| L(s) = 1 | + (−1.38 − 0.209i)2-s + (0.826 − 0.563i)3-s + (−0.0255 − 0.00789i)4-s + (0.148 + 1.98i)5-s + (−1.26 + 0.609i)6-s + (2.57 − 0.613i)7-s + (2.56 + 1.23i)8-s + (0.365 − 0.930i)9-s + (0.209 − 2.79i)10-s + (−0.119 − 0.305i)11-s + (−0.0255 + 0.00789i)12-s + (3.29 − 4.13i)13-s + (−3.70 + 0.313i)14-s + (1.24 + 1.55i)15-s + (−3.26 − 2.22i)16-s + (1.11 + 1.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.982 − 0.148i)2-s + (0.477 − 0.325i)3-s + (−0.0127 − 0.00394i)4-s + (0.0665 + 0.888i)5-s + (−0.516 + 0.248i)6-s + (0.972 − 0.231i)7-s + (0.906 + 0.436i)8-s + (0.121 − 0.310i)9-s + (0.0661 − 0.882i)10-s + (−0.0361 − 0.0921i)11-s + (−0.00738 + 0.00227i)12-s + (0.914 − 1.14i)13-s + (−0.989 + 0.0838i)14-s + (0.320 + 0.402i)15-s + (−0.815 − 0.555i)16-s + (0.269 + 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.827546 - 0.107542i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.827546 - 0.107542i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.826 + 0.563i)T \) |
| 7 | \( 1 + (-2.57 + 0.613i)T \) |
| good | 2 | \( 1 + (1.38 + 0.209i)T + (1.91 + 0.589i)T^{2} \) |
| 5 | \( 1 + (-0.148 - 1.98i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (0.119 + 0.305i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (-3.29 + 4.13i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.11 - 1.03i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.37 - 2.38i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.09 - 1.94i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.304 + 1.33i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (3.05 - 5.28i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (9.90 - 3.05i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (7.06 + 3.40i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (2.80 - 1.34i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (8.81 + 1.32i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-9.94 - 3.06i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-1.05 + 14.1i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (-5.87 + 1.81i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (5.78 - 10.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.497 - 2.17i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (2.03 - 0.307i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (4.58 + 7.94i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.72 + 3.41i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-2.56 + 6.54i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−13.22182535089862395508372573340, −11.71244231868200178745065351327, −10.61818354466453454582338324292, −10.14461947565427447498134528163, −8.610202549798905587574126887047, −8.069803370665895652392431788776, −7.01627601300624187448965279012, −5.34624246827014915984775380707, −3.46943289142867234738530526064, −1.58040664251094572716636806858,
1.60004918036024996838806168880, 4.11673757478131680598331463308, 5.14488528211163192279238684414, 7.10435709620326514328963728174, 8.389522729067367281695678237256, 8.762234032917818293899548839409, 9.667571405896692898475052559111, 10.86527331990381543143067659247, 11.94722700278841904771905979129, 13.30187886532414570839701727790
