L(s) = 1 | + (−0.809 + 0.587i)3-s + (−0.427 − 1.31i)5-s + (−3.42 − 2.48i)7-s + (0.309 − 0.951i)9-s + (−3.30 − 0.224i)11-s + (−0.690 + 2.12i)13-s + (1.11 + 0.812i)15-s + (−1.73 − 5.34i)17-s + (6.35 − 4.61i)19-s + 4.23·21-s − 8.70·23-s + (2.50 − 1.81i)25-s + (0.309 + 0.951i)27-s + (0.381 + 0.277i)29-s + (−2.5 + 7.69i)31-s + ⋯ |
L(s) = 1 | + (−0.467 + 0.339i)3-s + (−0.190 − 0.587i)5-s + (−1.29 − 0.941i)7-s + (0.103 − 0.317i)9-s + (−0.997 − 0.0676i)11-s + (−0.191 + 0.589i)13-s + (0.288 + 0.209i)15-s + (−0.421 − 1.29i)17-s + (1.45 − 1.05i)19-s + 0.924·21-s − 1.81·23-s + (0.500 − 0.363i)25-s + (0.0594 + 0.183i)27-s + (0.0709 + 0.0515i)29-s + (−0.449 + 1.38i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.218213 - 0.440164i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.218213 - 0.440164i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (3.30 + 0.224i)T \) |
good | 5 | \( 1 + (0.427 + 1.31i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (3.42 + 2.48i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (0.690 - 2.12i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.73 + 5.34i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-6.35 + 4.61i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 8.70T + 23T^{2} \) |
| 29 | \( 1 + (-0.381 - 0.277i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.5 - 7.69i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.19 - 0.865i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 3.47T + 43T^{2} \) |
| 47 | \( 1 + (-4.16 + 3.02i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.0278 + 0.0857i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.73 - 1.26i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.899 - 2.76i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + (1.28 + 3.94i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (9.09 + 6.60i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.83 + 14.8i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.69 - 5.20i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + (1.57 - 4.84i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.68792207795782359564286521155, −10.55650705310093707762641415722, −9.796052516437679525439806979451, −8.990555684732157807305655715321, −7.48688366224391219299138175652, −6.72522471042489428378072165777, −5.34837938465586817702858416110, −4.37086770225083655303521559767, −3.02700544777891569491488600746, −0.38158001623064104733664442583,
2.43040824008892156268821023382, 3.66119059307798381225726804831, 5.61865639120173605737251025895, 6.07754986535200997213033164830, 7.37385290319372791007784453275, 8.256444503846197228215747827569, 9.718950673255643679009956104632, 10.31209555751091452301792920191, 11.42868843753162788512976272934, 12.47856927487417298743941999922