L(s) = 1 | + (−0.835 + 0.549i)3-s + (−0.939 + 0.342i)4-s + (−0.939 + 0.342i)7-s + (0.396 − 0.918i)9-s + (0.597 − 0.802i)12-s + (−0.744 − 1.72i)13-s + (0.766 − 0.642i)16-s + (1.57 + 0.571i)19-s + (0.597 − 0.802i)21-s + (−0.686 + 0.727i)25-s + (0.173 + 0.984i)27-s + (0.766 − 0.642i)28-s + (1.86 − 0.218i)31-s + (−0.0581 + 0.998i)36-s + (−0.786 + 0.0919i)37-s + ⋯ |
L(s) = 1 | + (−0.835 + 0.549i)3-s + (−0.939 + 0.342i)4-s + (−0.939 + 0.342i)7-s + (0.396 − 0.918i)9-s + (0.597 − 0.802i)12-s + (−0.744 − 1.72i)13-s + (0.766 − 0.642i)16-s + (1.57 + 0.571i)19-s + (0.597 − 0.802i)21-s + (−0.686 + 0.727i)25-s + (0.173 + 0.984i)27-s + (0.766 − 0.642i)28-s + (1.86 − 0.218i)31-s + (−0.0581 + 0.998i)36-s + (−0.786 + 0.0919i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 - 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5567266884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5567266884\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.835 - 0.549i)T \) |
| 7 | \( 1 + (0.939 - 0.342i)T \) |
| 109 | \( 1 + (-0.173 + 0.984i)T \) |
good | 2 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (0.686 - 0.727i)T^{2} \) |
| 11 | \( 1 + (-0.597 + 0.802i)T^{2} \) |
| 13 | \( 1 + (0.744 + 1.72i)T + (-0.686 + 0.727i)T^{2} \) |
| 17 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 19 | \( 1 + (-1.57 - 0.571i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
| 31 | \( 1 + (-1.86 + 0.218i)T + (0.973 - 0.230i)T^{2} \) |
| 37 | \( 1 + (0.786 - 0.0919i)T + (0.973 - 0.230i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1.05 - 0.882i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.893 - 0.448i)T^{2} \) |
| 53 | \( 1 + (-0.973 - 0.230i)T^{2} \) |
| 59 | \( 1 + (0.993 + 0.116i)T^{2} \) |
| 61 | \( 1 + (-1.36 - 1.44i)T + (-0.0581 + 0.998i)T^{2} \) |
| 67 | \( 1 + (-0.993 + 0.116i)T + (0.973 - 0.230i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.512 - 0.257i)T + (0.597 - 0.802i)T^{2} \) |
| 79 | \( 1 + (0.227 - 0.526i)T + (-0.686 - 0.727i)T^{2} \) |
| 83 | \( 1 + (-0.597 - 0.802i)T^{2} \) |
| 89 | \( 1 + (-0.893 + 0.448i)T^{2} \) |
| 97 | \( 1 + (-1.86 - 0.218i)T + (0.973 + 0.230i)T^{2} \) |
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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
−9.709212594203701584413813099047, −8.572400129430174497864876071506, −7.80131532353915280553804901828, −6.95145326930984359027755973128, −5.77011405791363398472394003428, −5.44158058308378497125446491464, −4.60625198119360027993905514589, −3.48278107267100786548289316275, −3.03113729050813361823283238096, −0.77302799325488865562171610855,
0.71530486341201516784758423617, 2.07288758834603746361836107850, 3.50406628008638303046006059505, 4.55343672766466416164411750068, 5.05525976612517559362708246098, 6.08398791737533154872254007267, 6.73901813972314340034594324206, 7.35083986081024360299390399152, 8.379036247496729964633854825781, 9.293299869608253909345579848392