| L(s) = 1 | + i·2-s + 3-s − 4-s + 3.73i·5-s + i·6-s + 2.73i·7-s − i·8-s + 9-s − 3.73·10-s − 1.26i·11-s − 12-s − 2.73·14-s + 3.73i·15-s + 16-s + 5.73·17-s + i·18-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s + 1.66i·5-s + 0.408i·6-s + 1.03i·7-s − 0.353i·8-s + 0.333·9-s − 1.18·10-s − 0.382i·11-s − 0.288·12-s − 0.730·14-s + 0.963i·15-s + 0.250·16-s + 1.39·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.710221526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710221526\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.73iT - 5T^{2} \) |
| 7 | \( 1 - 2.73iT - 7T^{2} \) |
| 11 | \( 1 + 1.26iT - 11T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 - 4.73iT - 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 - 1.46iT - 31T^{2} \) |
| 37 | \( 1 + 3.53iT - 37T^{2} \) |
| 41 | \( 1 + 9.39iT - 41T^{2} \) |
| 43 | \( 1 - 9.66T + 43T^{2} \) |
| 47 | \( 1 + 2.19iT - 47T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 9.19T + 61T^{2} \) |
| 67 | \( 1 - 13.1iT - 67T^{2} \) |
| 71 | \( 1 - 4.73iT - 71T^{2} \) |
| 73 | \( 1 - 6.26iT - 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 + 0.196iT - 83T^{2} \) |
| 89 | \( 1 - 9.46iT - 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−10.15739175473434752809562957813, −9.495730171179944351419687545541, −8.502409067688451024404967714546, −7.71729589013903264774438913252, −7.12903314737018077618991785470, −5.95079464788934486621511728807, −5.66949197814614667171169777090, −3.88682029868656428418782459149, −3.17494057756455682483117360790, −2.10440536492097668068243114687,
0.75742029301580306283461037136, 1.72325044848115878292867966154, 3.23496183705299522066278176048, 4.30515398541692729113142390795, 4.77553777286908720560315081016, 5.94543613475437193748447700463, 7.52454646792504645429386169673, 7.947681083743630191523015879312, 8.986285342305806208053061244759, 9.534004997212959253691701630680
