L(s) = 1 | + (−0.448 − 1.67i)2-s + (−1.73 + 1.00i)4-s + (1.22 + 1.22i)8-s + (0.500 − 0.866i)16-s + (1.22 − 1.22i)17-s + i·19-s + (0.448 − 1.67i)23-s + (−0.5 − 0.866i)31-s + (−2.59 − 1.5i)34-s + (1.67 − 0.448i)38-s − 3·46-s + (0.866 − 0.5i)49-s + (−1.22 − 1.22i)53-s + (0.5 − 0.866i)61-s + (−1.22 + 1.22i)62-s + ⋯ |
L(s) = 1 | + (−0.448 − 1.67i)2-s + (−1.73 + 1.00i)4-s + (1.22 + 1.22i)8-s + (0.500 − 0.866i)16-s + (1.22 − 1.22i)17-s + i·19-s + (0.448 − 1.67i)23-s + (−0.5 − 0.866i)31-s + (−2.59 − 1.5i)34-s + (1.67 − 0.448i)38-s − 3·46-s + (0.866 − 0.5i)49-s + (−1.22 − 1.22i)53-s + (0.5 − 0.866i)61-s + (−1.22 + 1.22i)62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7917841437\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7917841437\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.448 + 1.67i)T + (-0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
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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
−9.308472815041178705342511533907, −8.440550779664476881237491693804, −7.80726871342299415625009643829, −6.74093396486084067469018485759, −5.55502646170678423085201476134, −4.62923061854536508681877531381, −3.70045061984745150653111003066, −2.91790504771444762260053635574, −2.00505530340631323964884036226, −0.75923996546691384249376715489,
1.34072867693316563923049959516, 3.15417557356665821861929249629, 4.27105207027614702451851778858, 5.30586596627721320757874152539, 5.76553350246512544902265646387, 6.65634200192204850595994274915, 7.41851266708255396250063298360, 7.894872816927261182084227609854, 8.800429725717111628234847774942, 9.301988265804572592912333372229