| L(s) = 1 | + 4.47i·3-s − 2.23i·5-s − 7·7-s − 11.0·9-s − 2·11-s + 13.4i·13-s + 10.0·15-s − 26.8i·17-s − 13.4i·19-s − 31.3i·21-s − 26·23-s − 5.00·25-s − 8.94i·27-s − 22·29-s − 53.6i·31-s + ⋯ |
| L(s) = 1 | + 1.49i·3-s − 0.447i·5-s − 7-s − 1.22·9-s − 0.181·11-s + 1.03i·13-s + 0.666·15-s − 1.57i·17-s − 0.706i·19-s − 1.49i·21-s − 1.13·23-s − 0.200·25-s − 0.331i·27-s − 0.758·29-s − 1.73i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3406418287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3406418287\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + 7T \) |
| good | 3 | \( 1 - 4.47iT - 9T^{2} \) |
| 11 | \( 1 + 2T + 121T^{2} \) |
| 13 | \( 1 - 13.4iT - 169T^{2} \) |
| 17 | \( 1 + 26.8iT - 289T^{2} \) |
| 19 | \( 1 + 13.4iT - 361T^{2} \) |
| 23 | \( 1 + 26T + 529T^{2} \) |
| 29 | \( 1 + 22T + 841T^{2} \) |
| 31 | \( 1 + 53.6iT - 961T^{2} \) |
| 37 | \( 1 - 14T + 1.36e3T^{2} \) |
| 41 | \( 1 - 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 34T + 1.84e3T^{2} \) |
| 47 | \( 1 + 26.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 34T + 2.80e3T^{2} \) |
| 59 | \( 1 + 40.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 93.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 14T + 4.48e3T^{2} \) |
| 71 | \( 1 + 62T + 5.04e3T^{2} \) |
| 73 | \( 1 - 53.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 38T + 6.24e3T^{2} \) |
| 83 | \( 1 - 40.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 26.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 26.8iT - 9.40e3T^{2} \) |
| 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
−10.04793462606254852248155938841, −9.410213748086958664638882805359, −9.154073451939256915553121998010, −7.74488605482507494585832078051, −6.56280472103517479617250412496, −5.48364209161872133506893961384, −4.52883190807371442812281487379, −3.78678625820937120452354907726, −2.55008658982862606074718514863, −0.12779721363889711470770518493,
1.48562548840913325532482908686, 2.73574460785639361706415047079, 3.80385629949481119769284773598, 5.81669745471889639274376696484, 6.17937658522725185502358640817, 7.23719304036575454984750404237, 7.88560447078840554153168128819, 8.770269462542081119546464760205, 10.14257304403491253082801767448, 10.63293182426061935263679722163
