| 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
−15.18163399572293079157516030686, −14.67676927380734026700343793409, −13.40642318207116159801467409173, −11.99522279543726530385163985466, −10.71985049205457766411654248707, −9.422241979189123808409556571398, −8.136974956325077170454378385563, −6.76902985254044016241760085869, −4.57533662717537470428190240860, −3.97175359149696364371862271458,
3.05437170889461401330356583545, 4.11360234924180208293182806047, 6.75798124013537162424019600619, 8.000826290697523718378470133666, 8.739976614003834073886907081007, 11.12762428149140010281909131877, 12.02991697490941012482122704134, 12.60749711792443812435576862752, 13.96401895132747954097437652140, 14.86036177640490283508033973678
