| L(s) = 1 | + 7.04·5-s − 2.85i·7-s + (10.9 + 0.824i)11-s + 21.7i·13-s − 11.6i·17-s − 36.8i·19-s − 1.09·23-s + 24.6·25-s + 44.1i·29-s + 56.1·31-s − 20.1i·35-s + 36.6·37-s − 38.1i·41-s + 43.5i·43-s − 47.4·47-s + ⋯ |
| L(s) = 1 | + 1.40·5-s − 0.407i·7-s + (0.997 + 0.0749i)11-s + 1.67i·13-s − 0.683i·17-s − 1.93i·19-s − 0.0477·23-s + 0.985·25-s + 1.52i·29-s + 1.81·31-s − 0.574i·35-s + 0.989·37-s − 0.930i·41-s + 1.01i·43-s − 1.00·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0749i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.979548560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.979548560\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-10.9 - 0.824i)T \) |
| good | 5 | \( 1 - 7.04T + 25T^{2} \) |
| 7 | \( 1 + 2.85iT - 49T^{2} \) |
| 13 | \( 1 - 21.7iT - 169T^{2} \) |
| 17 | \( 1 + 11.6iT - 289T^{2} \) |
| 19 | \( 1 + 36.8iT - 361T^{2} \) |
| 23 | \( 1 + 1.09T + 529T^{2} \) |
| 29 | \( 1 - 44.1iT - 841T^{2} \) |
| 31 | \( 1 - 56.1T + 961T^{2} \) |
| 37 | \( 1 - 36.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 43.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 47.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 51.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 2.67T + 3.48e3T^{2} \) |
| 61 | \( 1 - 50.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 48.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 18.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 60.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 98.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 89.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 114.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 60.5T + 9.40e3T^{2} \) |
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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
−9.162786860827647195249932958417, −8.904094843291032085261271785907, −7.37490683965740468904246289628, −6.61843907452858769319905374773, −6.27130243184523818392656107993, −4.92018324485638888007900309200, −4.43395370026170305553091809940, −3.00661508161147160011623512520, −2.00630006228020943484602750045, −1.02843121320113245125744795722,
1.05176782586386371060789345363, 2.05065314223867612105577131850, 3.08026687979565611017531940005, 4.17566150625918876780763646960, 5.42037760964640084638645111231, 6.04244554461863212711533991740, 6.40434234029069140194276024269, 7.961028174462281357138799533314, 8.298579909591077689078446440746, 9.490429875799277626982506644535
