| L(s) = 1 | + 3-s − 2·7-s + 9-s − 11-s − 5·13-s − 3·17-s + 4·19-s − 2·21-s + 3·23-s + 27-s − 8·29-s − 5·31-s − 33-s − 2·37-s − 5·39-s + 11·41-s − 43-s − 8·47-s − 3·49-s − 3·51-s − 11·53-s + 4·57-s + 4·59-s − 6·61-s − 2·63-s + 13·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s − 0.727·17-s + 0.917·19-s − 0.436·21-s + 0.625·23-s + 0.192·27-s − 1.48·29-s − 0.898·31-s − 0.174·33-s − 0.328·37-s − 0.800·39-s + 1.71·41-s − 0.152·43-s − 1.16·47-s − 3/7·49-s − 0.420·51-s − 1.51·53-s + 0.529·57-s + 0.520·59-s − 0.768·61-s − 0.251·63-s + 1.58·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9596846922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9596846922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| 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
−12.97513699343237, −12.59049278671856, −12.40137048231029, −11.50812370807582, −11.19790781281178, −10.71733701468203, −10.06759302330689, −9.573816101861285, −9.245984674712207, −9.134449631732888, −8.083802570956587, −7.838214803792263, −7.353269924050572, −6.826029675590218, −6.467789124422406, −5.698563166185877, −5.142315477284992, −4.820787168228277, −4.078644515207528, −3.433268339551901, −3.156081210307638, −2.341804573757964, −2.101225679025996, −1.191226540024727, −0.2640219162153231,
0.2640219162153231, 1.191226540024727, 2.101225679025996, 2.341804573757964, 3.156081210307638, 3.433268339551901, 4.078644515207528, 4.820787168228277, 5.142315477284992, 5.698563166185877, 6.467789124422406, 6.826029675590218, 7.353269924050572, 7.838214803792263, 8.083802570956587, 9.134449631732888, 9.245984674712207, 9.573816101861285, 10.06759302330689, 10.71733701468203, 11.19790781281178, 11.50812370807582, 12.40137048231029, 12.59049278671856, 12.97513699343237
