L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.258 − 0.965i)3-s + (−0.866 − 0.499i)4-s − 6-s + (−1.67 − 0.448i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)12-s + (−0.866 + 1.50i)14-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)18-s + 1.73i·21-s + (−0.258 − 0.965i)23-s + (0.866 + 0.5i)24-s + (0.707 + 0.707i)27-s + (1.22 + 1.22i)28-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.258 − 0.965i)3-s + (−0.866 − 0.499i)4-s − 6-s + (−1.67 − 0.448i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.258 + 0.965i)12-s + (−0.866 + 1.50i)14-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)18-s + 1.73i·21-s + (−0.258 − 0.965i)23-s + (0.866 + 0.5i)24-s + (0.707 + 0.707i)27-s + (1.22 + 1.22i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4607626606\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4607626606\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.67 + 0.448i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - 1.73T + T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
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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
−9.993335208131445856260060772654, −9.119818955612273617195353448957, −8.192982929860179834031867866538, −7.02385530911636248959973001261, −6.28489795345498618421339487505, −5.48142322459405010553792579996, −4.09519809062861879861920047295, −3.13077407824374318557409812754, −2.11048115993674712793970360595, −0.41250302076596937869340095831,
3.12301448348452906000480041253, 3.65227845385552693109210692253, 4.87899304092205341042690132242, 5.73730923688241347352091735887, 6.36792502306276868229884461037, 7.25518524508168251797704567230, 8.484460740565455916603289451122, 9.316423864422804424021194515264, 9.655242476093236685718681678672, 10.60049075453534581466313028065