L(s) = 1 | + (−1.28 − 2.21i)2-s + (1.84 + 3.19i)3-s + (0.719 − 1.24i)4-s + 0.561·5-s + (4.71 − 8.17i)6-s + (−9.08 + 15.7i)7-s − 24.1·8-s + (6.71 − 11.6i)9-s + (−0.719 − 1.24i)10-s + (−32.3 − 56.0i)11-s + 5.30·12-s + 46.5·14-s + (1.03 + 1.79i)15-s + (25.2 + 43.6i)16-s + (12.7 − 22.1i)17-s − 34.3·18-s + ⋯ |
L(s) = 1 | + (−0.452 − 0.784i)2-s + (0.354 + 0.614i)3-s + (0.0899 − 0.155i)4-s + 0.0502·5-s + (0.321 − 0.556i)6-s + (−0.490 + 0.849i)7-s − 1.06·8-s + (0.248 − 0.430i)9-s + (−0.0227 − 0.0393i)10-s + (−0.887 − 1.53i)11-s + 0.127·12-s + 0.888·14-s + (0.0178 + 0.0308i)15-s + (0.393 + 0.682i)16-s + (0.182 − 0.315i)17-s − 0.450·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.231529 - 0.886783i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.231529 - 0.886783i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (1.28 + 2.21i)T + (-4 + 6.92i)T^{2} \) |
| 3 | \( 1 + (-1.84 - 3.19i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 - 0.561T + 125T^{2} \) |
| 7 | \( 1 + (9.08 - 15.7i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (32.3 + 56.0i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-12.7 + 22.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-53.9 + 93.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (36.6 + 63.4i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (87.9 + 152. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (57.4 + 99.4i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-34.8 - 60.3i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (219. - 379. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 31.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 2.84T + 1.48e5T^{2} \) |
| 59 | \( 1 + (35.8 - 62.0i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-460. + 797. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-222. - 384. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-270. + 469. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 - 764.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 421.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-579. - 1.00e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (291. - 505. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.61497385325112472318886278447, −10.88584368082938731139863261068, −9.724504376464827791275988995464, −9.270299142840876560129056485539, −8.208238680157439429258784355755, −6.37051195911218099393106461508, −5.39699266518487194503548511801, −3.44947885884092282924320216215, −2.54321628636650256424530971474, −0.44194807365249133003897886939,
1.95652601494146955328175586963, 3.65808196524719961611417011260, 5.44835332201661153865513189541, 6.99527308058620390429088172851, 7.42982581603841182947961645868, 8.195272314577383850875855156911, 9.671714918396222049874846998149, 10.40122851724185191646898306236, 12.05048192882306585109859818240, 12.80610647486775650638328882253