| L(s) = 1 | − 5.39·3-s − 2.23i·5-s − 8.04i·7-s + 20.1·9-s − 13.7i·11-s − 25.1·13-s + 12.0i·15-s + 26.9i·17-s − 15.1i·19-s + 43.4i·21-s + (−21.4 − 8.29i)23-s − 5.00·25-s − 60.2·27-s − 19.5·29-s − 10.6·31-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.447i·5-s − 1.14i·7-s + 2.23·9-s − 1.25i·11-s − 1.93·13-s + 0.804i·15-s + 1.58i·17-s − 0.796i·19-s + 2.06i·21-s + (−0.932 − 0.360i)23-s − 0.200·25-s − 2.23·27-s − 0.674·29-s − 0.344·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03693692967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03693692967\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (21.4 + 8.29i)T \) |
| good | 3 | \( 1 + 5.39T + 9T^{2} \) |
| 7 | \( 1 + 8.04iT - 49T^{2} \) |
| 11 | \( 1 + 13.7iT - 121T^{2} \) |
| 13 | \( 1 + 25.1T + 169T^{2} \) |
| 17 | \( 1 - 26.9iT - 289T^{2} \) |
| 19 | \( 1 + 15.1iT - 361T^{2} \) |
| 29 | \( 1 + 19.5T + 841T^{2} \) |
| 31 | \( 1 + 10.6T + 961T^{2} \) |
| 37 | \( 1 + 53.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 3.97T + 1.68e3T^{2} \) |
| 43 | \( 1 - 14.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 81.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 81.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 31.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 56.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 0.391iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 60.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 17.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 46.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 144. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 115. iT - 9.40e3T^{2} \) |
| 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
−8.259993384724564096080992130758, −7.44193846006337217988929475727, −6.71888980432683168532532097571, −5.95010283659962468265843070003, −5.22106489111464511501482326269, −4.48067389001887032256463291703, −3.70021299853622327962876854372, −1.86410816176965546990733990528, −0.56635271533264852635983279412, −0.02220892477562411436682425902,
1.75429951839996097400034803594, 2.70049961597539276317851887152, 4.35163287446866986879718598966, 5.08087091942665781262689919314, 5.53281645772730318793045929879, 6.48157464701984043447111995130, 7.20495780334814611881997762485, 7.71898882563551695756378470595, 9.335678364885300727273308639423, 9.870817216746660342050897377633
