| L(s) = 1 | + (16.5 + 27.3i)2-s + 17.1i·3-s + (−475. + 906. i)4-s + (−469. + 283. i)6-s + 2.88e3i·7-s + (−3.27e4 + 1.98e3i)8-s + 5.87e4·9-s − 1.04e5i·11-s + (−1.55e4 − 8.14e3i)12-s + 1.45e5·13-s + (−7.89e4 + 4.77e4i)14-s + (−5.96e5 − 8.62e5i)16-s + 1.16e6·17-s + (9.72e5 + 1.60e6i)18-s − 4.52e6i·19-s + ⋯ |
| L(s) = 1 | + (0.517 + 0.855i)2-s + 0.0704i·3-s + (−0.464 + 0.885i)4-s + (−0.0603 + 0.0364i)6-s + 0.171i·7-s + (−0.998 + 0.0606i)8-s + 0.995·9-s − 0.651i·11-s + (−0.0624 − 0.0327i)12-s + 0.391·13-s + (−0.146 + 0.0887i)14-s + (−0.568 − 0.822i)16-s + 0.822·17-s + (0.514 + 0.851i)18-s − 1.82i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.53881 + 1.53512i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.53881 + 1.53512i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16.5 - 27.3i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 17.1iT - 5.90e4T^{2} \) |
| 7 | \( 1 - 2.88e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 1.04e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.45e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 1.16e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 4.52e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 3.50e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 2.68e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + 2.04e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 4.45e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 2.94e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 1.50e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 1.88e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 5.81e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 1.80e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 2.75e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 2.41e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 2.50e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 2.32e8T + 4.29e18T^{2} \) |
| 79 | \( 1 - 3.56e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 3.50e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 9.00e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 4.72e9T + 7.37e19T^{2} \) |
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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
−12.31791559887662465845387736911, −11.15268223530844711772031304854, −9.674511760047284509843734410357, −8.607381961767062668782388509040, −7.46051411342821450184093430535, −6.45490224165169878914764910479, −5.24711930199395451150335866936, −4.12645716922551167110417002430, −2.85609542264780362999190294272, −0.828746718191272511306223077951,
0.960412325443603645608824963726, 1.93056965028032914545065093800, 3.50446541821823918925237341493, 4.47803081185055790343572496447, 5.77759291359075967561943999538, 7.11649378074586828449619865700, 8.593747313412919488273899332530, 10.13443310666368400105612897896, 10.33534018820936313360269603380, 12.05312591070103022108055303047
