| L(s) = 1 | + (1 − 2.82i)3-s − 5.65i·5-s + 6·7-s + (−7.00 − 5.65i)9-s + 5.65i·11-s − 10·13-s + (−16.0 − 5.65i)15-s − 22.6i·17-s + 2·19-s + (6 − 16.9i)21-s + 11.3i·23-s − 7.00·25-s + (−23.0 + 14.1i)27-s − 16.9i·29-s + 22·31-s + ⋯ |
| L(s) = 1 | + (0.333 − 0.942i)3-s − 1.13i·5-s + 0.857·7-s + (−0.777 − 0.628i)9-s + 0.514i·11-s − 0.769·13-s + (−1.06 − 0.377i)15-s − 1.33i·17-s + 0.105·19-s + (0.285 − 0.808i)21-s + 0.491i·23-s − 0.280·25-s + (−0.851 + 0.523i)27-s − 0.585i·29-s + 0.709·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.947179 - 1.33951i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.947179 - 1.33951i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1 + 2.82i)T \) |
| good | 5 | \( 1 + 5.65iT - 25T^{2} \) |
| 7 | \( 1 - 6T + 49T^{2} \) |
| 11 | \( 1 - 5.65iT - 121T^{2} \) |
| 13 | \( 1 + 10T + 169T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 - 2T + 361T^{2} \) |
| 23 | \( 1 - 11.3iT - 529T^{2} \) |
| 29 | \( 1 + 16.9iT - 841T^{2} \) |
| 31 | \( 1 - 22T + 961T^{2} \) |
| 37 | \( 1 - 6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82T + 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 62.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 86T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2T + 4.48e3T^{2} \) |
| 71 | \( 1 - 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10T + 6.24e3T^{2} \) |
| 83 | \( 1 + 73.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.07533194086664122703912395565, −11.46749126503602838245904584622, −9.751796755197434385074848354339, −8.860349239253088693888270816133, −7.895983195278810667827954871521, −7.11015658890692328733828258675, −5.50803362391592836641157649931, −4.51932106171602055780806225932, −2.44784232092364310126463474209, −0.979802016761469266475686427400,
2.44172152162155152725027068676, 3.71510029959258957485593944174, 4.97674300282699422541812807906, 6.28255053067563499979673230672, 7.68884831408676266610086175556, 8.575045983740774084667128233337, 9.807868415336828821423410667152, 10.77243614813820501516033385695, 11.15273418806810480033068710746, 12.52270550173641442467250105425
