L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999i·8-s + (−0.866 + 0.5i)9-s − 0.999i·10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−1.86 − 0.5i)17-s − 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)25-s + (0.866 − 0.499i)26-s + (0.866 + 0.5i)29-s + (−0.866 + 0.499i)32-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999i·8-s + (−0.866 + 0.5i)9-s − 0.999i·10-s + (0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−1.86 − 0.5i)17-s − 0.999·18-s + (0.499 − 0.866i)20-s + (−0.499 + 0.866i)25-s + (0.866 − 0.499i)26-s + (0.866 + 0.5i)29-s + (−0.866 + 0.499i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.037232618\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.037232618\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−12.54739657420439308773875431613, −11.45067537985741691045765149004, −10.88454634280927802358843962927, −8.993290730411734548989439267759, −8.350591967226347338363630570593, −7.38751076847477049491947181761, −6.07082134860982987465290253900, −5.10093577754829827200131816323, −4.14635133008600651302510123851, −2.66755024029166648854087942914,
2.37832511883529102140432754542, 3.59014167840438340799161686076, 4.61523844011705598303519550043, 6.32036228321228221556375205602, 6.62305302343707836673622429906, 8.272424609776498631490998509494, 9.442658132145889442619498172885, 10.65818487342209344835552754668, 11.38353330184719115251364642657, 11.85587128203539149391768391387