| L(s) = 1 | − 0.613·3-s − 2.23i·5-s + 2.35i·7-s − 8.62·9-s − 0.167i·11-s − 23.3·13-s + 1.37i·15-s + 25.5i·17-s − 24.3i·19-s − 1.44i·21-s + (−9.84 + 20.7i)23-s − 5.00·25-s + 10.8·27-s + 15.2·29-s + 39.4·31-s + ⋯ |
| L(s) = 1 | − 0.204·3-s − 0.447i·5-s + 0.336i·7-s − 0.958·9-s − 0.0152i·11-s − 1.79·13-s + 0.0914i·15-s + 1.50i·17-s − 1.28i·19-s − 0.0687i·21-s + (−0.427 + 0.903i)23-s − 0.200·25-s + 0.400·27-s + 0.526·29-s + 1.27·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.150249143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150249143\) |
| \(L(2)\) |
|
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 | \( 1 + (9.84 - 20.7i)T \) |
| good | 3 | \( 1 + 0.613T + 9T^{2} \) |
| 7 | \( 1 - 2.35iT - 49T^{2} \) |
| 11 | \( 1 + 0.167iT - 121T^{2} \) |
| 13 | \( 1 + 23.3T + 169T^{2} \) |
| 17 | \( 1 - 25.5iT - 289T^{2} \) |
| 19 | \( 1 + 24.3iT - 361T^{2} \) |
| 29 | \( 1 - 15.2T + 841T^{2} \) |
| 31 | \( 1 - 39.4T + 961T^{2} \) |
| 37 | \( 1 - 4.29iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 29.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 50.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 15.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 58.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 93.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 97.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 14.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 64.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 110. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 4.10iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 87.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 61.4iT - 9.40e3T^{2} \) |
| 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
−8.878830620816476669894900339031, −8.383672861452509868524404663723, −7.46404055842732535814301159559, −6.59408908883770624853117991127, −5.64465356371416887520948187823, −5.07544321492334807699782599086, −4.15067838344031922979809944444, −2.89255633812130891477770758290, −2.07083582562190762805310011821, −0.49552051755791019605834084809,
0.63923949538595633017433937521, 2.40796653615345712661402397152, 2.93634067688410039869622862901, 4.27297437987466335793975292257, 5.06162451487176868051478306624, 5.91749391017245917705532134070, 6.80513688505307887458050330017, 7.50748040994218354857926361250, 8.231562865120139226466238497385, 9.194732995305855822458820513371
