L(s) = 1 | + (−0.0756 + 0.0549i)2-s + (0.453 + 1.39i)3-s + (−0.615 + 1.89i)4-s + (−0.809 − 0.587i)5-s + (−0.110 − 0.0806i)6-s + (1.39 − 4.30i)7-s + (−0.115 − 0.354i)8-s + (0.686 − 0.498i)9-s + 0.0935·10-s + (−2.39 + 2.29i)11-s − 2.92·12-s + (0.924 − 0.671i)13-s + (0.130 + 0.402i)14-s + (0.453 − 1.39i)15-s + (−3.19 − 2.32i)16-s + (−2.72 − 1.98i)17-s + ⋯ |
L(s) = 1 | + (−0.0534 + 0.0388i)2-s + (0.261 + 0.805i)3-s + (−0.307 + 0.946i)4-s + (−0.361 − 0.262i)5-s + (−0.0452 − 0.0329i)6-s + (0.528 − 1.62i)7-s + (−0.0407 − 0.125i)8-s + (0.228 − 0.166i)9-s + 0.0295·10-s + (−0.723 + 0.690i)11-s − 0.843·12-s + (0.256 − 0.186i)13-s + (0.0349 + 0.107i)14-s + (0.117 − 0.360i)15-s + (−0.798 − 0.580i)16-s + (−0.661 − 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.778152 + 0.301272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.778152 + 0.301272i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (2.39 - 2.29i)T \) |
good | 2 | \( 1 + (0.0756 - 0.0549i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.453 - 1.39i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + (-1.39 + 4.30i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.924 + 0.671i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (2.72 + 1.98i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.88 - 5.78i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 5.45T + 23T^{2} \) |
| 29 | \( 1 + (-1.02 + 3.15i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.44 - 1.05i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.460 + 1.41i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.539 + 1.66i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.263T + 43T^{2} \) |
| 47 | \( 1 + (-2.13 - 6.58i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.16 + 0.846i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.18 - 6.72i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.02 + 1.47i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 0.516T + 67T^{2} \) |
| 71 | \( 1 + (-8.68 - 6.30i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.75 - 5.40i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.14 + 6.64i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (3.62 + 2.63i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + (-2.71 + 1.97i)T + (29.9 - 92.2i)T^{2} \) |
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
−15.73545980252126709203189529169, −14.29918338904356413980797634033, −13.29373242917554276270222390737, −12.12882296882788741867914895213, −10.66186999550995766035570286007, −9.668427875746510150716741589468, −8.128342189611578373152660761357, −7.30773922762058708654097723428, −4.55888984588800133593494241046, −3.78641443683664671578735000907,
2.22958235079510487272992591066, 5.08338178394585410132280103937, 6.39099728462163925674248056550, 8.102407547247770772874187873860, 9.054457426749347955547235919320, 10.70392185370278958484396884066, 11.76187859983127453172267019501, 13.09229833418332536575947354181, 14.04239886742272692210702624474, 15.24538030413445673011047167735