L(s) = 1 | − 3.07·5-s − 3.99i·7-s + 1.89i·11-s + 1.06i·13-s + 7.08i·17-s − 3.73·19-s − 0.824·23-s + 4.46·25-s + 4.50·29-s + 5.84i·31-s + 12.2i·35-s + 6.91i·37-s + 3.79i·41-s − 2·43-s − 6.97·47-s + ⋯ |
L(s) = 1 | − 1.37·5-s − 1.50i·7-s + 0.572i·11-s + 0.296i·13-s + 1.71i·17-s − 0.856·19-s − 0.171·23-s + 0.892·25-s + 0.836·29-s + 1.04i·31-s + 2.07i·35-s + 1.13i·37-s + 0.593i·41-s − 0.304·43-s − 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.398 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.270599 + 0.412420i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.270599 + 0.412420i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.07T + 5T^{2} \) |
| 7 | \( 1 + 3.99iT - 7T^{2} \) |
| 11 | \( 1 - 1.89iT - 11T^{2} \) |
| 13 | \( 1 - 1.06iT - 13T^{2} \) |
| 17 | \( 1 - 7.08iT - 17T^{2} \) |
| 19 | \( 1 + 3.73T + 19T^{2} \) |
| 23 | \( 1 + 0.824T + 23T^{2} \) |
| 29 | \( 1 - 4.50T + 29T^{2} \) |
| 31 | \( 1 - 5.84iT - 31T^{2} \) |
| 37 | \( 1 - 6.91iT - 37T^{2} \) |
| 41 | \( 1 - 3.79iT - 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 6.97T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 1.89iT - 59T^{2} \) |
| 61 | \( 1 + 1.06iT - 61T^{2} \) |
| 67 | \( 1 - 9.19T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 5.92T + 73T^{2} \) |
| 79 | \( 1 - 6.12iT - 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 - 13.6iT - 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.56411116604606671451065272602, −9.789283347821775099439575030609, −8.316172303984689158207519295535, −8.067374063990042777312500957540, −7.00005384304301149855038776770, −6.44955703966816708358881980654, −4.69080046851218350005800373081, −4.15423042265553982984305965198, −3.35303581159794576460311244158, −1.44489562777165457114826293861,
0.24726067690025425058691021133, 2.42626453747348053072670578524, 3.34511029879566128483568629553, 4.52605534554818714567998922708, 5.44159350837571738116361173269, 6.42341808974145418262124301990, 7.50812635528284922901639708224, 8.265989881118638899327899455586, 8.895280660137932144343952516795, 9.753868706879189637100872329865