L(s) = 1 | + (−0.442 + 1.34i)2-s + (0.112 + 1.72i)3-s + (−1.60 − 1.18i)4-s + (0.989 + 2.00i)5-s + (−2.37 − 0.613i)6-s + (0.573 + 0.993i)7-s + (2.30 − 1.63i)8-s + (−2.97 + 0.388i)9-s + (−3.13 + 0.442i)10-s + (0.629 + 1.09i)11-s + (1.87 − 2.91i)12-s + (−4.39 − 2.53i)13-s + (−1.58 + 0.331i)14-s + (−3.35 + 1.93i)15-s + (1.17 + 3.82i)16-s + 3.29·17-s + ⋯ |
L(s) = 1 | + (−0.312 + 0.949i)2-s + (0.0649 + 0.997i)3-s + (−0.804 − 0.594i)4-s + (0.442 + 0.896i)5-s + (−0.968 − 0.250i)6-s + (0.216 + 0.375i)7-s + (0.815 − 0.578i)8-s + (−0.991 + 0.129i)9-s + (−0.990 + 0.140i)10-s + (0.189 + 0.328i)11-s + (0.540 − 0.841i)12-s + (−1.21 − 0.704i)13-s + (−0.424 + 0.0885i)14-s + (−0.866 + 0.499i)15-s + (0.294 + 0.955i)16-s + 0.799·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.189176 + 0.941363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.189176 + 0.941363i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.442 - 1.34i)T \) |
| 3 | \( 1 + (-0.112 - 1.72i)T \) |
| 5 | \( 1 + (-0.989 - 2.00i)T \) |
good | 7 | \( 1 + (-0.573 - 0.993i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.629 - 1.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.39 + 2.53i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 + 3.62iT - 19T^{2} \) |
| 23 | \( 1 + (-3.09 - 1.78i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.184 - 0.106i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-9.12 - 5.26i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.72iT - 37T^{2} \) |
| 41 | \( 1 + (-5.81 - 3.35i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.41 + 2.45i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.925 + 0.534i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 9.12T + 53T^{2} \) |
| 59 | \( 1 + (4.87 - 8.43i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.24 + 9.08i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.42 + 12.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.68T + 71T^{2} \) |
| 73 | \( 1 + 9.08iT - 73T^{2} \) |
| 79 | \( 1 + (5.78 - 3.34i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.93 - 2.84i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 2.26iT - 89T^{2} \) |
| 97 | \( 1 + (-6.69 + 3.86i)T + (48.5 - 84.0i)T^{2} \) |
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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
−13.53428670643115608516464792135, −11.99425681596288190610299297022, −10.65425781987320249158182573246, −9.968178511829532300565454961168, −9.209387782635874057283547459730, −7.977110309117070711417671677285, −6.86614091305233950137376707707, −5.60755477460206801828722761562, −4.74009054221091262253690801494, −2.94712121341270404416632699279,
1.07058242699742850647923908483, 2.48033662855235977892859771406, 4.34005335068441628815309184142, 5.69984362720324907343711363777, 7.35194470770274334184277362543, 8.276263317746711946472408265963, 9.241293805286978451401131381588, 10.18811326628392881645667950987, 11.56693811955718479870437256671, 12.22594600286932757786849375111