| L(s) = 1 | + (0.123 − 1.40i)2-s + (−2.99 + 0.103i)3-s + (−1.96 − 0.347i)4-s + (3.21 − 3.83i)5-s + (−0.223 + 4.23i)6-s + (7.14 + 5.00i)7-s + (−0.732 + 2.73i)8-s + (8.97 − 0.620i)9-s + (−5.00 − 4.99i)10-s + (−0.513 − 0.186i)11-s + (5.94 + 0.837i)12-s + (0.585 + 6.69i)13-s + (7.93 − 9.45i)14-s + (−9.22 + 11.8i)15-s + (3.75 + 1.36i)16-s + (−4.16 − 15.5i)17-s + ⋯ |
| L(s) = 1 | + (0.0616 − 0.704i)2-s + (−0.999 + 0.0344i)3-s + (−0.492 − 0.0868i)4-s + (0.642 − 0.766i)5-s + (−0.0372 + 0.706i)6-s + (1.02 + 0.715i)7-s + (−0.0915 + 0.341i)8-s + (0.997 − 0.0689i)9-s + (−0.500 − 0.499i)10-s + (−0.0466 − 0.0169i)11-s + (0.495 + 0.0697i)12-s + (0.0450 + 0.515i)13-s + (0.566 − 0.675i)14-s + (−0.615 + 0.788i)15-s + (0.234 + 0.0855i)16-s + (−0.245 − 0.914i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0452 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0452 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01518 - 0.970202i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01518 - 0.970202i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.123 + 1.40i)T \) |
| 3 | \( 1 + (2.99 - 0.103i)T \) |
| 5 | \( 1 + (-3.21 + 3.83i)T \) |
| good | 7 | \( 1 + (-7.14 - 5.00i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (0.513 + 0.186i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-0.585 - 6.69i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (4.16 + 15.5i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-16.6 + 9.60i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-28.2 + 19.7i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (31.8 + 37.9i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-1.03 + 5.89i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (0.836 + 3.12i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-2.89 - 2.43i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-22.5 - 48.4i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-50.4 - 35.3i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-54.4 - 54.4i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (11.8 + 32.6i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (14.9 + 84.9i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (124. - 10.9i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-29.1 + 50.4i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (14.1 - 52.8i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-15.4 - 18.4i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-86.8 - 7.59i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-79.5 + 45.9i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (23.6 - 11.0i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−11.52026061514542328686625512794, −10.82834222922836696413653690600, −9.522801058577310776313704442427, −9.012212075096953288087745903571, −7.59241368348300733913217282821, −6.07912835558502715520031086725, −5.10381043030278892870219139709, −4.52090167167503493470172400783, −2.30404231811052871928384728777, −0.949667108732143733323325108110,
1.40864474362648659985355707682, 3.73473049826261458690350040752, 5.16430072019476749951644740500, 5.78766530846223749212321076846, 7.08603812159504954997187131167, 7.53768780913811998392900324114, 9.051897597997219927063695564732, 10.38707751404934198828302421743, 10.75207114068298669124773989876, 11.81132643920846809966696505201
