| L(s) = 1 | + (−1.28 − 1.52i)2-s + (−0.680 + 3.94i)4-s − 3.40·5-s + 4.01i·7-s + (6.90 − 4.03i)8-s + (4.38 + 5.21i)10-s − 13.8i·11-s + 13.1·13-s + (6.13 − 5.16i)14-s + (−15.0 − 5.36i)16-s − 31.1·17-s + 4.35i·19-s + (2.31 − 13.4i)20-s + (−21.2 + 17.8i)22-s + 18.4i·23-s + ⋯ |
| L(s) = 1 | + (−0.644 − 0.764i)2-s + (−0.170 + 0.985i)4-s − 0.681·5-s + 0.572i·7-s + (0.863 − 0.504i)8-s + (0.438 + 0.521i)10-s − 1.26i·11-s + 1.00·13-s + (0.438 − 0.369i)14-s + (−0.942 − 0.335i)16-s − 1.83·17-s + 0.229i·19-s + (0.115 − 0.671i)20-s + (−0.964 + 0.811i)22-s + 0.800i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9596049298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9596049298\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 + 1.52i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 + 3.40T + 25T^{2} \) |
| 7 | \( 1 - 4.01iT - 49T^{2} \) |
| 11 | \( 1 + 13.8iT - 121T^{2} \) |
| 13 | \( 1 - 13.1T + 169T^{2} \) |
| 17 | \( 1 + 31.1T + 289T^{2} \) |
| 23 | \( 1 - 18.4iT - 529T^{2} \) |
| 29 | \( 1 - 43.7T + 841T^{2} \) |
| 31 | \( 1 - 27.9iT - 961T^{2} \) |
| 37 | \( 1 - 40.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 28.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 81.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 59.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 38.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 59.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 29.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 12.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 5.79iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 41.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.805iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 136. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 150.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 50.3T + 9.40e3T^{2} \) |
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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
−10.48872235567260162544384386679, −9.175246815531119214283952419528, −8.650515298331885323543284942801, −8.048546306953436137818031894334, −6.87149515329463868745436974803, −5.83296542963928845027454806314, −4.34936175415449108636119306770, −3.50429616826672707986820278755, −2.40754676925281621089761441952, −0.855335708888654821245613271960,
0.62226075218673611660221264322, 2.20444215356167339474147571869, 4.19340002268828156715093178047, 4.64148259732612498061460034347, 6.22301124300796728024814652433, 6.82630936520215496017245221944, 7.71901642492644020833675049940, 8.442551394049231251702230849102, 9.312206084306721669269414322697, 10.20623401929073375940209230302
