| L(s) = 1 | + 0.0840·3-s + 3.49i·5-s + i·7-s − 2.99·9-s + i·11-s + (−2.83 + 2.22i)13-s + 0.293i·15-s − 0.970·17-s + 6.96i·19-s + 0.0840i·21-s + 2.68·23-s − 7.20·25-s − 0.503·27-s + 3.14·29-s − 1.13i·31-s + ⋯ |
| L(s) = 1 | + 0.0485·3-s + 1.56i·5-s + 0.377i·7-s − 0.997·9-s + 0.301i·11-s + (−0.786 + 0.617i)13-s + 0.0758i·15-s − 0.235·17-s + 1.59i·19-s + 0.0183i·21-s + 0.559·23-s − 1.44·25-s − 0.0969·27-s + 0.583·29-s − 0.203i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7289624785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7289624785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (2.83 - 2.22i)T \) |
| good | 3 | \( 1 - 0.0840T + 3T^{2} \) |
| 5 | \( 1 - 3.49iT - 5T^{2} \) |
| 17 | \( 1 + 0.970T + 17T^{2} \) |
| 19 | \( 1 - 6.96iT - 19T^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 + 1.13iT - 31T^{2} \) |
| 37 | \( 1 - 1.41iT - 37T^{2} \) |
| 41 | \( 1 + 2.80iT - 41T^{2} \) |
| 43 | \( 1 - 3.04T + 43T^{2} \) |
| 47 | \( 1 - 1.87iT - 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 0.275iT - 59T^{2} \) |
| 61 | \( 1 - 4.89T + 61T^{2} \) |
| 67 | \( 1 + 0.948iT - 67T^{2} \) |
| 71 | \( 1 - 4.35iT - 71T^{2} \) |
| 73 | \( 1 + 3.72iT - 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 11.4iT - 83T^{2} \) |
| 89 | \( 1 - 4.56iT - 89T^{2} \) |
| 97 | \( 1 - 10.4iT - 97T^{2} \) |
| 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
−8.896762469352911457698519019120, −8.033663085811918618255613845788, −7.41831838712336373151114355660, −6.62829300346422133524286004703, −6.10376207444004648152624646864, −5.30472290274982270106526537966, −4.25091061311204088096282892928, −3.26964758456016172807539781341, −2.67797485353191582310518775484, −1.85880072535554216396452468592,
0.22574263910282295223569068756, 1.03713217048973245332412095549, 2.42708999541150752258750821029, 3.24397685001882327421102722062, 4.46281545911932921090233451081, 4.96322227830950804824950781774, 5.55479062686834927024617228695, 6.50086073877048672436751337716, 7.41040280845490964360456365308, 8.176214928134086311561201749157
