| L(s) = 1 | + i·2-s − 4-s + 1.08i·5-s − 0.351i·7-s − i·8-s − 1.08·10-s − 0.734i·11-s + (3.43 − 1.08i)13-s + 0.351·14-s + 16-s − 4.82·17-s + 4.79i·19-s − 1.08i·20-s + 0.734·22-s + 4.43·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.485i·5-s − 0.133i·7-s − 0.353i·8-s − 0.343·10-s − 0.221i·11-s + (0.953 − 0.301i)13-s + 0.0940·14-s + 0.250·16-s − 1.16·17-s + 1.09i·19-s − 0.242i·20-s + 0.156·22-s + 0.925·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.553142788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553142788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-3.43 + 1.08i)T \) |
| good | 5 | \( 1 - 1.08iT - 5T^{2} \) |
| 7 | \( 1 + 0.351iT - 7T^{2} \) |
| 11 | \( 1 + 0.734iT - 11T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 4.79iT - 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + 1.43T + 29T^{2} \) |
| 31 | \( 1 + 3.35iT - 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 + 3.03iT - 41T^{2} \) |
| 43 | \( 1 - 4.87T + 43T^{2} \) |
| 47 | \( 1 - 5.52iT - 47T^{2} \) |
| 53 | \( 1 + 0.258T + 53T^{2} \) |
| 59 | \( 1 - 14.4iT - 59T^{2} \) |
| 61 | \( 1 - 3.49T + 61T^{2} \) |
| 67 | \( 1 + 8.87iT - 67T^{2} \) |
| 71 | \( 1 - 7.05iT - 71T^{2} \) |
| 73 | \( 1 - 2.87iT - 73T^{2} \) |
| 79 | \( 1 + 6.43T + 79T^{2} \) |
| 83 | \( 1 - 3.61iT - 83T^{2} \) |
| 89 | \( 1 - 2.26iT - 89T^{2} \) |
| 97 | \( 1 - 2.78iT - 97T^{2} \) |
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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
−9.062730150111240426509732379455, −8.552179842150229227219507519158, −7.72554263282595253816531975556, −6.92568227925657492525575925450, −6.26724593783447003367865382835, −5.56837410757678182062104682429, −4.50950678399499793678080848284, −3.68678503585254253606825842306, −2.68040340949692152963727491799, −1.13806768194415315883905982845,
0.64898299550748686400781988610, 1.87139612750134877282762157668, 2.86414316361530903606490707485, 3.95172875693317266487826941955, 4.70210582525132362060560689192, 5.47000792563812280749280142253, 6.57051061831575666571430516748, 7.27126005966522362323820282758, 8.513204208608628474029935196920, 8.930936945461721581837586581084
