| 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
−8.601210787736251452283567114581, −7.64893602029280037813170848950, −7.05417338786111630305243556601, −6.08890248327757243666290427977, −5.91226467228514597381191208541, −5.08416728227276067570101076998, −4.29256231449631530588458288255, −3.19507717556796983472041937912, −2.43506758899085231630649874501, −0.957216608346604294854305150230,
0.43778990589117934658639755139, 1.37005452738535426922838548994, 2.27518197208888701136112060493, 3.66066401784555330183746257264, 4.07542643680189639343869062064, 4.94550006468922623269185147684, 5.50409982882993187220297422579, 6.54785454165669105975724525716, 7.36133899707079316986979821576, 7.74376961124057779346326971116
