L(s) = 1 | + (0.831 + 1.14i)2-s + (−0.618 + 1.90i)4-s − 2.68·5-s + 4.15i·7-s + (−2.68 + 0.874i)8-s + (−2.23 − 3.07i)10-s + 4.35·11-s + (3.53 − 0.726i)13-s + (−4.74 + 3.45i)14-s + (−3.23 − 2.35i)16-s − 5.87·17-s − 5.71·19-s + (1.66 − 5.11i)20-s + (3.61 + 4.97i)22-s − 3.62·23-s + ⋯ |
L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s − 1.20·5-s + 1.56i·7-s + (−0.951 + 0.309i)8-s + (−0.707 − 0.973i)10-s + 1.31·11-s + (0.979 − 0.201i)13-s + (−1.26 + 0.922i)14-s + (−0.809 − 0.587i)16-s − 1.42·17-s − 1.31·19-s + (0.371 − 1.14i)20-s + (0.771 + 1.06i)22-s − 0.756·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.254125 - 0.960974i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.254125 - 0.960974i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.831 - 1.14i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-3.53 + 0.726i)T \) |
good | 5 | \( 1 + 2.68T + 5T^{2} \) |
| 7 | \( 1 - 4.15iT - 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 + 5.71T + 19T^{2} \) |
| 23 | \( 1 + 3.62T + 23T^{2} \) |
| 29 | \( 1 + 3.08iT - 29T^{2} \) |
| 31 | \( 1 - 9.28iT - 31T^{2} \) |
| 37 | \( 1 + 2.69T + 37T^{2} \) |
| 41 | \( 1 + 11.1iT - 41T^{2} \) |
| 43 | \( 1 + 3.80iT - 43T^{2} \) |
| 47 | \( 1 - 4.91iT - 47T^{2} \) |
| 53 | \( 1 - 1.17iT - 53T^{2} \) |
| 59 | \( 1 - 2.29T + 59T^{2} \) |
| 61 | \( 1 - 7.05iT - 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 2.08iT - 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 9.73T + 83T^{2} \) |
| 89 | \( 1 - 12.1iT - 89T^{2} \) |
| 97 | \( 1 - 5.13iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.84181065189939636957697474094, −9.140062017304931571508445863887, −8.650560295715117811707542536139, −8.229987954863400757106142163783, −6.88816145641264327056889936906, −6.35933304535990635054021112568, −5.42944530010882991528065489839, −4.21765154583559986863690475918, −3.71879380418241093869117658793, −2.31259338697273058214411809940,
0.38310231208535953241476236557, 1.74295979865637434036364258280, 3.52651570670812064833009943094, 4.15067072342384397738091985355, 4.44495736514931066096488563979, 6.32418986165799489406932598502, 6.73320510270460509112371599961, 7.968947326307864592391076676801, 8.812735133087766610759077348707, 9.775319079882581865097653391085