| L(s) = 1 | + (0.0423 + 0.0113i)2-s + (−1.60 − 2.77i)3-s + (−3.46 − 1.99i)4-s + (−1.58 + 1.58i)5-s + (−0.0364 − 0.135i)6-s + (−8.09 + 2.16i)7-s + (−0.248 − 0.248i)8-s + (−0.646 + 1.11i)9-s + (−0.0849 + 0.0490i)10-s + (4.51 − 16.8i)11-s + 12.8i·12-s + (12.2 − 4.27i)13-s − 0.367·14-s + (6.92 + 1.85i)15-s + (7.98 + 13.8i)16-s + (−8.42 − 4.86i)17-s + ⋯ |
| L(s) = 1 | + (0.0211 + 0.00567i)2-s + (−0.534 − 0.926i)3-s + (−0.865 − 0.499i)4-s + (−0.316 + 0.316i)5-s + (−0.00607 − 0.0226i)6-s + (−1.15 + 0.309i)7-s + (−0.0310 − 0.0310i)8-s + (−0.0718 + 0.124i)9-s + (−0.00849 + 0.00490i)10-s + (0.410 − 1.53i)11-s + 1.06i·12-s + (0.944 − 0.329i)13-s − 0.0262·14-s + (0.461 + 0.123i)15-s + (0.499 + 0.864i)16-s + (−0.495 − 0.286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.153940 - 0.524044i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.153940 - 0.524044i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.58 - 1.58i)T \) |
| 13 | \( 1 + (-12.2 + 4.27i)T \) |
| good | 2 | \( 1 + (-0.0423 - 0.0113i)T + (3.46 + 2i)T^{2} \) |
| 3 | \( 1 + (1.60 + 2.77i)T + (-4.5 + 7.79i)T^{2} \) |
| 7 | \( 1 + (8.09 - 2.16i)T + (42.4 - 24.5i)T^{2} \) |
| 11 | \( 1 + (-4.51 + 16.8i)T + (-104. - 60.5i)T^{2} \) |
| 17 | \( 1 + (8.42 + 4.86i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (2.30 + 8.61i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (25.4 - 14.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-15.5 - 26.8i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-38.7 + 38.7i)T - 961iT^{2} \) |
| 37 | \( 1 + (-6.84 + 25.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + (11.9 + 3.18i)T + (1.45e3 + 840.5i)T^{2} \) |
| 43 | \( 1 + (17.8 + 10.3i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (17.0 + 17.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 - 41.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (47.8 - 12.8i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-3.44 + 5.96i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-104. - 28.0i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + (30.1 + 112. i)T + (-4.36e3 + 2.52e3i)T^{2} \) |
| 73 | \( 1 + (-11.7 - 11.7i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 122.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-23.5 + 23.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (1.57 - 5.88i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-3.85 - 14.3i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−13.73331714214300599620918281382, −13.30095438161337372480660072106, −12.06552715851803657250601956398, −10.93159771076261579305061720890, −9.499372908064892334583351085950, −8.339921810998323985167057674347, −6.51276857994221110470037760547, −5.86846080958512033904077040218, −3.61436955852351567218942645650, −0.54939939356462938279365767653,
3.89188623115179145698689512335, 4.64461471158323538950324988335, 6.52115672037613928150526235721, 8.264209170887481456236601782684, 9.615381807037100823009041022641, 10.22926497229135673606128534851, 11.90727761006148459080450335195, 12.81094877289714312396207963309, 13.86101164636511553938774409667, 15.39309310341004088320169323364
