L(s) = 1 | + (−1.53 + 0.411i)2-s + (0.454 − 0.262i)4-s + (0.0206 + 0.0769i)5-s + (−2.16 − 1.52i)7-s + (1.65 − 1.65i)8-s + (−0.0632 − 0.109i)10-s + (−4.08 − 1.09i)11-s + (−0.565 − 3.56i)13-s + (3.94 + 1.44i)14-s + (−2.38 + 4.13i)16-s + (2.90 + 5.02i)17-s + (1.36 + 5.11i)19-s + (0.0295 + 0.0295i)20-s + 6.71·22-s + (0.755 + 0.436i)23-s + ⋯ |
L(s) = 1 | + (−1.08 + 0.290i)2-s + (0.227 − 0.131i)4-s + (0.00921 + 0.0343i)5-s + (−0.818 − 0.575i)7-s + (0.585 − 0.585i)8-s + (−0.0200 − 0.0346i)10-s + (−1.23 − 0.329i)11-s + (−0.156 − 0.987i)13-s + (1.05 + 0.386i)14-s + (−0.596 + 1.03i)16-s + (0.704 + 1.21i)17-s + (0.314 + 1.17i)19-s + (0.00661 + 0.00661i)20-s + 1.43·22-s + (0.157 + 0.0909i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.280094 + 0.332692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.280094 + 0.332692i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.16 + 1.52i)T \) |
| 13 | \( 1 + (0.565 + 3.56i)T \) |
good | 2 | \( 1 + (1.53 - 0.411i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-0.0206 - 0.0769i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (4.08 + 1.09i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.90 - 5.02i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.36 - 5.11i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.755 - 0.436i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.362T + 29T^{2} \) |
| 31 | \( 1 + (1.34 + 0.361i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-1.00 - 3.76i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (7.70 - 7.70i)T - 41iT^{2} \) |
| 43 | \( 1 + 2.65iT - 43T^{2} \) |
| 47 | \( 1 + (-2.79 + 0.748i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.26 - 9.12i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.573 + 2.14i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-3.63 - 2.09i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.61 - 9.76i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (3.65 + 3.65i)T + 71iT^{2} \) |
| 73 | \( 1 + (3.08 - 11.4i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (4.27 - 7.40i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.91 + 4.91i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.78 + 2.08i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (6.04 - 6.04i)T - 97iT^{2} \) |
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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
−10.34663972628446483823324708149, −9.802162618303872953584285511666, −8.589204006759280167545462783304, −8.005907462869784614357827719075, −7.37066967436059352519829667493, −6.30304313876420938817701213272, −5.36132972426907067882861735976, −3.94929634597170319738874177462, −2.97845291701405458051289059334, −1.08146910189746664730126813038,
0.37134742496044003964777254698, 2.13829052733603810154929323885, 3.07116836272369116184954751076, 4.81119331667775108206220872713, 5.42412221467340503396067646341, 6.93603249901804397770539409412, 7.45422458907522370850708395952, 8.652137750412831046913346445123, 9.230611628143356659066389196285, 9.812254159737534607752442708775