| L(s) = 1 | − 7.53i·5-s + 5.18·7-s − 20.8i·11-s + 5.35·13-s + 8.47i·17-s − 4.35·19-s + 22.0i·23-s − 31.7·25-s − 34.4i·29-s − 35.1·31-s − 39.0i·35-s + 15.3·37-s + 19.8i·41-s + 24.9·43-s − 11.0i·47-s + ⋯ |
| L(s) = 1 | − 1.50i·5-s + 0.740·7-s − 1.89i·11-s + 0.412·13-s + 0.498i·17-s − 0.229·19-s + 0.956i·23-s − 1.27·25-s − 1.18i·29-s − 1.13·31-s − 1.11i·35-s + 0.413·37-s + 0.484i·41-s + 0.580·43-s − 0.236i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.676261592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676261592\) |
| \(L(2)\) |
|
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 | \( 1 + 4.35T \) |
| good | 5 | \( 1 + 7.53iT - 25T^{2} \) |
| 7 | \( 1 - 5.18T + 49T^{2} \) |
| 11 | \( 1 + 20.8iT - 121T^{2} \) |
| 13 | \( 1 - 5.35T + 169T^{2} \) |
| 17 | \( 1 - 8.47iT - 289T^{2} \) |
| 23 | \( 1 - 22.0iT - 529T^{2} \) |
| 29 | \( 1 + 34.4iT - 841T^{2} \) |
| 31 | \( 1 + 35.1T + 961T^{2} \) |
| 37 | \( 1 - 15.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 19.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 24.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 90.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 51.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 47.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.34T + 4.48e3T^{2} \) |
| 71 | \( 1 + 30.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 79.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 152.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 22.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 134. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 32.4T + 9.40e3T^{2} \) |
| 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
−9.847435170105766763567307448126, −8.818024528328823341679370665211, −8.445044459955360424388801977195, −7.68175548566284058410390756132, −6.05137313346055909383723393323, −5.50876112746500586172580539641, −4.46683024100138825893202938936, −3.46857254505810325195477424771, −1.68450523489865902457231668626, −0.60854912148259470129057392168,
1.80957691891340475328426439226, 2.78469577584006922933283510414, 4.08814112793701673860561289126, 5.04525063163241990846053122298, 6.34758890731985809799879243985, 7.16162268264113718085429954277, 7.64678996791361005043311092276, 8.931802246087450306027110763297, 9.881262683150835577208347773823, 10.69130262758632886004539926789
