L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.500 + 0.866i)8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.173 − 0.984i)16-s + (−1.87 + 0.684i)17-s + (−0.173 + 0.984i)22-s + (−0.766 − 0.642i)24-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)32-s + (0.766 + 0.642i)33-s + (1.53 − 1.28i)34-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.500 + 0.866i)8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.173 − 0.984i)16-s + (−1.87 + 0.684i)17-s + (−0.173 + 0.984i)22-s + (−0.766 − 0.642i)24-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)32-s + (0.766 + 0.642i)33-s + (1.53 − 1.28i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6649874703\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6649874703\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
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
−9.282853918995797156868021930839, −8.625721937804754221439754983364, −7.912300848932111071654926253697, −6.90267644255066307415390916447, −6.30783012334885212419549940503, −5.48519093779802920444481530902, −4.58378914598783496287340410022, −3.75739115220094840020928697475, −2.58776253955292085717042848389, −1.34219073662482448052155765958,
0.60910059503852922791290117459, 1.95828242719723953100016845398, 2.36719589324225337726965757802, 3.82228519154998574299724434764, 4.65551833989971582584397563549, 6.04900231678502745651434416881, 6.85078991123578969011660908325, 7.06092673917340485063895094668, 7.891914451069816741927300565796, 8.804620283198922509362667604369