| L(s) = 1 | + (−1.02 − 0.975i)2-s + (−0.578 − 1.06i)3-s + (0.0981 + 1.99i)4-s + (−1.97 + 1.05i)5-s + (−0.440 + 1.65i)6-s + (0.795 + 1.06i)7-s + (1.84 − 2.14i)8-s + (0.833 − 1.29i)9-s + (3.04 + 0.845i)10-s + (3.89 − 1.78i)11-s + (2.06 − 1.26i)12-s + (−2.86 − 2.14i)13-s + (0.221 − 1.86i)14-s + (2.25 + 1.48i)15-s + (−3.98 + 0.391i)16-s + (−6.12 + 0.437i)17-s + ⋯ |
| L(s) = 1 | + (−0.724 − 0.689i)2-s + (−0.334 − 0.612i)3-s + (0.0490 + 0.998i)4-s + (−0.882 + 0.470i)5-s + (−0.179 + 0.673i)6-s + (0.300 + 0.401i)7-s + (0.653 − 0.757i)8-s + (0.277 − 0.432i)9-s + (0.963 + 0.267i)10-s + (1.17 − 0.536i)11-s + (0.594 − 0.363i)12-s + (−0.794 − 0.594i)13-s + (0.0591 − 0.498i)14-s + (0.583 + 0.382i)15-s + (−0.995 + 0.0979i)16-s + (−1.48 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0304513 - 0.435541i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0304513 - 0.435541i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.02 + 0.975i)T \) |
| 5 | \( 1 + (1.97 - 1.05i)T \) |
| 23 | \( 1 + (2.67 + 3.97i)T \) |
| good | 3 | \( 1 + (0.578 + 1.06i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (-0.795 - 1.06i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (-3.89 + 1.78i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (2.86 + 2.14i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (6.12 - 0.437i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (0.0836 + 0.0965i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (5.36 + 4.65i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-1.01 - 3.47i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.76 - 8.10i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (0.209 - 0.134i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (1.81 + 3.32i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (-7.00 + 7.00i)T - 47iT^{2} \) |
| 53 | \( 1 + (6.50 + 8.69i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (1.78 + 12.4i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (12.7 - 3.75i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (3.71 + 9.97i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (-0.857 - 0.391i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (7.18 + 0.513i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-2.30 - 16.0i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-1.59 - 7.32i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (-0.0839 + 0.286i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-10.5 - 2.29i)T + (88.2 + 40.2i)T^{2} \) |
| 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
−10.84491943881232085888131343559, −9.774771664188466639576431051339, −8.770586768122370638383690496892, −8.028577385160380095350000399327, −6.99419586050386002371726045848, −6.38333535735325737817088771847, −4.47275785362462906534381526343, −3.45735411036111000443212941917, −2.05778835074518674128686526162, −0.35922084447031190153380580974,
1.70416163591929145683388144433, 4.37701212333234234041963027247, 4.48178541986092210182343484848, 5.93574906296284170586712908983, 7.31580668655958029419688389152, 7.54195774986764736824437416609, 9.112808915361849985754110159532, 9.297112909019880282232378681134, 10.60098997281652362763088895866, 11.23144801839564067820477612049
