L(s) = 1 | + (−0.5 − 0.866i)2-s − i·3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)6-s + 0.999·8-s − 9-s + 11-s + (0.866 + 0.499i)12-s + (−0.5 − 0.866i)16-s + (0.866 + 1.5i)17-s + (0.5 + 0.866i)18-s + (−0.866 + 1.5i)19-s + (−0.5 − 0.866i)22-s − 0.999i·24-s + 25-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s − i·3-s + (−0.499 + 0.866i)4-s + (−0.866 + 0.5i)6-s + 0.999·8-s − 9-s + 11-s + (0.866 + 0.499i)12-s + (−0.5 − 0.866i)16-s + (0.866 + 1.5i)17-s + (0.5 + 0.866i)18-s + (−0.866 + 1.5i)19-s + (−0.5 − 0.866i)22-s − 0.999i·24-s + 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9197816928\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9197816928\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
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
−8.574605527160711629369200975614, −8.059385924334129925998550503318, −7.44326334327915250342054211577, −6.37817451672988870434362248922, −5.96234637458455512018416966159, −4.61136534672556291796842267615, −3.72170849934600115620809703231, −2.97638854570145715238898231671, −1.75429709737129956512768424967, −1.26444431298471443799959088914,
0.74097185769453194174198283375, 2.43142907708803100381118666289, 3.56971840994745540480434544119, 4.52080284368441277699495124567, 5.05585346049678181027887119045, 5.82527307035005143609739969349, 6.76855734832522849102098998975, 7.22307545161002484410174962855, 8.317435577159660187440907300782, 8.980819439312248161163157150158