L(s) = 1 | + (−0.866 − 1.5i)2-s + (−1 + 1.73i)4-s + (−0.866 + 0.5i)5-s + 1.73·8-s + (1.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s − 2i·20-s + (1.73 + i)23-s + (0.499 − 0.866i)25-s − 1.73i·31-s + (−1.5 − 0.866i)34-s + (0.866 − 1.5i)38-s + (−1.49 + 0.866i)40-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.5i)2-s + (−1 + 1.73i)4-s + (−0.866 + 0.5i)5-s + 1.73·8-s + (1.5 + 0.866i)10-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s − 2i·20-s + (1.73 + i)23-s + (0.499 − 0.866i)25-s − 1.73i·31-s + (−1.5 − 0.866i)34-s + (0.866 − 1.5i)38-s + (−1.49 + 0.866i)40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5290020268\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5290020268\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 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 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + iT - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 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
−10.35793902610328640727258960769, −9.520250157275780507614410392095, −8.861534449069625825067462670756, −7.70791821694235048212026851105, −7.42612250556313384202836444730, −5.79869084309986985325411994483, −4.35875613181792266951122632210, −3.41660803359277330885890510384, −2.71147287593864817993574700140, −1.13702285324214844227838720468,
0.962928893502074783929844499154, 3.29450488986264267878724666422, 4.77898558258981877605675601868, 5.31021223291255225747593025575, 6.58642958121350416520659305547, 7.19004234320433817750689980426, 7.986794862866683600103741233044, 8.726953843729756419896084618602, 9.209923366301314620166924550311, 10.30253613488168865032810565882