| L(s) = 1 | + (0.617 + 0.774i)2-s + (−0.339 − 1.69i)3-s + (0.226 − 0.992i)4-s + (−1.92 − 1.31i)5-s + (1.10 − 1.31i)6-s + (1.23 + 0.713i)7-s + (2.69 − 1.29i)8-s + (−2.76 + 1.15i)9-s + (−0.172 − 2.30i)10-s + (2.11 − 0.482i)11-s + (−1.76 − 0.0478i)12-s + (−0.101 + 1.35i)13-s + (0.210 + 1.39i)14-s + (−1.57 + 3.71i)15-s + (0.834 + 0.401i)16-s + (2.52 + 3.69i)17-s + ⋯ |
| L(s) = 1 | + (0.436 + 0.547i)2-s + (−0.196 − 0.980i)3-s + (0.113 − 0.496i)4-s + (−0.860 − 0.586i)5-s + (0.451 − 0.535i)6-s + (0.467 + 0.269i)7-s + (0.952 − 0.458i)8-s + (−0.923 + 0.384i)9-s + (−0.0545 − 0.727i)10-s + (0.637 − 0.145i)11-s + (−0.508 − 0.0137i)12-s + (−0.0282 + 0.376i)13-s + (0.0563 + 0.373i)14-s + (−0.406 + 0.958i)15-s + (0.208 + 0.100i)16-s + (0.611 + 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11461 - 0.486894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11461 - 0.486894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.339 + 1.69i)T \) |
| 43 | \( 1 + (0.712 - 6.51i)T \) |
| good | 2 | \( 1 + (-0.617 - 0.774i)T + (-0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (1.92 + 1.31i)T + (1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (-1.23 - 0.713i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 0.482i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (0.101 - 1.35i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (-2.52 - 3.69i)T + (-6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.81 - 5.87i)T + (-15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (3.49 + 3.76i)T + (-1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.419 + 0.0632i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (0.183 - 0.468i)T + (-22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (-2.80 + 1.62i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.01 - 1.60i)T + (9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (9.94 + 2.26i)T + (42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-13.4 + 1.00i)T + (52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (6.23 - 12.9i)T + (-36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-9.98 + 3.91i)T + (44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (8.19 - 2.52i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (-8.00 - 7.42i)T + (5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (15.0 + 1.13i)T + (72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (-0.879 + 1.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.47 + 9.76i)T + (-79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (1.28 + 0.193i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (0.0861 + 0.377i)T + (-87.3 + 42.0i)T^{2} \) |
| 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
−13.22831051528796418447440227290, −12.15652143253990056533092348756, −11.58082996745738577028070274317, −10.19787709613618416193771278111, −8.479707279077184377467115944936, −7.74714326964019949600359267727, −6.47788404086222999564991990175, −5.54592126285772147922867154277, −4.15183411072957571101999212840, −1.48414893778959499133854589200,
3.04851485710584061656499112633, 3.99257288761261115748771191555, 5.10571037984344492086188184268, 7.08162671379317450918272851589, 8.082909372838907229050480872544, 9.502840870282871944921327011928, 10.74577947547162538509864617601, 11.54412476923779730088415786380, 11.92909155703856117943526401286, 13.51351682641050210780952203820
