L(s) = 1 | + (−0.5 − 0.866i)3-s + 1.73i·7-s + (−0.499 + 0.866i)9-s + (1.5 + 0.866i)13-s − 19-s + (1.49 − 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + 31-s − 1.73i·37-s − 1.73i·39-s + (1.5 − 0.866i)43-s − 1.99·49-s + (0.5 + 0.866i)57-s + (−0.5 + 0.866i)61-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + 1.73i·7-s + (−0.499 + 0.866i)9-s + (1.5 + 0.866i)13-s − 19-s + (1.49 − 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + 31-s − 1.73i·37-s − 1.73i·39-s + (1.5 − 0.866i)43-s − 1.99·49-s + (0.5 + 0.866i)57-s + (−0.5 + 0.866i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8413133056\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8413133056\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 - 1.73iT - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + 1.73iT - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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
−10.63329832787308778048650249878, −9.139731769298838817332191762202, −8.776584838130294206236292347636, −7.88594942305067594626713894947, −6.74832591917933938275961302806, −5.97886334414949892191968872077, −5.51526903210991131555756247306, −4.12793233970819820594824663808, −2.60526066064608882421087183903, −1.69137513154022773232491994232,
0.971879820341198687377397898927, 3.18690109174963381921795330804, 4.07414465839919399666334338079, 4.67805484962293904559876738960, 6.05667542956163776937233349902, 6.55157865660213038101024641484, 7.84930549095709127944490498619, 8.516162003757247352936647733747, 9.698371078832326721865306984382, 10.44517312024561282274350383685