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
−12.64815579653282929434773259583, −11.71469214125048062234118552952, −11.34426992340366138618463872218, −10.33724931024525970365106413864, −8.742061341287691153607917218282, −7.981305841010488882846606172808, −6.50584517969459528885727822104, −4.93999305218596607796568252699, −4.19294559950726812922150933347, −3.43044609570163414626320191195,
0.65810428244765256131131500165, 3.44802235559410542081735227883, 4.86446441886803969222261684343, 5.41943377928318302938511255420, 7.08598768892827793312833616129, 8.074511314654919331825546788009, 8.711156813909541668266148657881, 10.69200512354069794637478427546, 11.77313792431904950732595260299, 12.14831843025401993309565343158