| L(s) = 1 | + (−1 − i)2-s + 2i·4-s + (−2 + i)5-s + (2 − 2i)8-s − 3i·9-s + (3 + i)10-s + (−1 + i)13-s − 4·16-s + (3 + 3i)17-s + (−3 + 3i)18-s + (−2 − 4i)20-s + (3 − 4i)25-s + 2·26-s + 4i·29-s + (4 + 4i)32-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s + (−0.894 + 0.447i)5-s + (0.707 − 0.707i)8-s − i·9-s + (0.948 + 0.316i)10-s + (−0.277 + 0.277i)13-s − 16-s + (0.727 + 0.727i)17-s + (−0.707 + 0.707i)18-s + (−0.447 − 0.894i)20-s + (0.600 − 0.800i)25-s + 0.392·26-s + 0.742i·29-s + (0.707 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.412523 - 0.117189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.412523 - 0.117189i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + i)T \) |
| 5 | \( 1 + (2 - i)T \) |
| good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3 - 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (7 + 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (-9 + 9i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (11 - 11i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 16iT - 89T^{2} \) |
| 97 | \( 1 + (-13 - 13i)T + 97iT^{2} \) |
| 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
−18.54530994742234557988936814298, −17.36278766896153774645413833010, −15.98851315466574025753781465568, −14.60863390629778476709057892335, −12.57162116369725768064984496401, −11.62341391771111472288252712239, −10.22889472215849124414124733521, −8.640379848934142593144264292186, −7.08937157793967758816517109516, −3.63351761276323569547339575571,
5.09388682361178306352765485359, 7.36427401884056470781626500805, 8.476557285997462789360043827500, 10.19606565450750423970831368888, 11.72375060446644028152469300011, 13.61776821454158890200258592321, 15.12087754482813165213151877280, 16.18889154393944071424592053479, 17.05197663834121013206113169629, 18.63950642032327279407698589487
