| L(s) = 1 | + (−0.198 − 1.48i)3-s + (−0.0282 − 0.0125i)5-s + (2.06 − 4.44i)7-s + (0.715 − 0.194i)9-s + (1.41 − 1.81i)11-s + (−2.91 + 5.68i)13-s + (−0.0130 + 0.0445i)15-s + (6.97 + 0.529i)17-s + (0.209 − 0.314i)19-s + (−7.03 − 2.19i)21-s + (−8.13 − 3.23i)23-s + (−3.36 − 3.69i)25-s + (−2.17 − 5.18i)27-s + (−5.14 + 2.04i)29-s + (3.60 − 0.832i)31-s + ⋯ |
| L(s) = 1 | + (−0.114 − 0.860i)3-s + (−0.0126 − 0.00559i)5-s + (0.781 − 1.68i)7-s + (0.238 − 0.0647i)9-s + (0.426 − 0.547i)11-s + (−0.807 + 1.57i)13-s + (−0.00335 + 0.0115i)15-s + (1.69 + 0.128i)17-s + (0.0480 − 0.0722i)19-s + (−1.53 − 0.479i)21-s + (−1.69 − 0.674i)23-s + (−0.672 − 0.739i)25-s + (−0.418 − 0.997i)27-s + (−0.954 + 0.379i)29-s + (0.647 − 0.149i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.952418 - 1.23048i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.952418 - 1.23048i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (8.49 + 9.73i)T \) |
| good | 3 | \( 1 + (0.198 + 1.48i)T + (-2.89 + 0.785i)T^{2} \) |
| 5 | \( 1 + (0.0282 + 0.0125i)T + (3.36 + 3.69i)T^{2} \) |
| 7 | \( 1 + (-2.06 + 4.44i)T + (-4.51 - 5.35i)T^{2} \) |
| 11 | \( 1 + (-1.41 + 1.81i)T + (-2.67 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.91 - 5.68i)T + (-7.60 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-6.97 - 0.529i)T + (16.8 + 2.56i)T^{2} \) |
| 19 | \( 1 + (-0.209 + 0.314i)T + (-7.35 - 17.5i)T^{2} \) |
| 23 | \( 1 + (8.13 + 3.23i)T + (16.7 + 15.7i)T^{2} \) |
| 29 | \( 1 + (5.14 - 2.04i)T + (21.0 - 19.9i)T^{2} \) |
| 31 | \( 1 + (-3.60 + 0.832i)T + (27.8 - 13.6i)T^{2} \) |
| 37 | \( 1 + (-5.79 - 1.57i)T + (31.9 + 18.7i)T^{2} \) |
| 41 | \( 1 + (-7.28 + 3.55i)T + (25.2 - 32.3i)T^{2} \) |
| 43 | \( 1 + (0.603 - 6.36i)T + (-42.2 - 8.08i)T^{2} \) |
| 47 | \( 1 + (0.0246 + 1.30i)T + (-46.9 + 1.77i)T^{2} \) |
| 53 | \( 1 + (0.848 + 4.93i)T + (-49.9 + 17.7i)T^{2} \) |
| 59 | \( 1 + (1.08 - 0.0820i)T + (58.3 - 8.89i)T^{2} \) |
| 61 | \( 1 + (6.52 - 6.64i)T + (-1.15 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-7.57 + 3.34i)T + (45.0 - 49.5i)T^{2} \) |
| 71 | \( 1 + (-6.58 + 0.751i)T + (69.1 - 15.9i)T^{2} \) |
| 73 | \( 1 + (4.31 - 3.23i)T + (20.4 - 70.0i)T^{2} \) |
| 79 | \( 1 + (4.94 - 13.1i)T + (-59.4 - 52.0i)T^{2} \) |
| 83 | \( 1 + (0.616 - 0.855i)T + (-26.2 - 78.7i)T^{2} \) |
| 89 | \( 1 + (0.357 + 1.22i)T + (-75.0 + 47.8i)T^{2} \) |
| 97 | \( 1 + (-5.87 - 1.35i)T + (87.1 + 42.5i)T^{2} \) |
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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
−10.15851803094000106806855069120, −9.645092897566968571215030871848, −8.098339387718757751239141399938, −7.64208256449998778025566769560, −6.85800469240127733050835407909, −6.00906041064017367247470420954, −4.45742849063509682885754566476, −3.90577991608243167401993881730, −1.96248547133170342641587478764, −0.915291663997196811640442309841,
1.86897475513865359340637325205, 3.16526952874085975666064580606, 4.39921377178170747315887894459, 5.55747239493737158896004173830, 5.66783610609610204140807621911, 7.69816204371267489306560269543, 7.947662317630658837442489835751, 9.464030532920294124676617895924, 9.667555612788293169099583933747, 10.63129443805530687159451682574
