L(s) = 1 | + (0.866 + 1.5i)3-s + (1.73 − i)4-s + (0.5 + 2.59i)7-s + (−1.5 + 2.59i)9-s + (3 + 1.73i)12-s + (−0.866 − 3.5i)13-s + (1.99 − 3.46i)16-s + (−1.23 + 0.330i)19-s + (−3.46 + 3i)21-s + (4.33 − 2.5i)25-s − 5.19·27-s + (3.46 + 4i)28-s + (−2.86 + 10.6i)31-s + 6i·36-s + (−2.40 − 8.96i)37-s + ⋯ |
L(s) = 1 | + (0.499 + 0.866i)3-s + (0.866 − 0.5i)4-s + (0.188 + 0.981i)7-s + (−0.5 + 0.866i)9-s + (0.866 + 0.499i)12-s + (−0.240 − 0.970i)13-s + (0.499 − 0.866i)16-s + (−0.282 + 0.0757i)19-s + (−0.755 + 0.654i)21-s + (0.866 − 0.5i)25-s − 1.00·27-s + (0.654 + 0.755i)28-s + (−0.514 + 1.92i)31-s + i·36-s + (−0.394 − 1.47i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61766 + 0.614127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61766 + 0.614127i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
| 13 | \( 1 + (0.866 + 3.5i)T \) |
good | 2 | \( 1 + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.23 - 0.330i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (2.86 - 10.6i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (2.40 + 8.96i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 41iT^{2} \) |
| 43 | \( 1 + 12.1iT - 43T^{2} \) |
| 47 | \( 1 + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.46 + 6i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.205 + 0.767i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 71iT^{2} \) |
| 73 | \( 1 + (15.4 + 4.13i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.06 - 10.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-0.0717 - 0.0717i)T + 97iT^{2} \) |
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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
−11.92232093257829574765379571428, −10.76271192307170268882449048859, −10.34917130005651616853333308230, −9.147179441100633216981922985294, −8.353078646265069113477089438741, −7.13188597114265530636648900114, −5.73926421859207841314585106419, −5.03176937330993522126408462253, −3.27393135133182694244412332112, −2.20513457695094955499927868300,
1.61023001958516016286348661855, 2.96980371316808273796371616269, 4.25880083933450224478574868539, 6.20667821347275736761351383920, 7.04849407572659673790749751181, 7.67632174342558231133387800131, 8.666686229138527669370415668387, 9.894906298134057226113959307038, 11.19903684803345788404414984717, 11.70001205060572290706203760056