L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.766 + 0.642i)5-s + (0.5 + 0.866i)6-s + (−0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)10-s + (−0.766 − 0.642i)12-s + (0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (1.76 + 0.642i)17-s + (0.766 − 0.642i)18-s + (0.173 + 0.984i)19-s + (−0.173 + 0.984i)20-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (−0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.766 + 0.642i)5-s + (0.5 + 0.866i)6-s + (−0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)10-s + (−0.766 − 0.642i)12-s + (0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (1.76 + 0.642i)17-s + (0.766 − 0.642i)18-s + (0.173 + 0.984i)19-s + (−0.173 + 0.984i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5739738970\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5739738970\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 5 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
good | 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (0.439 + 1.20i)T + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.766 + 0.642i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19300592532233638510791369348, −8.896585396538226212018826830573, −8.077329621964971341892084011626, −7.67786720901028844723601911985, −6.83629109054342351852499722629, −6.15676387279241098184820537026, −5.24191633919443499673159492507, −3.51277184036533436500673191821, −2.45259707376716736381452179422, −1.04797317323592926310095591355,
0.991003962574094555346465805733, 2.98065265826437221623441681709, 3.62744954087301195012401871058, 4.82944844779882892378947252947, 5.61954190901784783595938253119, 7.03387969963769768194370800897, 7.76390850271010155414487900416, 8.605041070473554706069007144551, 9.353769100533018872638238039118, 9.799349936472566611060389157060