| L(s) = 1 | − 0.933·2-s + 1.42·3-s − 1.12·4-s + 3.66·5-s − 1.32·6-s + 3.96·7-s + 2.91·8-s − 0.974·9-s − 3.42·10-s − 4.36·11-s − 1.60·12-s − 3.69·14-s + 5.21·15-s − 0.465·16-s + 3.00·17-s + 0.909·18-s + 7.36·19-s − 4.13·20-s + 5.63·21-s + 4.07·22-s + 7.15·23-s + 4.15·24-s + 8.43·25-s − 5.65·27-s − 4.47·28-s − 8.93·29-s − 4.86·30-s + ⋯ |
| L(s) = 1 | − 0.659·2-s + 0.821·3-s − 0.564·4-s + 1.63·5-s − 0.542·6-s + 1.49·7-s + 1.03·8-s − 0.324·9-s − 1.08·10-s − 1.31·11-s − 0.463·12-s − 0.987·14-s + 1.34·15-s − 0.116·16-s + 0.728·17-s + 0.214·18-s + 1.68·19-s − 0.925·20-s + 1.23·21-s + 0.867·22-s + 1.49·23-s + 0.848·24-s + 1.68·25-s − 1.08·27-s − 0.845·28-s − 1.65·29-s − 0.888·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.657745586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.657745586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.933T + 2T^{2} \) |
| 3 | \( 1 - 1.42T + 3T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 17 | \( 1 - 3.00T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 - 7.15T + 23T^{2} \) |
| 29 | \( 1 + 8.93T + 29T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 0.259T + 41T^{2} \) |
| 43 | \( 1 - 0.991T + 43T^{2} \) |
| 47 | \( 1 - 8.92T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 - 8.53T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 + 2.75T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 1.28T + 73T^{2} \) |
| 79 | \( 1 - 4.40T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 3.69T + 89T^{2} \) |
| 97 | \( 1 + 7.38T + 97T^{2} \) |
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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
−8.279216206776973637284122300121, −7.69305902332186631315691751352, −7.23101757492791350295756620335, −5.59349382326196597853681371936, −5.39811986590597317872401966870, −4.84885915048250194311890149585, −3.48516828367850570248283856399, −2.58010119009353675715960097144, −1.81852672715530796878528152829, −1.02123635248196542467181386720,
1.02123635248196542467181386720, 1.81852672715530796878528152829, 2.58010119009353675715960097144, 3.48516828367850570248283856399, 4.84885915048250194311890149585, 5.39811986590597317872401966870, 5.59349382326196597853681371936, 7.23101757492791350295756620335, 7.69305902332186631315691751352, 8.279216206776973637284122300121
