L(s) = 1 | + (1.22 − 2.12i)2-s + (−1.99 − 3.46i)4-s + (1.22 − 2.12i)5-s + (−0.5 + 2.59i)7-s − 4.89·8-s + (−2.99 − 5.19i)10-s + (2.44 + 4.24i)11-s − 4·13-s + (4.89 + 4.24i)14-s + (−1.99 + 3.46i)16-s + (−1.22 − 2.12i)17-s + (0.5 − 0.866i)19-s − 9.79·20-s + 11.9·22-s + (1.22 − 2.12i)23-s + ⋯ |
L(s) = 1 | + (0.866 − 1.49i)2-s + (−0.999 − 1.73i)4-s + (0.547 − 0.948i)5-s + (−0.188 + 0.981i)7-s − 1.73·8-s + (−0.948 − 1.64i)10-s + (0.738 + 1.27i)11-s − 1.10·13-s + (1.30 + 1.13i)14-s + (−0.499 + 0.866i)16-s + (−0.297 − 0.514i)17-s + (0.114 − 0.198i)19-s − 2.19·20-s + 2.55·22-s + (0.255 − 0.442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.789900 - 1.59341i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.789900 - 1.59341i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 2 | \( 1 + (-1.22 + 2.12i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.22 + 2.12i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 2.12i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.34T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-1.22 + 2.12i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.22 + 2.12i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.89 + 8.48i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 + (-1.22 + 2.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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
−12.18622442931276928058398029653, −11.80219272717407753281539100758, −10.20709336435470499856745072923, −9.586716397881844047947620189160, −8.741912310300545181441345269219, −6.68920028234694589583250605058, −5.04974381995710404371216687153, −4.73718544340895095236392094312, −2.85818645579476803603353314665, −1.67767735000749869672040150176,
3.21494881838272153135875681200, 4.40056290502220437786994293229, 5.84067678402851654791515757030, 6.60590713056930433224715436838, 7.35260581050469289555805905078, 8.496341649764835168194205485844, 9.901328454573310241433489773297, 10.94647435286020024635283465488, 12.28572656341934930708937920279, 13.54067783652982157923554419864