| L(s) = 1 | + 0.732i·3-s + 2.73·5-s + (−2 − 1.73i)7-s + 2.46·9-s + 1.46·11-s + 1.26·13-s + 2i·15-s + 4i·17-s − 4.73i·19-s + (1.26 − 1.46i)21-s − 1.46i·23-s + 2.46·25-s + 4i·27-s − 6.92i·29-s + 6.92·31-s + ⋯ |
| L(s) = 1 | + 0.422i·3-s + 1.22·5-s + (−0.755 − 0.654i)7-s + 0.821·9-s + 0.441·11-s + 0.351·13-s + 0.516i·15-s + 0.970i·17-s − 1.08i·19-s + (0.276 − 0.319i)21-s − 0.305i·23-s + 0.492·25-s + 0.769i·27-s − 1.28i·29-s + 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.00058 + 0.0717254i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00058 + 0.0717254i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
| good | 3 | \( 1 - 0.732iT - 3T^{2} \) |
| 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 4.73iT - 19T^{2} \) |
| 23 | \( 1 + 1.46iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 9.46T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6.92iT - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 5.66T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 7.26iT - 83T^{2} \) |
| 89 | \( 1 - 1.07iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−9.996025536389139432858745888699, −9.578774410305349153868636643096, −8.686809451990751221017630383942, −7.45914817253780981243666476965, −6.46147214218717442219292574519, −6.02910932864788844230966864404, −4.65366531951765145214512223090, −3.90610014905891161189626883322, −2.59568178798192424842772104658, −1.21297243102133145705836811809,
1.34347767653339836483265897385, 2.39360175681318770337373424891, 3.59324964283399356011177373755, 4.99093364719918186989852795378, 5.95926781013282362279531303881, 6.52593883859654657262226011378, 7.39467132615601650124807765059, 8.585963619028463559145333955950, 9.466254575202283464592951797419, 9.869594483076249593744084452478
