L(s) = 1 | + (2.59 − 1.50i)3-s + (2.59 + 2.59i)5-s + 7.30i·7-s + (4.47 − 7.81i)9-s + (11.3 + 11.3i)11-s + (−0.746 − 0.746i)13-s + (10.6 + 2.83i)15-s − 6.67i·17-s + (−22.1 − 22.1i)19-s + (10.9 + 18.9i)21-s + 21.4·23-s − 11.4i·25-s + (−0.153 − 26.9i)27-s + (−1.54 + 1.54i)29-s + 14.6·31-s + ⋯ |
L(s) = 1 | + (0.865 − 0.501i)3-s + (0.519 + 0.519i)5-s + 1.04i·7-s + (0.496 − 0.867i)9-s + (1.02 + 1.02i)11-s + (−0.0574 − 0.0574i)13-s + (0.710 + 0.188i)15-s − 0.392i·17-s + (−1.16 − 1.16i)19-s + (0.523 + 0.902i)21-s + 0.932·23-s − 0.459i·25-s + (−0.00567 − 0.999i)27-s + (−0.0531 + 0.0531i)29-s + 0.471·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.18851 + 0.163080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18851 + 0.163080i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.59 + 1.50i)T \) |
good | 5 | \( 1 + (-2.59 - 2.59i)T + 25iT^{2} \) |
| 7 | \( 1 - 7.30iT - 49T^{2} \) |
| 11 | \( 1 + (-11.3 - 11.3i)T + 121iT^{2} \) |
| 13 | \( 1 + (0.746 + 0.746i)T + 169iT^{2} \) |
| 17 | \( 1 + 6.67iT - 289T^{2} \) |
| 19 | \( 1 + (22.1 + 22.1i)T + 361iT^{2} \) |
| 23 | \( 1 - 21.4T + 529T^{2} \) |
| 29 | \( 1 + (1.54 - 1.54i)T - 841iT^{2} \) |
| 31 | \( 1 - 14.6T + 961T^{2} \) |
| 37 | \( 1 + (50.1 - 50.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 15.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + (26.3 - 26.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 36.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (50.9 + 50.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (12.1 + 12.1i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (27.5 + 27.5i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-4.84 - 4.84i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 74.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 3.47iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 103.T + 6.24e3T^{2} \) |
| 83 | \( 1 + (31.7 - 31.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 78.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 61.5T + 9.40e3T^{2} \) |
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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
−12.43256775047311090744019222401, −11.57199061835317290610885638901, −10.10286036878089679460895515826, −9.202986536303506771567402831626, −8.499374450382331999531120253443, −6.99034621028536983862158871385, −6.41833781521215244288052365007, −4.68552332635246295721285210474, −2.95560224276130595477486772100, −1.91066951882842899972020099549,
1.51476947601740274125027970058, 3.47195046256651586162742952329, 4.39577463528879420280075149965, 5.91765379231045675329779578621, 7.27932971400306644075697610157, 8.519135827941578895235461636769, 9.153994275836546979169334055706, 10.28502920735615048719978290819, 10.99257087350625821183686921025, 12.52145958599359440528639716029