| L(s) = 1 | − 2.64i·2-s − 3.00·4-s − 11.2·7-s − 2.64i·8-s + 4.24i·11-s − 11.2·13-s + 29.6i·14-s − 18.9·16-s − 10.5i·17-s − 20·19-s + 11.2·22-s − 5.29i·23-s + 29.6i·26-s + 33.6·28-s + 8.48i·29-s + ⋯ |
| L(s) = 1 | − 1.32i·2-s − 0.750·4-s − 1.60·7-s − 0.330i·8-s + 0.385i·11-s − 0.863·13-s + 2.12i·14-s − 1.18·16-s − 0.622i·17-s − 1.05·19-s + 0.510·22-s − 0.230i·23-s + 1.14i·26-s + 1.20·28-s + 0.292i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.223323 + 0.431426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.223323 + 0.431426i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.64iT - 4T^{2} \) |
| 7 | \( 1 + 11.2T + 49T^{2} \) |
| 11 | \( 1 - 4.24iT - 121T^{2} \) |
| 13 | \( 1 + 11.2T + 169T^{2} \) |
| 17 | \( 1 + 10.5iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 + 5.29iT - 529T^{2} \) |
| 29 | \( 1 - 8.48iT - 841T^{2} \) |
| 31 | \( 1 - 26T + 961T^{2} \) |
| 37 | \( 1 - 33.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 55.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 21.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 84.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 46.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 22T + 3.72e3T^{2} \) |
| 67 | \( 1 + 89.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 67.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 14T + 6.24e3T^{2} \) |
| 83 | \( 1 - 74.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 89.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 22.4T + 9.40e3T^{2} \) |
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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
−11.46150065451047357546551835835, −10.28090421406943348219491344936, −9.825876896065461019817823060099, −8.921580936540546459963415627598, −7.20265095047199058899465486400, −6.31398144062691241059449284850, −4.56740235815045195560222265329, −3.31355597429707771266312885729, −2.31901649433285768253414893739, −0.24345738659617943704166198109,
2.78912513467632958469077964012, 4.41480822564967980399111077866, 5.91993959767732348168681032740, 6.45185028100608453922774452074, 7.46360821760587602184042979171, 8.529595333213368713920655111300, 9.516883697897747469991945002153, 10.48209391659518944642022635586, 11.84655325262051825042345438681, 12.92465990528761777438178058831
