L(s) = 1 | + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (0.835 − 0.732i)5-s + (0.923 − 0.382i)8-s + (−0.608 + 0.793i)9-s + (0.732 − 0.835i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.793 − 0.608i)17-s + (−0.499 + 0.866i)18-s + (0.617 − 0.923i)20-s + (0.0306 − 0.232i)25-s + (−1.12 − 0.860i)26-s + (−0.216 − 1.08i)29-s + (0.793 − 0.608i)32-s + ⋯ |
L(s) = 1 | + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (0.835 − 0.732i)5-s + (0.923 − 0.382i)8-s + (−0.608 + 0.793i)9-s + (0.732 − 0.835i)10-s + (−1 − i)13-s + (0.866 − 0.5i)16-s + (0.793 − 0.608i)17-s + (−0.499 + 0.866i)18-s + (0.617 − 0.923i)20-s + (0.0306 − 0.232i)25-s + (−1.12 − 0.860i)26-s + (−0.216 − 1.08i)29-s + (0.793 − 0.608i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.655765195\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.655765195\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.991 + 0.130i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.793 + 0.608i)T \) |
good | 3 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 5 | \( 1 + (-0.835 + 0.732i)T + (0.130 - 0.991i)T^{2} \) |
| 11 | \( 1 + (-0.991 + 0.130i)T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 19 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 23 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 29 | \( 1 + (0.216 + 1.08i)T + (-0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (-0.608 + 0.793i)T^{2} \) |
| 37 | \( 1 + (0.108 - 1.65i)T + (-0.991 - 0.130i)T^{2} \) |
| 41 | \( 1 + (-0.0761 + 0.382i)T + (-0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.12 - 1.46i)T + (-0.258 + 0.965i)T^{2} \) |
| 59 | \( 1 + (-0.965 - 0.258i)T^{2} \) |
| 61 | \( 1 + (-0.630 - 1.85i)T + (-0.793 + 0.608i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (1.57 + 0.534i)T + (0.793 + 0.608i)T^{2} \) |
| 79 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (1.63 - 0.324i)T + (0.923 - 0.382i)T^{2} \) |
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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
−8.643398774171405332026871177877, −7.79340509341281811360521437594, −7.25048524690780573808427788860, −6.07948821270604965305855747785, −5.45136593535720083868508903079, −5.12181641146545939122589498297, −4.23831885762843513765121234881, −2.93743532467631561619264879415, −2.44690739075334442720094291530, −1.24826894549178629095598830741,
1.75401684186257254299049495232, 2.56485190531222502261392815147, 3.39936150219773583379569857782, 4.18773484200273489544210493376, 5.32271259723078474361568681366, 5.77851115483670475186221545178, 6.69406988653687654377977071573, 6.96931963080234867251506297859, 7.972257529097417686756458556082, 8.946477700741616817725971939327