| L(s) = 1 | + 5.60i·3-s − 2.23·5-s − 1.32i·7-s − 22.4·9-s + 11.2i·11-s + 17.4·13-s − 12.5i·15-s + 18·17-s + 5.29i·19-s + 7.41·21-s + 15.1i·23-s + 5.00·25-s − 75.1i·27-s + 8.83·29-s − 42.1i·31-s + ⋯ |
| L(s) = 1 | + 1.86i·3-s − 0.447·5-s − 0.189i·7-s − 2.49·9-s + 1.01i·11-s + 1.33·13-s − 0.835i·15-s + 1.05·17-s + 0.278i·19-s + 0.353·21-s + 0.659i·23-s + 0.200·25-s − 2.78i·27-s + 0.304·29-s − 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.565442 + 0.979375i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.565442 + 0.979375i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 3 | \( 1 - 5.60iT - 9T^{2} \) |
| 7 | \( 1 + 1.32iT - 49T^{2} \) |
| 11 | \( 1 - 11.2iT - 121T^{2} \) |
| 13 | \( 1 - 17.4T + 169T^{2} \) |
| 17 | \( 1 - 18T + 289T^{2} \) |
| 19 | \( 1 - 5.29iT - 361T^{2} \) |
| 23 | \( 1 - 15.1iT - 529T^{2} \) |
| 29 | \( 1 - 8.83T + 841T^{2} \) |
| 31 | \( 1 + 42.1iT - 961T^{2} \) |
| 37 | \( 1 - 33.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 25.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 10.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 28.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 44.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 77.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 36.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 97.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 15.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + 112. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 92.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 59.6T + 7.92e3T^{2} \) |
| 97 | \( 1 - 108.T + 9.40e3T^{2} \) |
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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
−14.89989074908984062418778409037, −13.70757727970655256232665744298, −12.01416148725693276241280105773, −10.98878378402053937591684864785, −10.08604724120706180698544387737, −9.172700467010364855643923581895, −7.889510102332390559813042332016, −5.82781281753287519400421558846, −4.45975206250091000394148278049, −3.47249429946372439329020688835,
1.06904770890671644453700554979, 3.11445754944512377494061572657, 5.76249425566487254658174167997, 6.74053706327825409381364169442, 8.033853818234939696989364129111, 8.680187085025324165906565093447, 10.91787020413572878136658579023, 11.82116438070507778379948410018, 12.72610495171032290374555163714, 13.64826041661203036235922948515
