L(s) = 1 | + 1.73i·3-s − 1.99·9-s − 1.73·11-s − i·17-s − 1.73·19-s − 1.73i·27-s − 2.99i·33-s − 41-s − 49-s + 1.73·51-s − 2.99i·57-s + 1.73i·67-s + i·73-s + 0.999·81-s − 1.73i·83-s + ⋯ |
L(s) = 1 | + 1.73i·3-s − 1.99·9-s − 1.73·11-s − i·17-s − 1.73·19-s − 1.73i·27-s − 2.99i·33-s − 41-s − 49-s + 1.73·51-s − 2.99i·57-s + 1.73i·67-s + i·73-s + 0.999·81-s − 1.73i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1909982403\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1909982403\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 1.73iT - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + 1.73T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + iT - T^{2} \) |
| 19 | \( 1 + 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.73iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.73iT - T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - 2iT - T^{2} \) |
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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
−9.490500766666234831839415075838, −8.630020151629887469556433109339, −8.187190993340122413050511981255, −7.16124309371047992442225908326, −6.09279481991954981275129654767, −5.24116864240490708735342503743, −4.80596252749770486759981497688, −4.02249497625762682181511290640, −3.05909424569579090849823202821, −2.35749732806830664068271494801,
0.10337546939216796599228451727, 1.70215958604729267975548296727, 2.32326956396262588260463321118, 3.25668123468542451196049438812, 4.59004349436226075542588884414, 5.55821380405303154270464010164, 6.23404515549729927076674638751, 6.84887569111194782469599435438, 7.66559062196886698524791956676, 8.263351584569671659079884194421