| L(s) = 1 | + (2.67 − 0.403i)2-s + (−0.403 − 1.68i)3-s + (5.08 − 1.56i)4-s + (−3.22 − 2.19i)5-s + (−1.75 − 4.34i)6-s + (1.60 + 2.10i)7-s + (8.09 − 3.89i)8-s + (−2.67 + 1.35i)9-s + (−9.51 − 4.58i)10-s + (−1.46 + 0.220i)11-s + (−4.69 − 7.93i)12-s + (0.0379 − 0.0967i)13-s + (5.14 + 4.97i)14-s + (−2.40 + 6.31i)15-s + (11.2 − 7.70i)16-s + (0.650 + 2.84i)17-s + ⋯ |
| L(s) = 1 | + (1.89 − 0.285i)2-s + (−0.232 − 0.972i)3-s + (2.54 − 0.784i)4-s + (−1.44 − 0.983i)5-s + (−0.717 − 1.77i)6-s + (0.607 + 0.794i)7-s + (2.86 − 1.37i)8-s + (−0.891 + 0.452i)9-s + (−3.00 − 1.44i)10-s + (−0.441 + 0.0665i)11-s + (−1.35 − 2.28i)12-s + (0.0105 − 0.0268i)13-s + (1.37 + 1.32i)14-s + (−0.620 + 1.63i)15-s + (2.82 − 1.92i)16-s + (0.157 + 0.691i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0660 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0660 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.25332 - 2.40746i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25332 - 2.40746i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.403 + 1.68i)T \) |
| 7 | \( 1 + (-1.60 - 2.10i)T \) |
| good | 2 | \( 1 + (-2.67 + 0.403i)T + (1.91 - 0.589i)T^{2} \) |
| 5 | \( 1 + (3.22 + 2.19i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (1.46 - 0.220i)T + (10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (-0.0379 + 0.0967i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-0.650 - 2.84i)T + (-15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 + (-2.10 + 0.650i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (2.59 + 0.800i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-1.43 - 2.47i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.41 + 6.18i)T + (-33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (9.02 + 6.15i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-2.49 + 1.70i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (0.524 - 0.0790i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (1.45 - 6.35i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-0.920 - 12.2i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-3.86 - 1.19i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (4.17 + 7.23i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.988 - 4.33i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (7.14 - 8.96i)T + (-16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + (6.90 - 11.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.0759 - 0.193i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-3.79 + 4.75i)T + (-19.8 - 86.7i)T^{2} \) |
| 97 | \( 1 + (-1.71 + 2.97i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.53987775449195905703979696785, −10.73520041487237010014131239511, −8.711675916745170638628661928938, −7.75798900176225067009571443714, −7.09026401585011590842650890272, −5.61192529203420003767764207842, −5.21188312445548749521569966736, −4.09859083412370207849646897212, −2.89404937275276755333701432442, −1.44807544556510598912795927833,
3.06283554379926568547921801791, 3.54230804555345161973329594216, 4.57345113939436619157494029587, 5.18521218957306061585035661718, 6.56344366483872834297221926254, 7.39961901196783165198693357006, 8.064506160639566040509119858394, 10.10417597557912386744266086345, 10.97172364525731828307391589140, 11.57974080840388036591194664038
