| L(s) = 1 | + 2.03i·2-s − 0.127·4-s + 0.796i·5-s − 1.12·7-s + 7.86i·8-s − 1.61·10-s − 9.25·13-s − 2.28i·14-s − 16.4·16-s + 11.7i·17-s − 17.2·19-s − 0.101i·20-s − 5.11i·23-s + 24.3·25-s − 18.7i·26-s + ⋯ |
| L(s) = 1 | + 1.01i·2-s − 0.0317·4-s + 0.159i·5-s − 0.161·7-s + 0.983i·8-s − 0.161·10-s − 0.711·13-s − 0.163i·14-s − 1.03·16-s + 0.691i·17-s − 0.907·19-s − 0.00506i·20-s − 0.222i·23-s + 0.974·25-s − 0.723i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6599485604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6599485604\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.03iT - 4T^{2} \) |
| 5 | \( 1 - 0.796iT - 25T^{2} \) |
| 7 | \( 1 + 1.12T + 49T^{2} \) |
| 13 | \( 1 + 9.25T + 169T^{2} \) |
| 17 | \( 1 - 11.7iT - 289T^{2} \) |
| 19 | \( 1 + 17.2T + 361T^{2} \) |
| 23 | \( 1 + 5.11iT - 529T^{2} \) |
| 29 | \( 1 + 29.0iT - 841T^{2} \) |
| 31 | \( 1 + 52.7T + 961T^{2} \) |
| 37 | \( 1 - 37.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 75.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.36iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 34.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 69.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 24.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 77.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 55.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 64.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 120. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 110. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 14.0T + 9.40e3T^{2} \) |
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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
−10.14757603448679613429149470792, −9.217501822007095382354589138835, −8.282379840030265010695501295845, −7.71333185926373311800724200717, −6.72081226804023743488054492596, −6.26323388698159434175822723369, −5.25007003259947130243381452932, −4.36208381054812495490863213357, −2.98042363271584453514367179457, −1.87946633957535477012872002300,
0.18223111205969090886899805041, 1.59177747811205024505149652427, 2.60939685567580961909883463120, 3.51809779073078276316742084605, 4.56313406822614476917612312214, 5.56346588750448155873263563187, 6.82253543303562576483451262313, 7.27999799589828258982030921589, 8.595396414963862383250885237746, 9.336817183670115060934195664860
