L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·7-s + 0.999i·8-s + i·9-s + (1.36 − 1.36i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.499 + 1.86i)22-s − 1.73i·23-s + (−0.866 − 0.499i)28-s + (−0.366 − 0.366i)29-s + (0.866 + 0.499i)32-s + (0.866 + 0.499i)36-s + (−0.366 + 0.366i)37-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − i·7-s + 0.999i·8-s + i·9-s + (1.36 − 1.36i)11-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)18-s + (−0.499 + 1.86i)22-s − 1.73i·23-s + (−0.866 − 0.499i)28-s + (−0.366 − 0.366i)29-s + (0.866 + 0.499i)32-s + (0.866 + 0.499i)36-s + (−0.366 + 0.366i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8333887919\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8333887919\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 + 1.73iT - T^{2} \) |
| 29 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.580437093427732412155654898049, −8.357965508909631835516973292634, −7.46230189121604055678046263217, −6.63221714762607734721407553622, −6.21061330075575895872582058406, −5.12328174492143650455433284701, −4.28083942749858679936669280304, −3.18076547897161398174192070245, −1.88408178851038344834700896323, −0.77195613501369203812415190822,
1.39548302130376866648924344970, 2.15958436808587479820396933396, 3.41571398030758557080556803499, 3.96849873453520379295202759791, 5.22847631866372422830348488825, 6.26686877433515050027682524246, 6.93647851665496142405925755543, 7.54975804021047388505208163311, 8.674924498746235911296829225562, 9.156303259008504267990847613084