L(s) = 1 | − 3-s + (0.5 + 0.866i)4-s + 9-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s − 1.73i·17-s + (1.5 − 0.866i)23-s − 27-s + (1.5 + 0.866i)29-s + (0.5 + 0.866i)36-s + (−0.5 − 0.866i)39-s + (−1 + 1.73i)43-s + (0.499 − 0.866i)48-s + (−0.5 − 0.866i)49-s + ⋯ |
L(s) = 1 | − 3-s + (0.5 + 0.866i)4-s + 9-s + (−0.5 − 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.499 + 0.866i)16-s − 1.73i·17-s + (1.5 − 0.866i)23-s − 27-s + (1.5 + 0.866i)29-s + (0.5 + 0.866i)36-s + (−0.5 − 0.866i)39-s + (−1 + 1.73i)43-s + (0.499 − 0.866i)48-s + (−0.5 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.067096459\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.067096459\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + 1.73iT - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - 1.73iT - T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.015043374718758063103371153342, −8.257483593347088192303630590600, −7.24697137807904832190276273387, −6.79241352350971923805166997634, −6.27995356210081432159370329403, −4.95776978465840941817539044792, −4.61433594913154577511896827356, −3.41381767625906504493466417817, −2.55538959797408841099497814970, −1.18501616234644251416905445017,
0.932065941643655549928485974495, 1.84550035349281504514333328572, 3.21322203612308548681624232342, 4.29487297546706745934905611886, 5.26951181423591410598760244973, 5.73268323545178832073801250807, 6.47391478819925392283317905583, 7.03644320710034070766493430297, 8.026157913204464922194035801680, 8.849477355669511631535547934629