| L(s) = 1 | + (0.5 − 1.53i)2-s + (−3.16 − 0.672i)3-s + (−0.5 − 0.363i)4-s + (−0.5 − 0.866i)5-s + (−2.61 + 4.53i)6-s + (0.215 − 0.0960i)7-s + (1.80 − 1.31i)8-s + (6.82 + 3.03i)9-s + (−1.58 + 0.336i)10-s + (0.209 − 1.98i)11-s + (1.33 + 1.48i)12-s + (2.16 − 2.40i)13-s + (−0.0399 − 0.379i)14-s + (1 + 3.07i)15-s + (−1.50 − 4.61i)16-s + (0.0798 + 0.759i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 1.08i)2-s + (−1.82 − 0.388i)3-s + (−0.250 − 0.181i)4-s + (−0.223 − 0.387i)5-s + (−1.06 + 1.85i)6-s + (0.0815 − 0.0362i)7-s + (0.639 − 0.464i)8-s + (2.27 + 1.01i)9-s + (−0.500 + 0.106i)10-s + (0.0630 − 0.599i)11-s + (0.386 + 0.429i)12-s + (0.600 − 0.666i)13-s + (−0.0106 − 0.101i)14-s + (0.258 + 0.794i)15-s + (−0.375 − 1.15i)16-s + (0.0193 + 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.105237 + 0.973759i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105237 + 0.973759i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (-0.5 + 1.53i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (3.16 + 0.672i)T + (2.74 + 1.22i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.215 + 0.0960i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.209 + 1.98i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-2.16 + 2.40i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.0798 - 0.759i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (1.49 + 1.66i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-4.61 + 3.35i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.854 - 2.62i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.84 - 1.45i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (0.827 + 0.918i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.763 - 2.35i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.56 - 4.25i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (2.18 + 0.464i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 8.18T + 61T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.38 + 3.73i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.885 + 8.42i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (1.22 + 11.6i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (14.6 - 3.10i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-9.47 - 6.88i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-12.8 - 9.37i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.27136277838794688899868199713, −8.874918039595230097936793655513, −7.73811689747419667447122409016, −6.78329304887953857193893509408, −6.04593272225681182463781859014, −4.99409119905619728978412472071, −4.39448573548930614357429059339, −3.10556657297953477120340437343, −1.50705092370514977211226580736, −0.56906579653739646137767397002,
1.54690774430578196714892649507, 3.83449799193111635304736941703, 4.72042028204797129317005884926, 5.42326047624162397477317526747, 6.14200069447733079832408734410, 6.97874620203888313906928629832, 7.27516145714776935385383764415, 8.710563897516164088892071144164, 9.891661296549121398032222949144, 10.59215839523724529612017315659
