| L(s) = 1 | + (1.10 + 0.194i)2-s + (−0.867 − 1.49i)3-s + (−0.698 − 0.254i)4-s + (−3.85 − 1.40i)5-s + (−0.666 − 1.82i)6-s + (2.61 + 0.417i)7-s + (−2.66 − 1.53i)8-s + (−1.49 + 2.60i)9-s + (−3.97 − 2.29i)10-s + (−0.344 − 0.946i)11-s + (0.225 + 1.26i)12-s + (1.20 − 3.32i)13-s + (2.80 + 0.969i)14-s + (1.24 + 6.99i)15-s + (−1.50 − 1.26i)16-s + (3.07 − 5.32i)17-s + ⋯ |
| L(s) = 1 | + (0.780 + 0.137i)2-s + (−0.501 − 0.865i)3-s + (−0.349 − 0.127i)4-s + (−1.72 − 0.626i)5-s + (−0.272 − 0.744i)6-s + (0.987 + 0.157i)7-s + (−0.941 − 0.543i)8-s + (−0.497 + 0.867i)9-s + (−1.25 − 0.726i)10-s + (−0.103 − 0.285i)11-s + (0.0649 + 0.365i)12-s + (0.335 − 0.921i)13-s + (0.749 + 0.259i)14-s + (0.320 + 1.80i)15-s + (−0.375 − 0.315i)16-s + (0.746 − 1.29i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.635 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.356443 - 0.755304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.356443 - 0.755304i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.867 + 1.49i)T \) |
| 7 | \( 1 + (-2.61 - 0.417i)T \) |
| good | 2 | \( 1 + (-1.10 - 0.194i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (3.85 + 1.40i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (0.344 + 0.946i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.20 + 3.32i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.07 + 5.32i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.08 - 1.20i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.00 + 0.353i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 1.94i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.284 + 0.781i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 + (9.37 + 3.41i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.78 + 10.1i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (6.83 - 2.48i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.60 + 2.08i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.12 + 1.78i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.28 - 3.53i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.338 + 1.91i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.21 + 3.00i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.26iT - 73T^{2} \) |
| 79 | \( 1 + (-1.27 + 7.23i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.39 + 2.69i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.56 - 11.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.98 - 1.05i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.14831843025401993309565343158, −11.77313792431904950732595260299, −10.69200512354069794637478427546, −8.711156813909541668266148657881, −8.074511314654919331825546788009, −7.08598768892827793312833616129, −5.41943377928318302938511255420, −4.86446441886803969222261684343, −3.44802235559410542081735227883, −0.65810428244765256131131500165,
3.43044609570163414626320191195, 4.19294559950726812922150933347, 4.93999305218596607796568252699, 6.50584517969459528885727822104, 7.981305841010488882846606172808, 8.742061341287691153607917218282, 10.33724931024525970365106413864, 11.34426992340366138618463872218, 11.71469214125048062234118552952, 12.64815579653282929434773259583
