| L(s) = 1 | + (−1.67 + 0.450i)3-s + (−0.436 − 1.19i)5-s + (−3.53 + 0.622i)7-s + (2.59 − 1.50i)9-s + (4.59 + 1.67i)11-s + (1.75 + 1.47i)13-s + (1.27 + 1.80i)15-s + (0.393 + 0.227i)17-s + (5.43 − 3.13i)19-s + (5.62 − 2.63i)21-s + (−0.629 + 3.57i)23-s + (2.58 − 2.16i)25-s + (−3.66 + 3.68i)27-s + (6.09 + 7.26i)29-s + (0.352 + 0.0621i)31-s + ⋯ |
| L(s) = 1 | + (−0.965 + 0.260i)3-s + (−0.195 − 0.536i)5-s + (−1.33 + 0.235i)7-s + (0.864 − 0.502i)9-s + (1.38 + 0.504i)11-s + (0.485 + 0.407i)13-s + (0.327 + 0.467i)15-s + (0.0954 + 0.0551i)17-s + (1.24 − 0.719i)19-s + (1.22 − 0.574i)21-s + (−0.131 + 0.744i)23-s + (0.516 − 0.433i)25-s + (−0.704 + 0.709i)27-s + (1.13 + 1.34i)29-s + (0.0633 + 0.0111i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.926601 + 0.151286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.926601 + 0.151286i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.67 - 0.450i)T \) |
| good | 5 | \( 1 + (0.436 + 1.19i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (3.53 - 0.622i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-4.59 - 1.67i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.75 - 1.47i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.393 - 0.227i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.43 + 3.13i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.629 - 3.57i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.09 - 7.26i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.352 - 0.0621i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.46 + 4.26i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.66 - 5.56i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.37 + 6.52i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.475 - 2.69i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 4.30iT - 53T^{2} \) |
| 59 | \( 1 + (-9.78 + 3.56i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.68 - 15.2i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.26 + 1.50i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.92 + 5.06i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.44 + 5.96i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5.89 + 7.02i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (9.97 - 8.37i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-0.480 + 0.277i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.40 + 1.96i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.41221638959671778877856688151, −10.19490871644768359627335817252, −9.439480241112073769613595213716, −8.838584853726952182750810578144, −7.07448385962457584609867518567, −6.56193464157677750263542484021, −5.50553985483629939694060020553, −4.39791447873559504464013317528, −3.37099573448029210315853799612, −1.09687424730759802170543649448,
0.929539520172569096110741704084, 3.12606665801225466184446225407, 4.11629361669669946934994852042, 5.69593716600455819481629146559, 6.46246756611607821882988321505, 7.01309581126718229406150300420, 8.279368654059786627119662191487, 9.640220994646613808050622584491, 10.18791594075322648241556401626, 11.24084754379778853828803267528
