| L(s) = 1 | + (−0.235 − 0.971i)2-s + (0.327 + 0.945i)3-s + (−0.888 + 0.458i)4-s + (−3.77 + 1.51i)5-s + (0.841 − 0.540i)6-s + (1.17 − 2.36i)7-s + (0.654 + 0.755i)8-s + (−0.786 + 0.618i)9-s + (2.35 + 3.31i)10-s + (−0.371 + 1.53i)11-s + (−0.723 − 0.690i)12-s + (−0.735 + 1.60i)13-s + (−2.57 − 0.588i)14-s + (−2.66 − 3.07i)15-s + (0.580 − 0.814i)16-s + (0.161 + 3.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.166 − 0.687i)2-s + (0.188 + 0.545i)3-s + (−0.444 + 0.229i)4-s + (−1.68 + 0.676i)5-s + (0.343 − 0.220i)6-s + (0.445 − 0.895i)7-s + (0.231 + 0.267i)8-s + (−0.262 + 0.206i)9-s + (0.746 + 1.04i)10-s + (−0.112 + 0.461i)11-s + (−0.208 − 0.199i)12-s + (−0.203 + 0.446i)13-s + (−0.689 − 0.157i)14-s + (−0.687 − 0.793i)15-s + (0.145 − 0.203i)16-s + (0.0390 + 0.819i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.238 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.422036 - 0.538031i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.422036 - 0.538031i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.235 + 0.971i)T \) |
| 3 | \( 1 + (-0.327 - 0.945i)T \) |
| 7 | \( 1 + (-1.17 + 2.36i)T \) |
| 23 | \( 1 + (2.29 + 4.20i)T \) |
| good | 5 | \( 1 + (3.77 - 1.51i)T + (3.61 - 3.45i)T^{2} \) |
| 11 | \( 1 + (0.371 - 1.53i)T + (-9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (0.735 - 1.60i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.161 - 3.38i)T + (-16.9 + 1.61i)T^{2} \) |
| 19 | \( 1 + (-0.365 + 7.67i)T + (-18.9 - 1.80i)T^{2} \) |
| 29 | \( 1 + (-5.73 + 3.68i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (3.80 - 0.733i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (-2.40 + 1.89i)T + (8.72 - 35.9i)T^{2} \) |
| 41 | \( 1 + (-0.242 + 1.68i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.582 + 0.672i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (4.32 + 7.48i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.59 - 0.916i)T + (52.0 + 10.0i)T^{2} \) |
| 59 | \( 1 + (-1.74 - 2.45i)T + (-19.2 + 55.7i)T^{2} \) |
| 61 | \( 1 + (-4.01 + 11.5i)T + (-47.9 - 37.7i)T^{2} \) |
| 67 | \( 1 + (-9.41 + 8.97i)T + (3.18 - 66.9i)T^{2} \) |
| 71 | \( 1 + (14.3 + 4.21i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (12.5 - 6.48i)T + (42.3 - 59.4i)T^{2} \) |
| 79 | \( 1 + (3.17 - 0.303i)T + (77.5 - 14.9i)T^{2} \) |
| 83 | \( 1 + (1.40 + 9.77i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-9.41 - 1.81i)T + (82.6 + 33.0i)T^{2} \) |
| 97 | \( 1 + (2.02 - 14.0i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.13641241271304133183777807855, −8.888732530860401113294723369520, −8.188259681570739637762585382038, −7.40950747401989064584942364547, −6.70661867534517999033882361578, −4.78268190027691321981157612527, −4.26033243599181585223084502778, −3.53746700286590871255187289819, −2.40310548851767957064888648253, −0.38203067106285602139052174678,
1.19954537193522256677248225777, 3.05146183209280062856093542307, 4.12214873962790521937664716316, 5.18842866506577687602983119856, 5.90353444084673259116532152763, 7.25192475565711994853458123446, 7.82291634906244701463990283941, 8.380254866656423759594691454829, 8.946284243780166352792283742100, 10.09812127936784387419656155262
