L(s) = 1 | + (−1.41 + 0.0900i)2-s + (1.98 − 0.254i)4-s − 2.48i·5-s + (−2.77 + 0.537i)8-s + (0.224 + 3.51i)10-s + 4.60·11-s + 5.22·13-s + (3.87 − 1.00i)16-s − 5.61i·17-s + 3.19i·19-s + (−0.632 − 4.93i)20-s + (−6.50 + 0.414i)22-s + 0.718·23-s − 1.19·25-s + (−7.37 + 0.470i)26-s + ⋯ |
L(s) = 1 | + (−0.997 + 0.0636i)2-s + (0.991 − 0.127i)4-s − 1.11i·5-s + (−0.981 + 0.189i)8-s + (0.0708 + 1.11i)10-s + 1.38·11-s + 1.44·13-s + (0.967 − 0.252i)16-s − 1.36i·17-s + 0.732i·19-s + (−0.141 − 1.10i)20-s + (−1.38 + 0.0884i)22-s + 0.149·23-s − 0.238·25-s + (−1.44 + 0.0921i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.328059246\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.328059246\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0900i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.48iT - 5T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 - 5.22T + 13T^{2} \) |
| 17 | \( 1 + 5.61iT - 17T^{2} \) |
| 19 | \( 1 - 3.19iT - 19T^{2} \) |
| 23 | \( 1 - 0.718T + 23T^{2} \) |
| 29 | \( 1 - 4.53iT - 29T^{2} \) |
| 31 | \( 1 - 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 2.71T + 37T^{2} \) |
| 41 | \( 1 + 3.83iT - 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 5.41T + 47T^{2} \) |
| 53 | \( 1 - 2.06iT - 53T^{2} \) |
| 59 | \( 1 + 4.11T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 + 12.6iT - 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 - 9.63T + 73T^{2} \) |
| 79 | \( 1 - 8.83iT - 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 8.52iT - 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−9.227469749111170994452226405949, −8.572134592486617794286708100837, −7.86483023043571841715861581410, −6.85026239998081148908979821287, −6.17591531233360017594749587781, −5.26980233315528370788835737017, −4.14761354536484374141194598028, −3.11973604511272663866251480980, −1.55345536084745322797682502665, −0.915516864024767389906771201490,
1.11586910554042702533833135339, 2.22962146986591607316730576467, 3.39681407763862332012201230105, 4.03264125243277780327225504469, 5.88718588305287252623580524043, 6.40804919366943431995568975542, 6.96521760022345333795575849365, 7.924944224609429723564712063254, 8.740755062493517908717762321654, 9.256350345749907838851088911882