L(s) = 1 | + (0.173 + 0.984i)3-s + (−1.93 + 1.62i)5-s + (−0.386 − 0.223i)7-s + (−0.939 + 0.342i)9-s + (−2.02 + 1.16i)11-s + (0.358 + 0.0632i)13-s + (−1.93 − 1.62i)15-s + (−5.58 − 2.03i)17-s + (−0.354 − 4.34i)19-s + (0.152 − 0.419i)21-s + (2.14 − 2.55i)23-s + (0.245 − 1.39i)25-s + (−0.5 − 0.866i)27-s + (0.449 + 1.23i)29-s + (3.35 − 5.80i)31-s + ⋯ |
L(s) = 1 | + (0.100 + 0.568i)3-s + (−0.867 + 0.727i)5-s + (−0.146 − 0.0843i)7-s + (−0.313 + 0.114i)9-s + (−0.609 + 0.351i)11-s + (0.0994 + 0.0175i)13-s + (−0.500 − 0.420i)15-s + (−1.35 − 0.493i)17-s + (−0.0813 − 0.996i)19-s + (0.0333 − 0.0915i)21-s + (0.447 − 0.533i)23-s + (0.0490 − 0.278i)25-s + (−0.0962 − 0.166i)27-s + (0.0834 + 0.229i)29-s + (0.601 − 1.04i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 + (0.354 + 4.34i)T \) |
good | 5 | \( 1 + (1.93 - 1.62i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.386 + 0.223i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.02 - 1.16i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.358 - 0.0632i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (5.58 + 2.03i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.14 + 2.55i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.449 - 1.23i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.35 + 5.80i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 7.52iT - 37T^{2} \) |
| 41 | \( 1 + (11.2 - 1.99i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.52 + 5.39i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.66 + 7.33i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (1.13 - 1.34i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-5.69 - 2.07i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.19 - 4.35i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (8.71 - 3.17i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.11 - 3.45i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.591 + 3.35i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.442 + 2.50i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.39 + 0.802i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.14 + 1.08i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-0.368 + 1.01i)T + (-74.3 - 62.3i)T^{2} \) |
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
−9.969786130160871834538692497999, −8.871995998233588490799366179409, −8.214757328136465652881406242217, −7.08082586677646696753266020080, −6.64313153673585398895857140778, −5.12624894560113856133342796752, −4.40716727667572893556870344324, −3.33357606915733909208737220647, −2.44183304658287403772041005704, 0,
1.60153717205985677931377844336, 3.04934310921968617210941227109, 4.13410847826956390871703289852, 5.10987681480055099933915975474, 6.17410677997778884870963752208, 7.06475524252244403001340233127, 8.170969525023226694758043164972, 8.375307489315371126014184374377, 9.390209601293767978343409275530