L(s) = 1 | + 2-s + (0.866 − 0.5i)3-s + 4-s + (0.866 − 0.5i)6-s + 8-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.866 − 0.5i)12-s + 16-s + (−0.866 + 1.5i)17-s + (0.499 − 0.866i)18-s + (0.866 + 1.5i)19-s + (−0.5 − 0.866i)22-s + (0.866 − 0.5i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
L(s) = 1 | + 2-s + (0.866 − 0.5i)3-s + 4-s + (0.866 − 0.5i)6-s + 8-s + (0.499 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.866 − 0.5i)12-s + 16-s + (−0.866 + 1.5i)17-s + (0.499 − 0.866i)18-s + (0.866 + 1.5i)19-s + (−0.5 − 0.866i)22-s + (0.866 − 0.5i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.248865810\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.248865810\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + 1.73T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−8.273955735626948169204756260094, −8.039530847459218805006846518860, −7.18766473381800320978115891078, −6.18256351694547769078107074552, −5.95591483206267339170616089488, −4.76048138334343864681289687447, −3.76881990575735506633775637596, −3.35646294671467833176834480307, −2.28205846763283607213959957695, −1.50192276516152470065178035986,
1.75848866271229996519326139013, 2.72372848457672153679145536314, 3.17711767920968668380279876335, 4.37610181681956975406612854120, 4.83694006765172745712026721795, 5.43779420884832283705897081469, 6.80845903876148038916726024043, 7.21770322760282856111854020336, 7.86693213849887474889753796690, 8.898461096579013213885228000291