L(s) = 1 | + (−1.99 − 1.14i)2-s + (0.640 + 1.10i)4-s + (2.91 − 5.05i)5-s + 9.38·7-s + 6.24i·8-s + (−11.6 + 6.71i)10-s + 4.66·11-s + (−4.96 + 2.86i)13-s + (−18.6 − 10.7i)14-s + (9.74 − 16.8i)16-s + (9.49 − 16.4i)17-s + (−3.40 − 18.6i)19-s + 7.48·20-s + (−9.27 − 5.35i)22-s + (−6.41 − 11.1i)23-s + ⋯ |
L(s) = 1 | + (−0.995 − 0.574i)2-s + (0.160 + 0.277i)4-s + (0.583 − 1.01i)5-s + 1.34·7-s + 0.781i·8-s + (−1.16 + 0.671i)10-s + 0.423·11-s + (−0.381 + 0.220i)13-s + (−1.33 − 0.769i)14-s + (0.608 − 1.05i)16-s + (0.558 − 0.966i)17-s + (−0.178 − 0.983i)19-s + 0.374·20-s + (−0.421 − 0.243i)22-s + (−0.278 − 0.482i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.637596 - 0.817292i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.637596 - 0.817292i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 + (3.40 + 18.6i)T \) |
good | 2 | \( 1 + (1.99 + 1.14i)T + (2 + 3.46i)T^{2} \) |
| 5 | \( 1 + (-2.91 + 5.05i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 - 9.38T + 49T^{2} \) |
| 11 | \( 1 - 4.66T + 121T^{2} \) |
| 13 | \( 1 + (4.96 - 2.86i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-9.49 + 16.4i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (6.41 + 11.1i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (27.7 - 16.0i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + 26.9iT - 961T^{2} \) |
| 37 | \( 1 + 0.140iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (10.0 + 5.82i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (16.3 - 28.4i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-5.37 - 9.30i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-30.7 + 17.7i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-76.1 - 43.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-45.7 - 79.2i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-17.3 + 10.0i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-15.8 - 9.14i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-28.7 + 49.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (81.7 + 47.2i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 81.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + (17.3 - 10.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-117. - 67.8i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−11.81617137004164214439578791856, −11.28025716638490647117466757723, −10.05378277347953618205871427232, −9.169248642115547428270740944657, −8.545707529589182139600880457017, −7.41277019334216731966583379251, −5.46839668526564745790163962025, −4.68384979318780714016290946091, −2.17748319796286909649577001952, −0.984429271599371427248141287420,
1.73969298035697342705600428661, 3.78264458977982860086894509254, 5.61193080221241883558133922262, 6.77501356203908892047664447343, 7.78751275834297169782434391433, 8.515321209622089811368016338200, 9.851832669303005840885972542056, 10.46838873940152871143868303101, 11.58646207537183504039423297621, 12.78976669406548686153600024329