L(s) = 1 | + (1.20 − 2.09i)2-s + (−1.91 − 3.31i)4-s + (−0.5 + 0.866i)5-s − 3.82·7-s − 4.41·8-s + (1.20 + 2.09i)10-s − 2.82·11-s + (−1.91 − 3.31i)13-s + (−4.62 + 8.00i)14-s + (−1.49 + 2.59i)16-s + (−3.41 + 5.91i)17-s + (4 − 1.73i)19-s + 3.82·20-s + (−3.41 + 5.91i)22-s + (2.41 + 4.18i)23-s + ⋯ |
L(s) = 1 | + (0.853 − 1.47i)2-s + (−0.957 − 1.65i)4-s + (−0.223 + 0.387i)5-s − 1.44·7-s − 1.56·8-s + (0.381 + 0.661i)10-s − 0.852·11-s + (−0.530 − 0.919i)13-s + (−1.23 + 2.13i)14-s + (−0.374 + 0.649i)16-s + (−0.828 + 1.43i)17-s + (0.917 − 0.397i)19-s + 0.856·20-s + (−0.727 + 1.26i)22-s + (0.503 + 0.871i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.278820 + 0.307543i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.278820 + 0.307543i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 2 | \( 1 + (-1.20 + 2.09i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + (1.91 + 3.31i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.41 - 5.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.41 - 4.18i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.828 + 1.43i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 7.82T + 37T^{2} \) |
| 41 | \( 1 + (1.41 - 2.44i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.08 - 1.88i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.41 + 7.64i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.15 + 12.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.74 - 9.94i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5 + 8.66i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.74 + 6.48i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.32 + 12.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + (2.24 + 3.88i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.949191021937475986026007195661, −9.194408617558616987773922792443, −7.85655746134008778644626033773, −6.78177968712170771112649359570, −5.71111025177151462283641598065, −4.91284907153272873165746223580, −3.47933169873033164158730829236, −3.25065695249925714829790988553, −2.05070944851746593685893097010, −0.14091225595846314830869586172,
2.77931632868215756296614681487, 3.83131101300964818251869368282, 4.89282408308823625802978593287, 5.47604792205812285833211971385, 6.72688556460919425717174805658, 6.98427089782899625206182493341, 7.933425913025809496941666197780, 9.046892409740634023086598909624, 9.529725391583880066031135354326, 10.77768192069703597719020214192