| L(s) = 1 | + (1.73 + i)2-s + (1.99 + 3.46i)4-s + (−2.96 + 5.13i)5-s + 7.99i·8-s + (−10.2 + 5.92i)10-s + (0.261 − 0.150i)11-s − 72.5i·13-s + (−8 + 13.8i)16-s + (22.1 + 38.4i)17-s + (73.1 + 42.2i)19-s − 23.7·20-s + 0.603·22-s + (54.7 + 31.6i)23-s + (44.9 + 77.8i)25-s + (72.5 − 125. i)26-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.265 + 0.459i)5-s + 0.353i·8-s + (−0.324 + 0.187i)10-s + (0.00715 − 0.00413i)11-s − 1.54i·13-s + (−0.125 + 0.216i)16-s + (0.316 + 0.548i)17-s + (0.882 + 0.509i)19-s − 0.265·20-s + 0.00584·22-s + (0.496 + 0.286i)23-s + (0.359 + 0.622i)25-s + (0.547 − 0.948i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.409323096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.409323096\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.73 - i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (2.96 - 5.13i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-0.261 + 0.150i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 72.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-22.1 - 38.4i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-73.1 - 42.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-54.7 - 31.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 183. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (43.5 - 25.1i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (142. - 247. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + (211. - 366. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (175. - 101. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (367. + 636. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (548. + 316. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (66.9 + 116. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.04e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-131. + 76.1i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (409. - 709. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 583.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-656. + 1.13e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 409. iT - 9.12e5T^{2} \) |
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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
−10.20118134990841353182289681215, −9.111587459794003479289443940754, −8.001836339690535369518973782380, −7.52747112021261147175217683259, −6.53253181039456596559339028919, −5.58960428616262949871908515633, −4.89190915192521333851970366514, −3.43686123911496020362214785038, −3.06521828361391796910200237597, −1.30377369817636422554437187870,
0.52534878228010648313480751303, 1.83694807333605274553021046033, 3.00248003303234479834978147087, 4.19685620145636942544353088137, 4.79563779774189517769385634587, 5.85174863362351482606363359519, 6.85116010684826319559652466582, 7.62433074833438161136214389264, 8.892117439637679372028370038758, 9.396812628841601479039674489684
