| L(s) = 1 | + (−0.784 + 1.17i)2-s + (−1.17 − 2.84i)3-s + (−0.769 − 1.84i)4-s + (−0.453 − 0.891i)5-s + (4.26 + 0.843i)6-s + (0.0665 − 0.277i)7-s + (2.77 + 0.543i)8-s + (−4.56 + 4.56i)9-s + (1.40 + 0.164i)10-s + (2.73 + 3.19i)11-s + (−4.33 + 4.35i)12-s + (−2.05 + 3.35i)13-s + (0.273 + 0.295i)14-s + (−1.99 + 2.33i)15-s + (−2.81 + 2.84i)16-s + (−0.281 + 3.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.554 + 0.832i)2-s + (−0.679 − 1.63i)3-s + (−0.384 − 0.923i)4-s + (−0.203 − 0.398i)5-s + (1.74 + 0.344i)6-s + (0.0251 − 0.104i)7-s + (0.981 + 0.191i)8-s + (−1.52 + 1.52i)9-s + (0.444 + 0.0520i)10-s + (0.823 + 0.964i)11-s + (−1.25 + 1.25i)12-s + (−0.571 + 0.931i)13-s + (0.0731 + 0.0790i)14-s + (−0.515 + 0.603i)15-s + (−0.704 + 0.710i)16-s + (−0.0681 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.612575 + 0.217744i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612575 + 0.217744i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.784 - 1.17i)T \) |
| 5 | \( 1 + (0.453 + 0.891i)T \) |
| 41 | \( 1 + (6.33 + 0.910i)T \) |
| good | 3 | \( 1 + (1.17 + 2.84i)T + (-2.12 + 2.12i)T^{2} \) |
| 7 | \( 1 + (-0.0665 + 0.277i)T + (-6.23 - 3.17i)T^{2} \) |
| 11 | \( 1 + (-2.73 - 3.19i)T + (-1.72 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.05 - 3.35i)T + (-5.90 - 11.5i)T^{2} \) |
| 17 | \( 1 + (0.281 - 3.57i)T + (-16.7 - 2.65i)T^{2} \) |
| 19 | \( 1 + (1.97 - 1.20i)T + (8.62 - 16.9i)T^{2} \) |
| 23 | \( 1 + (-6.89 + 5.00i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.343 - 4.36i)T + (-28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (-1.94 - 5.98i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.20 + 3.71i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.661 + 4.17i)T + (-40.8 - 13.2i)T^{2} \) |
| 47 | \( 1 + (-0.360 - 1.50i)T + (-41.8 + 21.3i)T^{2} \) |
| 53 | \( 1 + (9.44 - 0.743i)T + (52.3 - 8.29i)T^{2} \) |
| 59 | \( 1 + (-2.88 - 3.96i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.174 - 1.10i)T + (-58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (-4.20 - 3.59i)T + (10.4 + 66.1i)T^{2} \) |
| 71 | \( 1 + (0.838 - 0.716i)T + (11.1 - 70.1i)T^{2} \) |
| 73 | \( 1 + (-6.86 - 6.86i)T + 73iT^{2} \) |
| 79 | \( 1 + (1.64 - 0.683i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 13.5iT - 83T^{2} \) |
| 89 | \( 1 + (6.38 + 1.53i)T + (79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (-12.3 - 10.5i)T + (15.1 + 95.8i)T^{2} \) |
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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
−10.35220889043189046503151505791, −9.078202695192845297702987066357, −8.513820234512480197770496724318, −7.48120229656406291679545853143, −6.84217635011355601723375061821, −6.46329861865527369425041940342, −5.27440376411887013695987173795, −4.40825994762796957647183517017, −2.01558977100948371701316227435, −1.13440031348791535111182799722,
0.50907773948107134774400566282, 2.88676525569048851882340296427, 3.55286702647387517994042200173, 4.56902017240331092091873976960, 5.40501241096032525626720208691, 6.60635420227499012149155252183, 7.890919351283160770173841985514, 8.882759903839022115750398082621, 9.580982253291079860313525972159, 10.09343580932413464313466856467
