| L(s) = 1 | + 2.73i·5-s − 7-s + 0.732i·11-s − 4i·13-s + 6.19·17-s + 3.46i·19-s + 1.26·23-s − 2.46·25-s + 0.535i·29-s + 2·31-s − 2.73i·35-s − 2i·37-s + 7.66·41-s + 4.92i·43-s − 4·47-s + ⋯ |
| L(s) = 1 | + 1.22i·5-s − 0.377·7-s + 0.220i·11-s − 1.10i·13-s + 1.50·17-s + 0.794i·19-s + 0.264·23-s − 0.492·25-s + 0.0995i·29-s + 0.359·31-s − 0.461i·35-s − 0.328i·37-s + 1.19·41-s + 0.751i·43-s − 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.798023868\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.798023868\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2.73iT - 5T^{2} \) |
| 11 | \( 1 - 0.732iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6.19T + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 1.26T + 23T^{2} \) |
| 29 | \( 1 - 0.535iT - 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 7.66T + 41T^{2} \) |
| 43 | \( 1 - 4.92iT - 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 12.3iT - 61T^{2} \) |
| 67 | \( 1 - 14.3iT - 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 8.39T + 79T^{2} \) |
| 83 | \( 1 + 9.46iT - 83T^{2} \) |
| 89 | \( 1 + 1.80T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.433545826608503612915304501846, −7.69001716919089168510283737742, −7.27805561752105664466127826878, −6.29890659887889077741483902102, −5.82454729175353031089920187215, −4.94246263796692860256904512562, −3.69280272499413787295038250588, −3.19494830078205628904325098399, −2.41895393389129519303718974419, −1.02035178158320458750160937867,
0.63532690000773709421286009237, 1.56065878533319112200417076147, 2.77615096394674914971849600223, 3.76364435770939470697878748974, 4.58609954319951104595334145496, 5.22357653826211261477315976618, 6.01332261410957202028278571776, 6.82173260601758721881127608690, 7.64025328939077313918388685116, 8.373516571893996168930470112992
