| L(s) = 1 | − 1.73i·3-s − 13.2i·7-s − 2.99·9-s − 11.4i·11-s + 13-s − 31.8·17-s + 13.2i·19-s − 22.8·21-s − 32.2i·23-s + 5.19i·27-s − 4.10·29-s + 37.4i·31-s − 19.8·33-s + 17.7·37-s − 1.73i·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.88i·7-s − 0.333·9-s − 1.04i·11-s + 0.0769·13-s − 1.87·17-s + 0.695i·19-s − 1.09·21-s − 1.40i·23-s + 0.192i·27-s − 0.141·29-s + 1.20i·31-s − 0.603·33-s + 0.481·37-s − 0.0444i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8598292669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8598292669\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 13.2iT - 49T^{2} \) |
| 11 | \( 1 + 11.4iT - 121T^{2} \) |
| 13 | \( 1 - T + 169T^{2} \) |
| 17 | \( 1 + 31.8T + 289T^{2} \) |
| 19 | \( 1 - 13.2iT - 361T^{2} \) |
| 23 | \( 1 + 32.2iT - 529T^{2} \) |
| 29 | \( 1 + 4.10T + 841T^{2} \) |
| 31 | \( 1 - 37.4iT - 961T^{2} \) |
| 37 | \( 1 - 17.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 53.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 24.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 89.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 6.00iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 76.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 108. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 1.09iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 30.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 2.54iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 72.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 96T + 7.92e3T^{2} \) |
| 97 | \( 1 + 114.T + 9.40e3T^{2} \) |
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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
−8.859344019980271955934732559442, −8.264975759698833570988319720778, −7.28791559902322594947768415116, −6.73240413369469656936485314518, −5.93555534487130117035247429836, −4.53796929044133029215729572488, −3.89736788738035378110435729772, −2.67482218844643362020482257945, −1.23634716382824823794420554066, −0.25975923096981413355183217389,
2.05244300377577439458732532789, 2.67230849062657525695360543005, 4.09838168431322764300947046303, 4.94424379050699019770267870460, 5.73807250567505873719895707088, 6.57023917370287673199650605696, 7.66288136504223451762599830198, 8.680339953881010146105122446292, 9.303152827528694359651923682871, 9.665148927890511663690362358281
