L(s) = 1 | + (−0.326 + 1.85i)2-s + (−2.37 − 0.866i)4-s + (0.766 − 0.642i)5-s + (1.43 − 2.49i)8-s + (0.939 + 1.62i)10-s + (2.20 + 1.85i)16-s + (−0.173 − 0.300i)17-s + (−0.173 + 0.300i)19-s + (−2.37 + 0.866i)20-s + (−1.43 − 0.524i)23-s + (0.173 − 0.984i)25-s + (1.76 + 0.642i)31-s + (−1.93 + 1.62i)32-s + (0.613 − 0.223i)34-s + (−0.5 − 0.419i)38-s + ⋯ |
L(s) = 1 | + (−0.326 + 1.85i)2-s + (−2.37 − 0.866i)4-s + (0.766 − 0.642i)5-s + (1.43 − 2.49i)8-s + (0.939 + 1.62i)10-s + (2.20 + 1.85i)16-s + (−0.173 − 0.300i)17-s + (−0.173 + 0.300i)19-s + (−2.37 + 0.866i)20-s + (−1.43 − 0.524i)23-s + (0.173 − 0.984i)25-s + (1.76 + 0.642i)31-s + (−1.93 + 1.62i)32-s + (0.613 − 0.223i)34-s + (−0.5 − 0.419i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.011648677\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011648677\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
good | 2 | \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 11 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
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
−8.561364466608054065793248401387, −8.265741890858837523552086543522, −7.34603048394051273406385165147, −6.57241418570824096082603694745, −6.03595237564324473860447159782, −5.36591504470912314451047059121, −4.66832609433507449553983366227, −3.93721153015069067916793801777, −2.25369818063018688770212636884, −0.811097466293185967335315788683,
1.10495264906303686108661848055, 2.25385205013438496565366681181, 2.61169357018305344432707520978, 3.72890808550084697479666423444, 4.31021443371172629316023055738, 5.40371039676906504403199385019, 6.17993867863070230854920016061, 7.25318443636047307597611134456, 8.208736672409372398890366521904, 8.794794635544691458534582365592