| L(s) = 1 | + (0.671 − 1.88i)2-s + (−3.09 − 2.53i)4-s + 4.54·5-s + 5.76i·7-s + (−6.84 + 4.13i)8-s + (3.05 − 8.56i)10-s + 10.1i·11-s − 13.5·13-s + (10.8 + 3.87i)14-s + (3.19 + 15.6i)16-s − 7.96·17-s + 4.35i·19-s + (−14.0 − 11.5i)20-s + (19.1 + 6.81i)22-s + 38.6i·23-s + ⋯ |
| L(s) = 1 | + (0.335 − 0.941i)2-s + (−0.774 − 0.632i)4-s + 0.909·5-s + 0.823i·7-s + (−0.855 + 0.517i)8-s + (0.305 − 0.856i)10-s + 0.921i·11-s − 1.04·13-s + (0.775 + 0.276i)14-s + (0.199 + 0.979i)16-s − 0.468·17-s + 0.229i·19-s + (−0.704 − 0.575i)20-s + (0.868 + 0.309i)22-s + 1.68i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.342196557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.342196557\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.671 + 1.88i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 5 | \( 1 - 4.54T + 25T^{2} \) |
| 7 | \( 1 - 5.76iT - 49T^{2} \) |
| 11 | \( 1 - 10.1iT - 121T^{2} \) |
| 13 | \( 1 + 13.5T + 169T^{2} \) |
| 17 | \( 1 + 7.96T + 289T^{2} \) |
| 23 | \( 1 - 38.6iT - 529T^{2} \) |
| 29 | \( 1 + 37.2T + 841T^{2} \) |
| 31 | \( 1 - 14.8iT - 961T^{2} \) |
| 37 | \( 1 - 6.53T + 1.36e3T^{2} \) |
| 41 | \( 1 + 34.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 40.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 50.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 31.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 12.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 108.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 67.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 17.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 121.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 103. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 41.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 74.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 74.3T + 9.40e3T^{2} \) |
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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
−10.27930591496191401364531311259, −9.536967468710199449478925909331, −9.257003984988787828957132991580, −7.919216518566132605390649447649, −6.66983398980431456964431735668, −5.50292954406880904161505623578, −5.06454609114327181821571740629, −3.70838749530317613771253492352, −2.37299656671615438536642816671, −1.76273504250252996377420114350,
0.40216484778506188900019287360, 2.43911622314540925696211511096, 3.80694874777030215258601182601, 4.81369742542873328843035548755, 5.74326580329913903857960898309, 6.57134210476938427127533764421, 7.35416734847341040809753151939, 8.320903175771962017499841107681, 9.206058193066636079574013413336, 9.984594978315323602775353657743
