| L(s) = 1 | + (0.289 − 3.30i)2-s + (−6.90 − 1.21i)4-s + (−2.82 + 4.12i)5-s + (−7.54 − 5.28i)7-s + (−2.59 + 9.67i)8-s + (12.8 + 10.5i)10-s + (13.3 + 4.85i)11-s + (1.78 + 20.3i)13-s + (−19.6 + 23.4i)14-s + (4.85 + 1.76i)16-s + (1.84 + 6.90i)17-s + (−5.44 + 3.14i)19-s + (24.5 − 25.0i)20-s + (19.9 − 42.7i)22-s + (11.4 − 7.98i)23-s + ⋯ |
| L(s) = 1 | + (0.144 − 1.65i)2-s + (−1.72 − 0.304i)4-s + (−0.565 + 0.824i)5-s + (−1.07 − 0.754i)7-s + (−0.323 + 1.20i)8-s + (1.28 + 1.05i)10-s + (1.21 + 0.441i)11-s + (0.137 + 1.56i)13-s + (−1.40 + 1.67i)14-s + (0.303 + 0.110i)16-s + (0.108 + 0.406i)17-s + (−0.286 + 0.165i)19-s + (1.22 − 1.25i)20-s + (0.905 − 1.94i)22-s + (0.495 − 0.347i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.951016 - 0.0891916i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.951016 - 0.0891916i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.82 - 4.12i)T \) |
| good | 2 | \( 1 + (-0.289 + 3.30i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (7.54 + 5.28i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-13.3 - 4.85i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-1.78 - 20.3i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-1.84 - 6.90i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (5.44 - 3.14i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-11.4 + 7.98i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-15.3 - 18.2i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (2.53 - 14.3i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-10.8 - 40.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-37.1 - 31.1i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (21.6 + 46.5i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (-16.4 - 11.5i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (34.0 + 34.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (1.34 + 3.70i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-19.1 - 108. i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (101. - 8.86i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (25.4 - 44.1i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (7.97 - 29.7i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-48.7 - 58.0i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (23.3 + 2.04i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (131. - 75.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (112. - 52.5i)T + (6.04e3 - 7.20e3i)T^{2} \) |
| 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
−11.07800427418363426609633549433, −10.27614672372916930953516414407, −9.600856655354368791242713423372, −8.717346890012475096634024032546, −6.97844850328991636596383541904, −6.55946528176885788743265497074, −4.33761952282605356963178448114, −3.86470967678138867887097984618, −2.84324713120048360162695789171, −1.35901553456489009297981196826,
0.43759384676364397021768255939, 3.26872288219811053734565259925, 4.47487670548528754443040715236, 5.66100985508391857086668495082, 6.15442342590109654054239639358, 7.32039723743521917083658497289, 8.181725662197643441989935454634, 9.000433635707759441462673129202, 9.548654432950770216453195654365, 11.17912110708757605701739610183
