| L(s) = 1 | + (16 − 27.7i)2-s + (−511. − 886. i)4-s + (5.86e3 + 1.01e4i)5-s + (2.50e4 − 4.33e4i)7-s − 3.27e4·8-s + 3.75e5·10-s + (2.65e5 − 4.60e5i)11-s + (−6.66e5 − 1.15e6i)13-s + (−8.00e5 − 1.38e6i)14-s + (−5.24e5 + 9.08e5i)16-s − 5.10e6·17-s + 2.90e6·19-s + (6.00e6 − 1.04e7i)20-s + (−8.50e6 − 1.47e7i)22-s + (−1.52e7 − 2.64e7i)23-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.839 + 1.45i)5-s + (0.562 − 0.973i)7-s − 0.353·8-s + 1.18·10-s + (0.497 − 0.861i)11-s + (−0.497 − 0.862i)13-s + (−0.397 − 0.688i)14-s + (−0.125 + 0.216i)16-s − 0.872·17-s + 0.268·19-s + (0.419 − 0.726i)20-s + (−0.351 − 0.609i)22-s + (−0.495 − 0.858i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.863331986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.863331986\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-16 + 27.7i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-5.86e3 - 1.01e4i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 7 | \( 1 + (-2.50e4 + 4.33e4i)T + (-9.88e8 - 1.71e9i)T^{2} \) |
| 11 | \( 1 + (-2.65e5 + 4.60e5i)T + (-1.42e11 - 2.47e11i)T^{2} \) |
| 13 | \( 1 + (6.66e5 + 1.15e6i)T + (-8.96e11 + 1.55e12i)T^{2} \) |
| 17 | \( 1 + 5.10e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.90e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + (1.52e7 + 2.64e7i)T + (-4.76e14 + 8.25e14i)T^{2} \) |
| 29 | \( 1 + (-3.85e7 + 6.66e7i)T + (-6.10e15 - 1.05e16i)T^{2} \) |
| 31 | \( 1 + (-1.19e8 - 2.07e8i)T + (-1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + 7.85e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + (2.05e8 + 3.56e8i)T + (-2.75e17 + 4.76e17i)T^{2} \) |
| 43 | \( 1 + (1.75e8 - 3.04e8i)T + (-4.64e17 - 8.04e17i)T^{2} \) |
| 47 | \( 1 + (4.79e7 - 8.29e7i)T + (-1.23e18 - 2.14e18i)T^{2} \) |
| 53 | \( 1 + 1.46e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + (2.81e9 + 4.86e9i)T + (-1.50e19 + 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-5.23e9 + 9.07e9i)T + (-2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (2.25e9 + 3.91e9i)T + (-6.10e19 + 1.05e20i)T^{2} \) |
| 71 | \( 1 + 8.50e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.01e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + (-1.11e10 + 1.92e10i)T + (-3.73e20 - 6.47e20i)T^{2} \) |
| 83 | \( 1 + (3.16e9 - 5.48e9i)T + (-6.43e20 - 1.11e21i)T^{2} \) |
| 89 | \( 1 + 5.01e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (4.74e10 - 8.21e10i)T + (-3.57e21 - 6.19e21i)T^{2} \) |
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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
−10.60843026767972041665876683293, −9.820911804785949219817018678184, −8.364200950393660726503938142335, −6.99629766783566371158005925720, −6.22570424015754687123092059819, −4.94065777795543270541342857699, −3.59674140694440897121189737901, −2.72189719944180540749090391332, −1.60858152426066079150737028902, −0.29965402353129158785408771687,
1.45811714043626704273725803939, 2.23179615005087631549914984234, 4.27756384531033751811124285197, 4.99807678360089459143813601613, 5.81565064679861598581820998625, 7.01319643841195564076273885658, 8.406715835744285363472020298259, 9.072398798973634859905351335849, 9.807858166782701866405104947932, 11.77071405254608029873804211023
