| L(s) = 1 | + (0.212 − 0.977i)2-s + (−2.69 + 2.02i)3-s + (−0.909 − 0.415i)4-s + (0.823 − 2.07i)5-s + (1.40 + 3.06i)6-s + (−0.114 + 1.60i)7-s + (−0.599 + 0.800i)8-s + (2.35 − 8.02i)9-s + (−1.85 − 1.24i)10-s + (2.59 − 4.03i)11-s + (3.29 − 0.716i)12-s + (3.73 − 0.267i)13-s + (1.54 + 0.453i)14-s + (1.97 + 7.27i)15-s + (0.654 + 0.755i)16-s + (1.05 − 2.83i)17-s + ⋯ |
| L(s) = 1 | + (0.150 − 0.690i)2-s + (−1.55 + 1.16i)3-s + (−0.454 − 0.207i)4-s + (0.368 − 0.929i)5-s + (0.571 + 1.25i)6-s + (−0.0434 + 0.607i)7-s + (−0.211 + 0.283i)8-s + (0.785 − 2.67i)9-s + (−0.586 − 0.394i)10-s + (0.782 − 1.21i)11-s + (0.951 − 0.206i)12-s + (1.03 − 0.0741i)13-s + (0.413 + 0.121i)14-s + (0.510 + 1.87i)15-s + (0.163 + 0.188i)16-s + (0.256 − 0.686i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.677941 - 0.452288i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.677941 - 0.452288i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.212 + 0.977i)T \) |
| 5 | \( 1 + (-0.823 + 2.07i)T \) |
| 23 | \( 1 + (0.218 + 4.79i)T \) |
| good | 3 | \( 1 + (2.69 - 2.02i)T + (0.845 - 2.87i)T^{2} \) |
| 7 | \( 1 + (0.114 - 1.60i)T + (-6.92 - 0.996i)T^{2} \) |
| 11 | \( 1 + (-2.59 + 4.03i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.73 + 0.267i)T + (12.8 - 1.85i)T^{2} \) |
| 17 | \( 1 + (-1.05 + 2.83i)T + (-12.8 - 11.1i)T^{2} \) |
| 19 | \( 1 + (0.432 - 0.947i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (4.46 - 2.03i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (0.275 + 1.91i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.12 - 5.72i)T + (-20.0 + 31.1i)T^{2} \) |
| 41 | \( 1 + (-5.94 + 1.74i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (3.04 + 4.07i)T + (-12.1 + 41.2i)T^{2} \) |
| 47 | \( 1 + (-1.23 + 1.23i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.08 + 0.363i)T + (52.4 + 7.54i)T^{2} \) |
| 59 | \( 1 + (-6.22 - 5.39i)T + (8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (6.21 - 0.893i)T + (58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (8.37 + 1.82i)T + (60.9 + 27.8i)T^{2} \) |
| 71 | \( 1 + (-2.20 + 1.41i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.257 + 0.0961i)T + (55.1 - 47.8i)T^{2} \) |
| 79 | \( 1 + (-6.22 + 7.18i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (2.59 - 1.41i)T + (44.8 - 69.8i)T^{2} \) |
| 89 | \( 1 + (-0.560 + 3.89i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (5.75 + 3.14i)T + (52.4 + 81.6i)T^{2} \) |
| 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
−11.81448046308979388777830056494, −11.19111125357121945234088236374, −10.31246008329987096830394069770, −9.264849603372393142462947172646, −8.745011721102029560142003305684, −6.18017285138026101425468355850, −5.70544047007182937126181222124, −4.65038309216924485016237775440, −3.60581844582042853612581768291, −0.880146260593970593356331720575,
1.60963296110582117537289811241, 4.14504747543064979058301428277, 5.63709572503372680212295669182, 6.36845626463606974028635925701, 7.09377342049072084774124806567, 7.76543245601696769901549427742, 9.653178595079931858460706006206, 10.75925746876965678483786373716, 11.41918273885466260140838871076, 12.50243721708727789495742327597
