| L(s) = 1 | − 2i·3-s − 4i·7-s − 9-s + 2·11-s + 2i·13-s + 2i·17-s + 2·19-s − 8·21-s − 4i·23-s − 4i·27-s + 6·29-s − 4i·33-s − 10i·37-s + 4·39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s − 1.51i·7-s − 0.333·9-s + 0.603·11-s + 0.554i·13-s + 0.485i·17-s + 0.458·19-s − 1.74·21-s − 0.834i·23-s − 0.769i·27-s + 1.11·29-s − 0.696i·33-s − 1.64i·37-s + 0.640·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.805160529\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.805160529\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
| 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
−8.269697574941950769499911919014, −7.33441252050001999945408146987, −6.93912965052025744754997615192, −6.48444055013073773531210707930, −5.38424556257030379627518165271, −4.21826392876047855227623197106, −3.78401417149802955330425078583, −2.39109042500610377097435741136, −1.40899889727374605756137633540, −0.60036600929581647792231505936,
1.44133962431065759803786858619, 2.82483649072711603000692070442, 3.32993007022582528094463001743, 4.50922004302943122149772704130, 5.06451357858188723652999491533, 5.79944925600103075498291299965, 6.55630753306782811995993570802, 7.62401279918674081669303650657, 8.511897890900031397586163017766, 9.052016631423950708058146424336
