L(s) = 1 | + (1.72 + 2.98i)5-s + (−0.5 + 0.866i)7-s + (−1 + 1.73i)11-s + (2.44 + 4.24i)13-s − 2·17-s − 7.44·19-s + (0.5 + 0.866i)23-s + (−3.44 + 5.97i)25-s + (1.44 − 2.51i)29-s + (3 + 5.19i)31-s − 3.44·35-s − 7.79·37-s + (−4.89 − 8.48i)41-s + (−1.44 + 2.51i)43-s + (4.89 − 8.48i)47-s + ⋯ |
L(s) = 1 | + (0.771 + 1.33i)5-s + (−0.188 + 0.327i)7-s + (−0.301 + 0.522i)11-s + (0.679 + 1.17i)13-s − 0.485·17-s − 1.70·19-s + (0.104 + 0.180i)23-s + (−0.689 + 1.19i)25-s + (0.269 − 0.466i)29-s + (0.538 + 0.933i)31-s − 0.583·35-s − 1.28·37-s + (−0.765 − 1.32i)41-s + (−0.221 + 0.382i)43-s + (0.714 − 1.23i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.312437827\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.312437827\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.72 - 2.98i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 7.44T + 19T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.44 + 2.51i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.79T + 37T^{2} \) |
| 41 | \( 1 + (4.89 + 8.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.44 - 2.51i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.89 + 8.48i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.72 - 9.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.55 + 2.68i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 + (-3.94 + 6.84i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + (-3.44 + 5.97i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.952624360315928896598724661350, −8.557328132270988421940719362337, −7.29103358144423426654618226280, −6.62004574339685855107489384451, −6.36342228030710423016367168209, −5.34991950241466356671536204392, −4.31703896368332522902798102218, −3.43711485177730441972578419761, −2.34188251176951576050834979200, −1.88651164777162190135914651856,
0.38626747172944161336706018056, 1.42631884643770872397490246610, 2.53563875895485583929757130106, 3.66740409410783654673776884485, 4.61494208318840483764078017493, 5.27343396883740495907600097923, 6.08541880532994953123462513614, 6.62155698540137283550764014903, 7.999058606039638107417423496361, 8.373293775134805274076345813916