| L(s) = 1 | + (−0.142 − 0.989i)3-s + (−0.768 − 0.225i)5-s + (−0.186 + 0.214i)7-s + (−0.959 + 0.281i)9-s + (2.11 − 1.35i)11-s + (−3.65 − 4.21i)13-s + (−0.114 + 0.793i)15-s + (1.96 − 4.30i)17-s + (0.447 + 0.980i)19-s + (0.239 + 0.153i)21-s + (−4.37 − 1.95i)23-s + (−3.66 − 2.35i)25-s + (0.415 + 0.909i)27-s + (3.56 − 7.81i)29-s + (−0.721 + 5.01i)31-s + ⋯ |
| L(s) = 1 | + (−0.0821 − 0.571i)3-s + (−0.343 − 0.100i)5-s + (−0.0703 + 0.0812i)7-s + (−0.319 + 0.0939i)9-s + (0.637 − 0.409i)11-s + (−1.01 − 1.16i)13-s + (−0.0294 + 0.204i)15-s + (0.476 − 1.04i)17-s + (0.102 + 0.224i)19-s + (0.0522 + 0.0335i)21-s + (−0.912 − 0.408i)23-s + (−0.733 − 0.471i)25-s + (0.0799 + 0.175i)27-s + (0.662 − 1.45i)29-s + (−0.129 + 0.901i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.494150 - 0.855324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494150 - 0.855324i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (4.37 + 1.95i)T \) |
| good | 5 | \( 1 + (0.768 + 0.225i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.186 - 0.214i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 1.35i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.65 + 4.21i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.96 + 4.30i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.447 - 0.980i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.56 + 7.81i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.721 - 5.01i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (7.59 - 2.23i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.420 - 0.123i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.861 + 5.99i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 1.21T + 47T^{2} \) |
| 53 | \( 1 + (-4.94 + 5.71i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.68 - 4.25i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-1.36 + 9.51i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 6.96i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.46 - 3.50i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (1.56 + 3.43i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.567 - 0.655i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (-0.408 + 0.119i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (1.24 + 8.66i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.70 - 1.96i)T + (81.6 + 52.4i)T^{2} \) |
| 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
−10.38881464965577786447060604576, −9.732552256735932392594450618354, −8.500719275352214284307640213167, −7.83492147962457136666136319507, −6.94977055856598231136402764942, −5.90827536692696180371881782440, −4.96329145750192702706855668561, −3.59396963125195223871088263501, −2.38743080829736437199832707610, −0.56701842987162450895589158916,
1.91336226741550253060205000263, 3.57287744757333428275543587132, 4.32823864827731976882381784003, 5.43029631792948394840179716388, 6.60816556586088896647363921894, 7.43782511790273616853387790728, 8.534875919352368407061529080913, 9.489930551585420288982459048383, 10.06147207693676301367977663800, 11.09630255451323818392239688902
