L(s) = 1 | + (−0.409 − 0.409i)3-s + (0.738 − 2.54i)7-s − 2.66i·9-s − 3.54·11-s + (−2.95 − 2.95i)13-s + (−5.59 + 5.59i)17-s + 3.59·19-s + (−1.34 + 0.737i)21-s + (−0.0472 + 0.0472i)23-s + (−2.31 + 2.31i)27-s + 5.34i·29-s + 10.3i·31-s + (1.45 + 1.45i)33-s + (−7.80 − 7.80i)37-s + 2.41i·39-s + ⋯ |
L(s) = 1 | + (−0.236 − 0.236i)3-s + (0.278 − 0.960i)7-s − 0.888i·9-s − 1.07·11-s + (−0.818 − 0.818i)13-s + (−1.35 + 1.35i)17-s + 0.824·19-s + (−0.292 + 0.160i)21-s + (−0.00986 + 0.00986i)23-s + (−0.446 + 0.446i)27-s + 0.993i·29-s + 1.86i·31-s + (0.252 + 0.252i)33-s + (−1.28 − 1.28i)37-s + 0.386i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2217868650\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2217868650\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.738 + 2.54i)T \) |
good | 3 | \( 1 + (0.409 + 0.409i)T + 3iT^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 + (2.95 + 2.95i)T + 13iT^{2} \) |
| 17 | \( 1 + (5.59 - 5.59i)T - 17iT^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 23 | \( 1 + (0.0472 - 0.0472i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.34iT - 29T^{2} \) |
| 31 | \( 1 - 10.3iT - 31T^{2} \) |
| 37 | \( 1 + (7.80 + 7.80i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.63iT - 41T^{2} \) |
| 43 | \( 1 + (-6.93 + 6.93i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.44 - 3.44i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.646 - 0.646i)T - 53iT^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 - 3.51iT - 61T^{2} \) |
| 67 | \( 1 + (-1.70 - 1.70i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.54T + 71T^{2} \) |
| 73 | \( 1 + (2.43 + 2.43i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.72iT - 79T^{2} \) |
| 83 | \( 1 + (2.04 + 2.04i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.18T + 89T^{2} \) |
| 97 | \( 1 + (6.23 - 6.23i)T - 97iT^{2} \) |
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
−9.022365290791404722842436986072, −8.285400115331816323456182057671, −7.29040065002347414614437634459, −6.89663652975613813596452945803, −5.72235728558090388758014382616, −4.96372717008901228444160826056, −3.90573292633960391293201444051, −2.94535392817943356239820289035, −1.49954679558364279101131053492, −0.087999694707906043070170666112,
2.20174439261894660208248078753, 2.63725977781755023276932359284, 4.37804939646396601493788084241, 5.03185507760697807029079125297, 5.64200936289053196855353169881, 6.83106217698654193649666400361, 7.65825807814069354170286798000, 8.361270678850634738806628473095, 9.390148028419538772379016240769, 9.832331924195459011251569967401