195, −4.15818412658941366018679918096, −3.07746983479082501983340861249, −2.12892934662501223887555448571, −1.67046892216272626226768580044,
1.67046892216272626226768580044, 2.12892934662501223887555448571, 3.07746983479082501983340861249, 4.15818412658941366018679918096, 4.97576235086074684566455379195, 6.03616163689913395473623472538, 6.35898262174812305638618170521, 7.22818613385843860098231981997, 8.552259333490708427860288525877, 8.906123619163605899284352327526