L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s − 3i·7-s + i·8-s + 9-s − 10-s + 3i·11-s + 12-s − 3·14-s + i·15-s + 16-s − i·18-s + 3i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 1.13i·7-s + 0.353i·8-s + 0.333·9-s − 0.316·10-s + 0.904i·11-s + 0.288·12-s − 0.801·14-s + 0.258i·15-s + 0.250·16-s − 0.235i·18-s + 0.688i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.171770500\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.171770500\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 9iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 14iT - 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 3iT - 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−7.74376961124057779346326971116, −7.36133899707079316986979821576, −6.54785454165669105975724525716, −5.50409982882993187220297422579, −4.94550006468922623269185147684, −4.07542643680189639343869062064, −3.66066401784555330183746257264, −2.27518197208888701136112060493, −1.37005452738535426922838548994, −0.43778990589117934658639755139,
0.957216608346604294854305150230, 2.43506758899085231630649874501, 3.19507717556796983472041937912, 4.29256231449631530588458288255, 5.08416728227276067570101076998, 5.91226467228514597381191208541, 6.08890248327757243666290427977, 7.05417338786111630305243556601, 7.64893602029280037813170848950, 8.601210787736251452283567114581