| L(s) = 1 | + (0.167 − 2.24i)3-s + (1.72 + 1.17i)5-s + (−2.19 − 1.48i)7-s + (−2.02 − 0.305i)9-s + (−3.20 + 0.482i)11-s + (−2.26 − 2.83i)13-s + (2.91 − 3.66i)15-s + (−3.38 − 1.04i)17-s + (−1.35 − 2.34i)19-s + (−3.69 + 4.66i)21-s + (0.439 − 0.135i)23-s + (−0.241 − 0.615i)25-s + (0.474 − 2.07i)27-s + (−0.802 − 3.51i)29-s + (1.87 − 3.24i)31-s + ⋯ |
| L(s) = 1 | + (0.0969 − 1.29i)3-s + (0.769 + 0.524i)5-s + (−0.828 − 0.560i)7-s + (−0.676 − 0.101i)9-s + (−0.966 + 0.145i)11-s + (−0.627 − 0.787i)13-s + (0.753 − 0.945i)15-s + (−0.820 − 0.253i)17-s + (−0.311 − 0.538i)19-s + (−0.805 + 1.01i)21-s + (0.0916 − 0.0282i)23-s + (−0.0483 − 0.123i)25-s + (0.0912 − 0.399i)27-s + (−0.149 − 0.653i)29-s + (0.336 − 0.583i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.231201 - 1.04899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.231201 - 1.04899i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (2.19 + 1.48i)T \) |
| good | 3 | \( 1 + (-0.167 + 2.24i)T + (-2.96 - 0.447i)T^{2} \) |
| 5 | \( 1 + (-1.72 - 1.17i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (3.20 - 0.482i)T + (10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (2.26 + 2.83i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (3.38 + 1.04i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (1.35 + 2.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.439 + 0.135i)T + (19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (0.802 + 3.51i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-1.87 + 3.24i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.25 + 5.80i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (7.91 - 3.81i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-9.23 - 4.44i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.153 + 0.391i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-3.48 - 3.23i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (5.32 - 3.62i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-5.97 + 5.53i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (7.97 - 13.8i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.653 + 2.86i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (4.67 + 11.9i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (4.83 + 8.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.56 - 3.22i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-12.7 - 1.92i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 6.30T + 97T^{2} \) |
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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
−10.07249662180165302858903931042, −9.125876490167932145899554870904, −7.82567434853593076099921364845, −7.37083694129440871570020435905, −6.46888480668416929596308186203, −5.88599743877879409669105968215, −4.50307831077024430609082682960, −2.81254978371553304512195079016, −2.27402341596394490094674051726, −0.49057331842062456392939358751,
2.11777287352957436027555231620, 3.26777911236134006052008997855, 4.44202909570156642524691293115, 5.21254316265320718974204011886, 6.02100172278892902707012448013, 7.11623401606419803762599565747, 8.589234823164085005657977319021, 9.074329596095508866729787520848, 9.878398583082118955023767885097, 10.29186126174095413912202774694
