| L(s) = 1 | + (0.152 + 1.40i)2-s + (0.0477 + 1.73i)3-s + (−1.95 + 0.429i)4-s + (−0.988 + 2.71i)5-s + (−2.42 + 0.331i)6-s + (0.984 + 0.173i)7-s + (−0.902 − 2.68i)8-s + (−2.99 + 0.165i)9-s + (−3.96 − 0.974i)10-s + (−3.71 + 1.35i)11-s + (−0.837 − 3.36i)12-s + (0.633 − 0.531i)13-s + (−0.0936 + 1.41i)14-s + (−4.74 − 1.58i)15-s + (3.63 − 1.67i)16-s + (−0.212 + 0.122i)17-s + ⋯ |
| L(s) = 1 | + (0.108 + 0.994i)2-s + (0.0275 + 0.999i)3-s + (−0.976 + 0.214i)4-s + (−0.441 + 1.21i)5-s + (−0.990 + 0.135i)6-s + (0.372 + 0.0656i)7-s + (−0.319 − 0.947i)8-s + (−0.998 + 0.0551i)9-s + (−1.25 − 0.308i)10-s + (−1.12 + 0.408i)11-s + (−0.241 − 0.970i)12-s + (0.175 − 0.147i)13-s + (−0.0250 + 0.377i)14-s + (−1.22 − 0.408i)15-s + (0.907 − 0.419i)16-s + (−0.0514 + 0.0297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.539747 - 0.601364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.539747 - 0.601364i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.152 - 1.40i)T \) |
| 3 | \( 1 + (-0.0477 - 1.73i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (0.988 - 2.71i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (3.71 - 1.35i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.633 + 0.531i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.212 - 0.122i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.07 - 3.50i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.46 + 8.32i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (6.46 - 7.70i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.64 - 0.290i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.90 - 8.50i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.575 + 0.685i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.711 - 1.95i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.44 + 8.21i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 0.346iT - 53T^{2} \) |
| 59 | \( 1 + (7.21 + 2.62i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.228 - 1.29i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.53 + 6.59i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.35 - 9.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.27 - 3.93i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.39 + 7.61i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.88 + 4.94i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-9.44 - 5.45i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.36 - 3.04i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.61974234606097994436452741511, −10.21721988753600135865432393334, −9.214262690188442434509219045851, −8.169936709339894979137135726794, −7.60833627710754828378175359594, −6.62405287658354470047731037208, −5.55663239758663455004595502380, −4.82790537836166146172683571407, −3.70599065292578006753203888209, −2.86865100705168436553470201496,
0.40927214920666226129967768814, 1.54052310326804290005742551260, 2.80178850003169570488548850645, 4.04153188573262225472565253080, 5.25944335302000601630667132343, 5.70640390799974634532069982498, 7.72750771569189563827003467517, 7.80359246188309848944496525073, 9.020047363518057521827981904380, 9.504147728597709288898212906808
