L(s) = 1 | + (1.84 − 0.493i)2-s + (1.41 − 0.818i)4-s + (0.885 + 3.30i)5-s + (−2.39 + 1.12i)7-s + (−0.489 + 0.489i)8-s + (3.26 + 5.65i)10-s + (−1.66 − 0.445i)11-s + (3.57 − 0.501i)13-s + (−3.84 + 3.26i)14-s + (−2.29 + 3.97i)16-s + (−1.22 − 2.12i)17-s + (1.34 + 5.03i)19-s + (3.96 + 3.96i)20-s − 3.28·22-s + (3.97 + 2.29i)23-s + ⋯ |
L(s) = 1 | + (1.30 − 0.349i)2-s + (0.708 − 0.409i)4-s + (0.396 + 1.47i)5-s + (−0.904 + 0.426i)7-s + (−0.173 + 0.173i)8-s + (1.03 + 1.78i)10-s + (−0.501 − 0.134i)11-s + (0.990 − 0.138i)13-s + (−1.02 + 0.871i)14-s + (−0.574 + 0.994i)16-s + (−0.297 − 0.515i)17-s + (0.309 + 1.15i)19-s + (0.885 + 0.885i)20-s − 0.700·22-s + (0.828 + 0.478i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.18563 + 1.50637i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18563 + 1.50637i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.39 - 1.12i)T \) |
| 13 | \( 1 + (-3.57 + 0.501i)T \) |
good | 2 | \( 1 + (-1.84 + 0.493i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-0.885 - 3.30i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (1.66 + 0.445i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.34 - 5.03i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.97 - 2.29i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.184T + 29T^{2} \) |
| 31 | \( 1 + (-2.46 - 0.659i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-0.0563 - 0.210i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-4.63 + 4.63i)T - 41iT^{2} \) |
| 43 | \( 1 - 0.562iT - 43T^{2} \) |
| 47 | \( 1 + (3.72 - 0.998i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.67 - 4.63i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.73 + 13.9i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (1.30 + 0.754i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.78 + 6.67i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (1.70 + 1.70i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.15 + 11.7i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.48 - 2.57i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.504 + 0.504i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.20 + 1.92i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-12.0 + 12.0i)T - 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.67473397910154811907560092032, −9.806101045590175710723562573789, −8.824604849029398637699163204044, −7.56729301203215177465814703552, −6.44317810373634674756527309279, −6.05609412425595251721090393120, −5.12421792787509530665346932822, −3.61041743696542012345012642623, −3.17741844820625456058121066502, −2.24469506527201965320212075000,
0.873506291583413989050938032970, 2.77502069393836613738492688475, 3.98142989183080706376883789533, 4.69593506205901766129205791812, 5.52262716956689420971302632693, 6.30776606663951387820699455631, 7.13998704675741043486902593265, 8.500328936690324805392854640279, 9.151894288761024643711733475854, 9.959563090436702248922259138833