| L(s) = 1 | + (−2.08 + 0.182i)2-s + (2.35 − 0.414i)4-s + (1.98 − 1.03i)5-s + (3.38 − 2.36i)7-s + (−0.784 + 0.210i)8-s + (−3.95 + 2.51i)10-s + (−1.66 − 4.58i)11-s + (−0.223 + 2.55i)13-s + (−6.62 + 5.55i)14-s + (−2.88 + 1.05i)16-s + (2.36 + 0.632i)17-s + (−4.05 − 2.34i)19-s + (4.23 − 3.24i)20-s + (4.32 + 9.26i)22-s + (−2.13 + 3.04i)23-s + ⋯ |
| L(s) = 1 | + (−1.47 + 0.129i)2-s + (1.17 − 0.207i)4-s + (0.887 − 0.461i)5-s + (1.27 − 0.894i)7-s + (−0.277 + 0.0743i)8-s + (−1.24 + 0.795i)10-s + (−0.503 − 1.38i)11-s + (−0.0619 + 0.707i)13-s + (−1.77 + 1.48i)14-s + (−0.722 + 0.262i)16-s + (0.572 + 0.153i)17-s + (−0.930 − 0.537i)19-s + (0.947 − 0.726i)20-s + (0.921 + 1.97i)22-s + (−0.445 + 0.635i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.709196 - 0.410050i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.709196 - 0.410050i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.98 + 1.03i)T \) |
| good | 2 | \( 1 + (2.08 - 0.182i)T + (1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (-3.38 + 2.36i)T + (2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (1.66 + 4.58i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.223 - 2.55i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-2.36 - 0.632i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (4.05 + 2.34i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.13 - 3.04i)T + (-7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (2.44 + 2.05i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.272 + 1.54i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.592 + 2.21i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-4.38 - 5.22i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.553 + 1.18i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-0.220 - 0.314i)T + (-16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (0.177 + 0.177i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.797 - 0.290i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.38 + 7.86i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.09 - 0.708i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-7.83 + 4.52i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.19 - 11.9i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.94 - 8.27i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.325 + 3.71i)T + (-81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (7.58 - 13.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.25 + 0.583i)T + (62.3 + 74.3i)T^{2} \) |
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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
−10.96778965830991777242760888937, −10.06632570908010364419521493279, −9.226694687783858561078314195285, −8.325340956292425760918672674741, −7.83659739608219186493190403722, −6.63645979537143530327724178136, −5.46967881461648687915320216461, −4.23305000085811715922196653649, −2.08572313219063958829177620738, −0.923658218986730734117082146200,
1.73784292491883236756955304743, 2.44609318984811425890376709279, 4.76688185316529411712753745186, 5.75942796767878647772050162705, 7.11512379125046597486083745829, 7.945057941085433313769359330823, 8.690634418516386656738182949826, 9.664220143576723876866754759661, 10.34611801017572638679441481940, 10.94862175471008320391190354218
