| L(s) = 1 | + 2-s − 2.26·3-s − 4-s + 2.11·5-s − 2.26·6-s + 3.37·7-s − 3·8-s + 2.11·9-s + 2.11·10-s + 11-s + 2.26·12-s + 3.37·14-s − 4.78·15-s − 16-s − 2.14·17-s + 2.11·18-s + 3.14·19-s − 2.11·20-s − 7.63·21-s + 22-s + 4.52·23-s + 6.78·24-s − 0.523·25-s + 2·27-s − 3.37·28-s + 3.49·29-s − 4.78·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.30·3-s − 0.5·4-s + 0.946·5-s − 0.923·6-s + 1.27·7-s − 1.06·8-s + 0.705·9-s + 0.669·10-s + 0.301·11-s + 0.652·12-s + 0.902·14-s − 1.23·15-s − 0.250·16-s − 0.520·17-s + 0.498·18-s + 0.721·19-s − 0.473·20-s − 1.66·21-s + 0.213·22-s + 0.943·23-s + 1.38·24-s − 0.104·25-s + 0.384·27-s − 0.638·28-s + 0.648·29-s − 0.873·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.735382206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735382206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 - 2.11T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 - 3.49T + 29T^{2} \) |
| 31 | \( 1 + 9.27T + 31T^{2} \) |
| 37 | \( 1 + 9.16T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 + 1.96T + 47T^{2} \) |
| 53 | \( 1 - 7.75T + 53T^{2} \) |
| 59 | \( 1 + 6.03T + 59T^{2} \) |
| 61 | \( 1 - 9.78T + 61T^{2} \) |
| 67 | \( 1 - 4.03T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 - 5.66T + 83T^{2} \) |
| 89 | \( 1 - 9.34T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−9.220822859763181901693564849652, −8.649141908575515032063860078028, −7.45170764337181489237579460172, −6.49803109170442741828055835556, −5.72793952772154734029994259383, −5.17950351876856038900869941581, −4.74197369643940032784104159827, −3.63230809694465598780711437982, −2.15229335217131980244979435527, −0.894329385019462081369376727628,
0.894329385019462081369376727628, 2.15229335217131980244979435527, 3.63230809694465598780711437982, 4.74197369643940032784104159827, 5.17950351876856038900869941581, 5.72793952772154734029994259383, 6.49803109170442741828055835556, 7.45170764337181489237579460172, 8.649141908575515032063860078028, 9.220822859763181901693564849652
