| L(s) = 1 | − 5.38i·3-s + 7-s − 19.9·9-s + 14·11-s − 16.1i·13-s − 23·17-s + (10 + 16.1i)19-s − 5.38i·21-s + 23-s + 59.2i·27-s − 48.4i·29-s − 32.3i·31-s − 75.3i·33-s − 32.3i·37-s − 86.9·39-s + ⋯ |
| L(s) = 1 | − 1.79i·3-s + 0.142·7-s − 2.22·9-s + 1.27·11-s − 1.24i·13-s − 1.35·17-s + (0.526 + 0.850i)19-s − 0.256i·21-s + 0.0434·23-s + 2.19i·27-s − 1.67i·29-s − 1.04i·31-s − 2.28i·33-s − 0.873i·37-s − 2.23·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9304119833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9304119833\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-10 - 16.1i)T \) |
| good | 3 | \( 1 + 5.38iT - 9T^{2} \) |
| 7 | \( 1 - T + 49T^{2} \) |
| 11 | \( 1 - 14T + 121T^{2} \) |
| 13 | \( 1 + 16.1iT - 169T^{2} \) |
| 17 | \( 1 + 23T + 289T^{2} \) |
| 23 | \( 1 - T + 529T^{2} \) |
| 29 | \( 1 + 48.4iT - 841T^{2} \) |
| 31 | \( 1 + 32.3iT - 961T^{2} \) |
| 37 | \( 1 + 32.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 32.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68T + 1.84e3T^{2} \) |
| 47 | \( 1 + 26T + 2.20e3T^{2} \) |
| 53 | \( 1 - 80.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 16.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 40T + 3.72e3T^{2} \) |
| 67 | \( 1 - 16.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 32.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 32T + 6.88e3T^{2} \) |
| 89 | \( 1 + 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 96.9iT - 9.40e3T^{2} \) |
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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
−8.185332624376198663458020955672, −7.80008402404225782701676376365, −6.91654640209174289938154260310, −6.22422608353853491382922903383, −5.70853333100271228607214661923, −4.35460927806022191141160858860, −3.16814915360480175547527920534, −2.14686116736693669525550679826, −1.28278985330950208206832815018, −0.23825991601652280007690044659,
1.70826750013309485902982299127, 3.14958904491427795788910584467, 3.83464977360317413857383659072, 4.75698720140338544383408796040, 5.03816048119746040433090629646, 6.49143340267224120843207149295, 6.87951244473126057049337685813, 8.522502019931019057223303193042, 8.940141667813688088281223377477, 9.437091758882549098576925397707
