L(s) = 1 | + (0.939 − 0.342i)2-s + (0.173 + 0.984i)5-s + (−0.499 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.173 + 0.984i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.766 + 0.642i)31-s + (0.766 + 0.642i)34-s + (0.173 − 0.984i)38-s + (−0.939 − 0.342i)40-s + (−0.5 + 0.866i)46-s + (1.53 + 1.28i)47-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (0.173 + 0.984i)5-s + (−0.499 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.173 + 0.984i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.766 + 0.642i)23-s + (−0.939 + 0.342i)25-s + (−0.766 + 0.642i)31-s + (0.766 + 0.642i)34-s + (0.173 − 0.984i)38-s + (−0.939 − 0.342i)40-s + (−0.5 + 0.866i)46-s + (1.53 + 1.28i)47-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.749373472\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.749373472\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.173 - 0.984i)T \) |
good | 2 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 11 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 13 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.939 + 0.342i)T^{2} \) |
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
−9.005991920492229339493829908394, −7.920527616216865570425610152624, −7.46122311847928670237367592430, −6.38099735873271990158097828731, −5.84201981046362160391332642691, −5.05596704782149599322642516965, −4.12012722682835764452730413348, −3.41266129674094107368156179787, −2.77051637224199330263393336784, −1.76840038390502273896940685863,
0.74330842508823620032946020676, 2.06688156982795190444875373392, 3.37762823176824652585141770970, 4.10432322931937498003178646044, 4.83493874406097764959002377095, 5.55873974686308667335075022640, 5.94936231229480138365758779245, 6.96740405448110871931394193234, 7.74973932907133942584934794344, 8.549064766565170851309974833994