L(s) = 1 | + (1.19 + 2.06i)2-s + (1.34 − 1.08i)3-s + (−1.84 + 3.20i)4-s + (1.46 − 2.52i)5-s + (3.85 + 1.49i)6-s − 4.05·8-s + (0.637 − 2.93i)9-s + 6.97·10-s + (0.676 + 1.17i)11-s + (0.987 + 6.32i)12-s + (−0.733 + 1.26i)13-s + (−0.779 − 4.99i)15-s + (−1.13 − 1.96i)16-s − 3.31·17-s + (6.82 − 2.18i)18-s + 2.20·19-s + ⋯ |
L(s) = 1 | + (0.843 + 1.46i)2-s + (0.778 − 0.627i)3-s + (−0.924 + 1.60i)4-s + (0.653 − 1.13i)5-s + (1.57 + 0.608i)6-s − 1.43·8-s + (0.212 − 0.977i)9-s + 2.20·10-s + (0.204 + 0.353i)11-s + (0.284 + 1.82i)12-s + (−0.203 + 0.352i)13-s + (−0.201 − 1.29i)15-s + (−0.284 − 0.492i)16-s − 0.802·17-s + (1.60 − 0.514i)18-s + 0.506·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.46112 + 1.35679i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.46112 + 1.35679i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.34 + 1.08i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1.19 - 2.06i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.46 + 2.52i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.676 - 1.17i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.733 - 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 3.31T + 17T^{2} \) |
| 19 | \( 1 - 2.20T + 19T^{2} \) |
| 23 | \( 1 + (1.31 - 2.27i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.521 - 0.903i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.63 - 2.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + (0.904 - 1.56i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.98 + 3.44i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6.45T + 53T^{2} \) |
| 59 | \( 1 + (-6.10 + 10.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.279 + 0.484i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.40 - 11.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + (0.383 + 0.664i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.983 + 1.70i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6.40T + 89T^{2} \) |
| 97 | \( 1 + (4.14 + 7.17i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.83151480104775796097877538778, −9.930028942644466127170472916282, −8.895000824456389354143385935309, −8.502985882327170149331792932979, −7.31412799055537705283872153772, −6.71417034522240047383998360799, −5.59665394305336976432095297699, −4.76613075754981307508919051029, −3.63293664451329669973882684955, −1.79144653416798600015110309099,
2.01915495759851157004178041024, 2.86145403290015530706922731350, 3.67054182822774135195028996306, 4.77977937570827080987610021662, 5.89152174502276492377736251844, 7.21700438775901506193914524493, 8.654881484304614714604805266238, 9.667915885137344199638800108454, 10.31272382724822596043325709878, 10.85271525755819690604871292999