| L(s) = 1 | + 0.732·5-s + 2.36·7-s + (−3.23 − 0.732i)11-s + 2.36i·13-s + 6.46i·17-s + 6.46·19-s + 4.73i·23-s − 4.46·25-s − 2i·31-s + 1.73·35-s + 7.46·37-s + 4.73i·41-s + 6.46·43-s − 6.19i·47-s − 1.39·49-s + ⋯ |
| L(s) = 1 | + 0.327·5-s + 0.895·7-s + (−0.975 − 0.220i)11-s + 0.656i·13-s + 1.56i·17-s + 1.48·19-s + 0.986i·23-s − 0.892·25-s − 0.359i·31-s + 0.293·35-s + 1.22·37-s + 0.739i·41-s + 0.986·43-s − 0.903i·47-s − 0.198·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.857796067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.857796067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (3.23 + 0.732i)T \) |
| good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 13 | \( 1 - 2.36iT - 13T^{2} \) |
| 17 | \( 1 - 6.46iT - 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 - 4.73iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 - 4.73iT - 41T^{2} \) |
| 43 | \( 1 - 6.46T + 43T^{2} \) |
| 47 | \( 1 + 6.19iT - 47T^{2} \) |
| 53 | \( 1 - 7.26T + 53T^{2} \) |
| 59 | \( 1 - 2.53iT - 59T^{2} \) |
| 61 | \( 1 - 15.3iT - 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 7.26iT - 71T^{2} \) |
| 73 | \( 1 - 4.73iT - 73T^{2} \) |
| 79 | \( 1 + 2.36T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 1.46T + 97T^{2} \) |
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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
−9.603087128908774590497969888194, −8.659197328860345829194765808953, −7.85863156582412435419468724888, −7.38141100102109596065697467907, −6.03515530719143611420031591447, −5.54171515033721502476657193420, −4.55750414418656188164161851743, −3.59341633309367026680212410662, −2.33092366736308160613157996120, −1.33950757465119075105545555396,
0.788870169905196896422651729223, 2.26593259916043741528726448722, 3.07137461336176476959472328365, 4.50000055807521621915864640427, 5.20665587758625698825001106260, 5.82504162679511704520928160779, 7.15413297351646408093907666918, 7.69036189641843372576028560922, 8.396663238338436338061053641816, 9.478598673420947694368964785473
