| L(s) = 1 | + (−0.413 − 0.459i)2-s + (0.169 − 1.60i)4-s + (−1.75 + 1.94i)5-s + (2.74 − 1.22i)7-s + (−1.80 + 1.31i)8-s + 1.61·10-s + (−3.31 + 0.0378i)11-s + (−1.72 + 0.366i)13-s + (−1.69 − 0.754i)14-s + (−1.81 − 0.385i)16-s + (0.5 + 1.53i)17-s + (−4.73 + 3.44i)19-s + (2.83 + 3.14i)20-s + (1.38 + 1.50i)22-s + (−1.73 − 3.00i)23-s + ⋯ |
| L(s) = 1 | + (−0.292 − 0.324i)2-s + (0.0845 − 0.804i)4-s + (−0.783 + 0.870i)5-s + (1.03 − 0.461i)7-s + (−0.639 + 0.464i)8-s + 0.511·10-s + (−0.999 + 0.0114i)11-s + (−0.478 + 0.101i)13-s + (−0.452 − 0.201i)14-s + (−0.453 − 0.0963i)16-s + (0.121 + 0.373i)17-s + (−1.08 + 0.789i)19-s + (0.633 + 0.703i)20-s + (0.296 + 0.321i)22-s + (−0.361 − 0.626i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0916411 + 0.177079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0916411 + 0.177079i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (3.31 - 0.0378i)T \) |
| good | 2 | \( 1 + (0.413 + 0.459i)T + (-0.209 + 1.98i)T^{2} \) |
| 5 | \( 1 + (1.75 - 1.94i)T + (-0.522 - 4.97i)T^{2} \) |
| 7 | \( 1 + (-2.74 + 1.22i)T + (4.68 - 5.20i)T^{2} \) |
| 13 | \( 1 + (1.72 - 0.366i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 1.53i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.73 + 3.00i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.08 + 1.81i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (2.79 - 0.593i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (10.9 + 4.85i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (3.11 - 5.40i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.169 + 1.60i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (2.97 - 9.14i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (1.07 - 10.2i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (7.68 + 1.63i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-4.78 - 8.28i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.71 + 5.29i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.61 - 1.90i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-6.33 - 7.03i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (0.692 + 0.147i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 - 0.527T + 89T^{2} \) |
| 97 | \( 1 + (9.39 + 10.4i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.56299705212249245130226827318, −9.964787034228389846394675532973, −8.583099688700708566417150750593, −7.969118327762431040530148159161, −7.11466458655602225577463591479, −6.14619260568166271869502341942, −5.06632658723388647345000050588, −4.16986916589665293680428434535, −2.80322057047166928682795845695, −1.70679342235085118965729824256,
0.099245298131578317865140551183, 2.14806709587009655147082975173, 3.41037307289947161918938431501, 4.68470661094258659774781300409, 5.12285455618462304151794068437, 6.61543401195706997478151145431, 7.60690586665240071533010670225, 8.239519955814472085550976227822, 8.567582584816788377212023668987, 9.583045780717151366373077394098
