| L(s) = 1 | + (−1.17 − 1.61i)2-s + 3.80·3-s + (−1.23 + 3.80i)4-s + (−4.47 − 6.15i)6-s + 8.50·7-s + (7.60 − 2.47i)8-s + 5.47·9-s + 1.79i·11-s + (−4.70 + 14.4i)12-s − 0.472i·13-s + (−10 − 13.7i)14-s + (−12.9 − 9.40i)16-s − 23.8i·17-s + (−6.43 − 8.85i)18-s + 9.40i·19-s + ⋯ |
| L(s) = 1 | + (−0.587 − 0.809i)2-s + 1.26·3-s + (−0.309 + 0.951i)4-s + (−0.745 − 1.02i)6-s + 1.21·7-s + (0.951 − 0.309i)8-s + 0.608·9-s + 0.163i·11-s + (−0.391 + 1.20i)12-s − 0.0363i·13-s + (−0.714 − 0.983i)14-s + (−0.809 − 0.587i)16-s − 1.40i·17-s + (−0.357 − 0.491i)18-s + 0.494i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.40504 - 0.575750i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40504 - 0.575750i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.17 + 1.61i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.80T + 9T^{2} \) |
| 7 | \( 1 - 8.50T + 49T^{2} \) |
| 11 | \( 1 - 1.79iT - 121T^{2} \) |
| 13 | \( 1 + 0.472iT - 169T^{2} \) |
| 17 | \( 1 + 23.8iT - 289T^{2} \) |
| 19 | \( 1 - 9.40iT - 361T^{2} \) |
| 23 | \( 1 + 16.1T + 529T^{2} \) |
| 29 | \( 1 + 6.94T + 841T^{2} \) |
| 31 | \( 1 - 47.4iT - 961T^{2} \) |
| 37 | \( 1 - 26.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 2.00T + 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 21.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 73.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 26.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 88.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 21.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 39.1iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.72803982536647690573117073507, −12.27791488690945970965294379857, −11.36029471660326473757615332425, −10.13605327875007426899349361721, −9.051625806217410537500868758853, −8.229114880490813399891529881321, −7.39053354919497603436076529370, −4.75619411817136900890651551872, −3.22239260105341489064110735930, −1.83294213452325698456044128655,
1.93511886143013495556028971815, 4.20252071309061292069213299962, 5.81962446780900527127868874899, 7.49814066720000872931958905770, 8.261354737138215866893702684627, 8.970476207122598750921389394809, 10.23281928435025752409849729257, 11.40314405976500843902110590050, 13.22271031157439731228675179941, 14.12433680780106399133678084180
