| L(s) = 1 | + 1.73·2-s + 0.999·4-s − 3.46·5-s + 7-s − 1.73·8-s − 5.99·10-s + 3.46·11-s + 2·13-s + 1.73·14-s − 5·16-s + 3.46·17-s − 4·19-s − 3.46·20-s + 5.99·22-s − 3.46·23-s + 6.99·25-s + 3.46·26-s + 0.999·28-s − 4·31-s − 5.19·32-s + 5.99·34-s − 3.46·35-s + 2·37-s − 6.92·38-s + 6.00·40-s + 10.3·41-s − 4·43-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s − 1.54·5-s + 0.377·7-s − 0.612·8-s − 1.89·10-s + 1.04·11-s + 0.554·13-s + 0.462·14-s − 1.25·16-s + 0.840·17-s − 0.917·19-s − 0.774·20-s + 1.27·22-s − 0.722·23-s + 1.39·25-s + 0.679·26-s + 0.188·28-s − 0.718·31-s − 0.918·32-s + 1.02·34-s − 0.585·35-s + 0.328·37-s − 1.12·38-s + 0.948·40-s + 1.62·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.256487160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.256487160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6.92T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−14.81885324888586223700140222590, −14.06926098054719753555946497209, −12.62008946584846744802697956037, −11.92091688160406750074517919675, −11.02479730124605003313336867740, −8.970685852658744454090100871251, −7.70923034018336810485556238538, −6.15911066831456100244343094530, −4.44828181879637253814536405983, −3.60365734514966092871602602517,
3.60365734514966092871602602517, 4.44828181879637253814536405983, 6.15911066831456100244343094530, 7.70923034018336810485556238538, 8.970685852658744454090100871251, 11.02479730124605003313336867740, 11.92091688160406750074517919675, 12.62008946584846744802697956037, 14.06926098054719753555946497209, 14.81885324888586223700140222590
