| L(s) = 1 | + (0.866 − 0.5i)2-s + (0.793 + 0.608i)3-s + (0.499 − 0.866i)4-s + (0.991 + 0.130i)6-s − 0.999i·8-s + (0.258 + 0.965i)9-s + (−0.448 + 0.258i)11-s + (0.923 − 0.382i)12-s + (−0.5 − 0.866i)16-s + 1.98·17-s + (0.707 + 0.707i)18-s + 1.58i·19-s + (−0.258 + 0.448i)22-s + (0.608 − 0.793i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
| L(s) = 1 | + (0.866 − 0.5i)2-s + (0.793 + 0.608i)3-s + (0.499 − 0.866i)4-s + (0.991 + 0.130i)6-s − 0.999i·8-s + (0.258 + 0.965i)9-s + (−0.448 + 0.258i)11-s + (0.923 − 0.382i)12-s + (−0.5 − 0.866i)16-s + 1.98·17-s + (0.707 + 0.707i)18-s + 1.58i·19-s + (−0.258 + 0.448i)22-s + (0.608 − 0.793i)24-s + (−0.5 − 0.866i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.833376249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.833376249\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.793 - 0.608i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 1.98T + T^{2} \) |
| 19 | \( 1 - 1.58iT - T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.130 + 0.226i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.608 - 1.05i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 1.21iT - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + 1.84T + T^{2} \) |
| 97 | \( 1 + (-0.226 + 0.130i)T + (0.5 - 0.866i)T^{2} \) |
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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
−8.787953042268719559540231666477, −7.86412928756774927252737743857, −7.45110333569572128547824116047, −6.19381480848950290652857662123, −5.48770321937187280507957615810, −4.84054348237634495395962879527, −3.80310658545165585453114999895, −3.43857435055484385113814972945, −2.42620307081950177388392138754, −1.52775688435436435369141385290,
1.40881303361777353639516534741, 2.71620069194163141556831464289, 3.15986994181416274705576819987, 4.06226401448649985626430482396, 5.11112474747394538850362220865, 5.74665398215646335989429598426, 6.66712061338354724195162180584, 7.26794657517654587814965611468, 7.944448097543634624504937744047, 8.399544592739904302807555286369
