| L(s) = 1 | + (−2.82 − i)3-s − 5.65·5-s + 6·7-s + (7.00 + 5.65i)9-s − 5.65·11-s − 10i·13-s + (16.0 + 5.65i)15-s − 22.6i·17-s − 2i·19-s + (−16.9 − 6i)21-s + 11.3i·23-s + 7.00·25-s + (−14.1 − 23.0i)27-s + 16.9·29-s − 22·31-s + ⋯ |
| L(s) = 1 | + (−0.942 − 0.333i)3-s − 1.13·5-s + 0.857·7-s + (0.777 + 0.628i)9-s − 0.514·11-s − 0.769i·13-s + (1.06 + 0.377i)15-s − 1.33i·17-s − 0.105i·19-s + (−0.808 − 0.285i)21-s + 0.491i·23-s + 0.280·25-s + (−0.523 − 0.851i)27-s + 0.585·29-s − 0.709·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2784150704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2784150704\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 + i)T \) |
| good | 5 | \( 1 + 5.65T + 25T^{2} \) |
| 7 | \( 1 - 6T + 49T^{2} \) |
| 11 | \( 1 + 5.65T + 121T^{2} \) |
| 13 | \( 1 + 10iT - 169T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 + 2iT - 361T^{2} \) |
| 23 | \( 1 - 11.3iT - 529T^{2} \) |
| 29 | \( 1 - 16.9T + 841T^{2} \) |
| 31 | \( 1 + 22T + 961T^{2} \) |
| 37 | \( 1 + 6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 62.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 86iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10T + 6.24e3T^{2} \) |
| 83 | \( 1 + 73.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−10.72113254253092819139375889143, −9.692224103763087356636129407903, −8.401392339078873072296468854523, −7.60690529114794596056638213542, −7.22380535663291817756595380499, −5.84111864986540933802151906691, −5.01491595816041265019629702989, −4.25620396100775565137892687840, −2.79983367555593361104335153374, −1.11745288644848325512040077586,
0.12816175011364755588063338365, 1.73923210825111024281738051487, 3.65756871604926836701494980801, 4.38962434286882916640837104821, 5.21464189783455695377190238288, 6.30079371806039156889915468798, 7.25452655274907011559472504234, 8.102016393216827622052017491635, 8.874017711825118644279461264619, 10.18184609961949945862848294983
