| L(s) = 1 | + (0.227 − 2.60i)2-s + (−2.79 − 0.492i)4-s + (−2.63 + 4.25i)5-s + (4.66 + 3.26i)7-s + (0.786 − 2.93i)8-s + (10.4 + 7.82i)10-s + (−12.9 − 4.70i)11-s + (−0.714 − 8.16i)13-s + (9.56 − 11.3i)14-s + (−18.1 − 6.59i)16-s + (−7.23 − 26.9i)17-s + (26.2 − 15.1i)19-s + (9.45 − 10.5i)20-s + (−15.1 + 32.5i)22-s + (14.8 − 10.3i)23-s + ⋯ |
| L(s) = 1 | + (0.113 − 1.30i)2-s + (−0.698 − 0.123i)4-s + (−0.526 + 0.850i)5-s + (0.665 + 0.466i)7-s + (0.0982 − 0.366i)8-s + (1.04 + 0.782i)10-s + (−1.17 − 0.427i)11-s + (−0.0549 − 0.628i)13-s + (0.683 − 0.814i)14-s + (−1.13 − 0.412i)16-s + (−0.425 − 1.58i)17-s + (1.37 − 0.796i)19-s + (0.472 − 0.529i)20-s + (−0.690 + 1.48i)22-s + (0.643 − 0.450i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.343i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.938 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.239966 - 1.35283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239966 - 1.35283i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.63 - 4.25i)T \) |
| good | 2 | \( 1 + (-0.227 + 2.60i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-4.66 - 3.26i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (12.9 + 4.70i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.714 + 8.16i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (7.23 + 26.9i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-26.2 + 15.1i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-14.8 + 10.3i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (21.2 + 25.2i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-4.67 + 26.5i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-10.0 - 37.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-0.528 - 0.443i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (3.77 + 8.09i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-32.6 - 22.8i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-29.6 - 29.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-36.4 - 100. i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-10.9 - 62.1i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (76.4 - 6.68i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-16.7 + 29.0i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-3.38 + 12.6i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (45.0 + 53.7i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (18.4 + 1.61i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-71.9 + 41.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (24.7 - 11.5i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−10.85899205530222242678896834383, −10.08848880183159995642360664345, −9.083334193277542263960168163572, −7.79151358507510256784139755723, −7.11801345024833376974982769994, −5.51631953270868525132976220564, −4.46577939868088205934648960995, −2.92360296486789825195490688517, −2.62050276111018318329398914607, −0.57605280961427611037008796905,
1.67534044437551417564796382689, 3.88012132450059260228671637931, 4.97912728258767684325334097105, 5.54537198681432416211456040026, 6.96725061344806619677421765413, 7.73622989620048914469845569789, 8.273080403829879797830364844817, 9.263443679805490534343438708503, 10.60998654698697138393105637163, 11.39808714594159473170722312965
