| L(s) = 1 | − 1.96e3·5-s − 2.80e4i·7-s − 4.35e4i·11-s − 1.39e5·13-s − 2.12e6·17-s − 2.98e6i·19-s − 4.60e6i·23-s − 5.92e6·25-s + 2.90e7·29-s − 1.52e7i·31-s + 5.49e7i·35-s − 3.18e7·37-s + 2.12e8·41-s − 1.02e8i·43-s + 3.05e8i·47-s + ⋯ |
| L(s) = 1 | − 0.627·5-s − 1.66i·7-s − 0.270i·11-s − 0.376·13-s − 1.49·17-s − 1.20i·19-s − 0.715i·23-s − 0.606·25-s + 1.41·29-s − 0.533i·31-s + 1.04i·35-s − 0.458·37-s + 1.83·41-s − 0.695i·43-s + 1.33i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.2403587049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2403587049\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.96e3T + 9.76e6T^{2} \) |
| 7 | \( 1 + 2.80e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 4.35e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 1.39e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.12e6T + 2.01e12T^{2} \) |
| 19 | \( 1 + 2.98e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 4.60e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 2.90e7T + 4.20e14T^{2} \) |
| 31 | \( 1 + 1.52e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 + 3.18e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 2.12e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + 1.02e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 3.05e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 2.21e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 6.12e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 6.10e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 5.40e7iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 2.86e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 2.92e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 2.00e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 3.29e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 8.15e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 5.41e9T + 7.37e19T^{2} \) |
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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
−10.75867817826547624415326579000, −9.535235860394199000460195889728, −8.289052233196049108338235063720, −7.28993315503581457948524842217, −6.50860175969295903956449155148, −4.64383900754208080084094241085, −4.03851117039852078754000562544, −2.59196333920336905572996154431, −0.865309133601005994236264245340, −0.06597278351902878770069007014,
1.79655735162747452544049566806, 2.83675334339202204357197665862, 4.26115181343379902336867020865, 5.44613175961390188566140306817, 6.50819218248733575524281131219, 7.86900241758015645191187503532, 8.745387517559462418047374767642, 9.674241786688198459529810661989, 11.02517847926816217091410182843, 12.03608779746183593138348062073
