| L(s) = 1 | + 3.89·3-s − 8.85·5-s + 6.18·9-s + 3.64i·11-s + 7.79·13-s − 34.5·15-s + 10.4i·17-s − 10.7·19-s + 12.9·23-s + 53.4·25-s − 10.9·27-s − 17.2i·29-s − 30.2i·31-s + 14.2i·33-s − 39.5i·37-s + ⋯ |
| L(s) = 1 | + 1.29·3-s − 1.77·5-s + 0.686·9-s + 0.331i·11-s + 0.599·13-s − 2.30·15-s + 0.616i·17-s − 0.567·19-s + 0.561·23-s + 2.13·25-s − 0.406·27-s − 0.594i·29-s − 0.975i·31-s + 0.430i·33-s − 1.06i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0175 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0175 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.442204629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442204629\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.89T + 9T^{2} \) |
| 5 | \( 1 + 8.85T + 25T^{2} \) |
| 11 | \( 1 - 3.64iT - 121T^{2} \) |
| 13 | \( 1 - 7.79T + 169T^{2} \) |
| 17 | \( 1 - 10.4iT - 289T^{2} \) |
| 19 | \( 1 + 10.7T + 361T^{2} \) |
| 23 | \( 1 - 12.9T + 529T^{2} \) |
| 29 | \( 1 + 17.2iT - 841T^{2} \) |
| 31 | \( 1 + 30.2iT - 961T^{2} \) |
| 37 | \( 1 + 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 73.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 41.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 6.41iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 15.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 12.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 7.79iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 41.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 89.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 70.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 60.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + 27.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 3.26iT - 9.40e3T^{2} \) |
| 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
−8.691966711613669903221028462650, −8.377830799969833627522412404581, −7.57361363630980704281499600403, −7.07288175206441338289104141229, −5.78893493203211796162742488298, −4.34243282149568937098032608885, −3.90653001419666996500654329960, −3.12819132428205516055036785085, −2.01655681296746932803892201158, −0.36294827736457026489711566429,
1.15846384164679823702635697136, 2.82294074653992597481609288925, 3.34371269067737866831510999822, 4.15034918346876546084066468064, 5.01715690389331171282421606658, 6.53283700586548704044870239948, 7.28715321087183607397986829360, 8.104455735633145429696863587922, 8.463719592436782664174334843500, 9.092677969378322182124646382527
