| L(s) = 1 | + 2.23i·3-s + 3·5-s − 2·7-s − 2.00·9-s − 2.23i·11-s + 13-s + 6.70i·15-s + 4.47i·17-s − 4.47i·21-s − 6·23-s + 4·25-s + 2.23i·27-s + (3 + 4.47i)29-s + 6.70i·31-s + 5.00·33-s + ⋯ |
| L(s) = 1 | + 1.29i·3-s + 1.34·5-s − 0.755·7-s − 0.666·9-s − 0.674i·11-s + 0.277·13-s + 1.73i·15-s + 1.08i·17-s − 0.975i·21-s − 1.25·23-s + 0.800·25-s + 0.430i·27-s + (0.557 + 0.830i)29-s + 1.20i·31-s + 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839566987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839566987\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + (-3 - 4.47i)T \) |
| good | 3 | \( 1 - 2.23iT - 3T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 4.47iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 31 | \( 1 - 6.70iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 - 6.70iT - 43T^{2} \) |
| 47 | \( 1 - 2.23iT - 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 6.70iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 4.47iT - 89T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−9.718401207174014503172660308893, −8.891608137321788966076522715303, −8.337639359831187560210062794260, −6.88168922737932766630796569940, −6.04657607767764937239700105556, −5.62079058345781217666466243667, −4.57954744281582734532647341384, −3.66161620111455688895882748141, −2.87200808338468111820701320434, −1.52432755453134803901369365234,
0.66310018716206193729112951394, 2.05622677713784741169377606313, 2.40323605185343480024635260788, 3.87378433375589717569562278782, 5.18847466938266158498358514536, 6.04463265710505315401007838528, 6.53452948088980873338242776023, 7.24842491198480076559381025712, 8.034641470385678920492917201201, 9.042285192399608718153223202057
