L(s) = 1 | + (−0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.439 + 0.524i)5-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.984i)12-s + (0.439 − 0.524i)15-s + (0.766 + 0.642i)16-s + (−0.233 − 0.642i)20-s + (0.592 − 1.62i)23-s + (0.0923 − 0.524i)25-s + (0.5 + 0.866i)27-s + (0.326 + 0.118i)31-s + 0.999·36-s + (−0.939 − 1.62i)37-s + (−0.592 − 0.342i)45-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.439 + 0.524i)5-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.984i)12-s + (0.439 − 0.524i)15-s + (0.766 + 0.642i)16-s + (−0.233 − 0.642i)20-s + (0.592 − 1.62i)23-s + (0.0923 − 0.524i)25-s + (0.5 + 0.866i)27-s + (0.326 + 0.118i)31-s + 0.999·36-s + (−0.939 − 1.62i)37-s + (−0.592 − 0.342i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8285332546\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8285332546\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.439 - 0.524i)T + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 13 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.592 + 1.62i)T + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 37 | \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + 1.96iT - T^{2} \) |
| 59 | \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \) |
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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
−8.566205431770709856485264672456, −7.950286028258937349553921911758, −6.86700539859842131713615084433, −6.48940886095712393827728523943, −5.57235417199948006801225353451, −4.97214837081066650101141859059, −3.90255912381001307393038261701, −2.78007155965612752466602809950, −1.88656625822407242026446373898, −0.57480218064960379523749741278,
1.29875893146418546164645615957, 2.97473811417611200024454703847, 3.65019555839717160049966558737, 4.54345887630728799729863111730, 5.18324758556152947685935760475, 5.64785702612269900454509086233, 6.75289819809402636083907571815, 7.86025401553258196937696328288, 8.500649698285444552211723301284, 9.212309334705313436622465708168