| L(s) = 1 | + (0.745 − 0.625i)2-s + (1.09 − 1.34i)3-s + (−0.182 + 1.03i)4-s + (−3.79 + 0.668i)5-s + (−0.0221 − 1.68i)6-s + (−0.469 + 0.813i)7-s + (1.48 + 2.57i)8-s + (−0.598 − 2.93i)9-s + (−2.40 + 2.87i)10-s + (3.44 − 1.99i)11-s + (1.18 + 1.38i)12-s + (−0.353 − 0.970i)13-s + (0.158 + 0.901i)14-s + (−3.25 + 5.81i)15-s + (0.740 + 0.269i)16-s + (−0.804 − 0.958i)17-s + ⋯ |
| L(s) = 1 | + (0.527 − 0.442i)2-s + (0.632 − 0.774i)3-s + (−0.0913 + 0.518i)4-s + (−1.69 + 0.298i)5-s + (−0.00905 − 0.688i)6-s + (−0.177 + 0.307i)7-s + (0.525 + 0.909i)8-s + (−0.199 − 0.979i)9-s + (−0.761 + 0.907i)10-s + (1.03 − 0.600i)11-s + (0.343 + 0.398i)12-s + (−0.0979 − 0.269i)13-s + (0.0424 + 0.240i)14-s + (−0.841 + 1.50i)15-s + (0.185 + 0.0673i)16-s + (−0.195 − 0.232i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00017 - 0.313320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00017 - 0.313320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.09 + 1.34i)T \) |
| 19 | \( 1 + (4.22 - 1.09i)T \) |
| good | 2 | \( 1 + (-0.745 + 0.625i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (3.79 - 0.668i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.469 - 0.813i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.44 + 1.99i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.353 + 0.970i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.804 + 0.958i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.477 + 0.0842i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.32 - 2.79i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.26 - 2.46i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.10iT - 37T^{2} \) |
| 41 | \( 1 + (-2.04 - 0.745i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.00 + 5.68i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (4.96 - 5.91i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.34 + 7.60i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (5.26 - 4.41i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.740 + 4.19i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.918 + 1.09i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.438 - 2.48i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-8.28 - 3.01i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (1.50 - 4.12i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.66 + 2.11i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.56 + 2.75i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (6.91 + 8.24i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−14.86036177640490283508033973678, −13.96401895132747954097437652140, −12.60749711792443812435576862752, −12.02991697490941012482122704134, −11.12762428149140010281909131877, −8.739976614003834073886907081007, −8.000826290697523718378470133666, −6.75798124013537162424019600619, −4.11360234924180208293182806047, −3.05437170889461401330356583545,
3.97175359149696364371862271458, 4.57533662717537470428190240860, 6.76902985254044016241760085869, 8.136974956325077170454378385563, 9.422241979189123808409556571398, 10.71985049205457766411654248707, 11.99522279543726530385163985466, 13.40642318207116159801467409173, 14.67676927380734026700343793409, 15.18163399572293079157516030686
