| L(s) = 1 | − 0.606·5-s + 2.85i·7-s + 2.41i·11-s − 20.3·13-s − 13.5·17-s − 4.35i·19-s + 12.0i·23-s − 24.6·25-s + 24.8·29-s − 18.4i·31-s − 1.73i·35-s + 1.16·37-s + 29.4·41-s − 33.7i·43-s − 13.7i·47-s + ⋯ |
| L(s) = 1 | − 0.121·5-s + 0.407i·7-s + 0.219i·11-s − 1.56·13-s − 0.794·17-s − 0.229i·19-s + 0.523i·23-s − 0.985·25-s + 0.855·29-s − 0.593i·31-s − 0.0495i·35-s + 0.0315·37-s + 0.718·41-s − 0.785i·43-s − 0.293i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.313152693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.313152693\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
| good | 5 | \( 1 + 0.606T + 25T^{2} \) |
| 7 | \( 1 - 2.85iT - 49T^{2} \) |
| 11 | \( 1 - 2.41iT - 121T^{2} \) |
| 13 | \( 1 + 20.3T + 169T^{2} \) |
| 17 | \( 1 + 13.5T + 289T^{2} \) |
| 23 | \( 1 - 12.0iT - 529T^{2} \) |
| 29 | \( 1 - 24.8T + 841T^{2} \) |
| 31 | \( 1 + 18.4iT - 961T^{2} \) |
| 37 | \( 1 - 1.16T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 13.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 51.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 17.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 43.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 30.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 64.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 80.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 128. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 163.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 25.1T + 9.40e3T^{2} \) |
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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
−8.663010926484352470782030972204, −7.69082402376572793997357866758, −7.21070577789114931546166329745, −6.30017598808475213718460310756, −5.42749251474219863880809629910, −4.69040138029988534080312546123, −3.87450208111559206609529160049, −2.64012560477142152628387468196, −2.04105669026994418371706947116, −0.43790211322413688467613618798,
0.69328731786298724002666962962, 2.10038374026084706634595463101, 2.91063188268256886052824554474, 4.09004540241946293142269414352, 4.68330107888039322380359566798, 5.59870297552711846440378988541, 6.53362959019376008968403792607, 7.22581748877225516494257015419, 7.88601367689197301080703853058, 8.696203689848261987642640274228
