L(s) = 1 | + (4.81 − 2.97i)2-s + (−6.36 + 6.36i)3-s + (14.3 − 28.6i)4-s + (−30.6 − 30.6i)5-s + (−11.7 + 49.5i)6-s − 92.2i·7-s + (−16.1 − 180. i)8-s − 81i·9-s + (−239. − 56.4i)10-s + (−246. − 246. i)11-s + (90.9 + 273. i)12-s + (296. − 296. i)13-s + (−274. − 444. i)14-s + 390.·15-s + (−613. − 819. i)16-s + 554.·17-s + ⋯ |
L(s) = 1 | + (0.850 − 0.525i)2-s + (−0.408 + 0.408i)3-s + (0.447 − 0.894i)4-s + (−0.549 − 0.549i)5-s + (−0.132 + 0.561i)6-s − 0.711i·7-s + (−0.0890 − 0.996i)8-s − 0.333i·9-s + (−0.755 − 0.178i)10-s + (−0.615 − 0.615i)11-s + (0.182 + 0.547i)12-s + (0.486 − 0.486i)13-s + (−0.374 − 0.605i)14-s + 0.448·15-s + (−0.599 − 0.800i)16-s + 0.465·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.910381 - 1.59898i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.910381 - 1.59898i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-4.81 + 2.97i)T \) |
| 3 | \( 1 + (6.36 - 6.36i)T \) |
good | 5 | \( 1 + (30.6 + 30.6i)T + 3.12e3iT^{2} \) |
| 7 | \( 1 + 92.2iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (246. + 246. i)T + 1.61e5iT^{2} \) |
| 13 | \( 1 + (-296. + 296. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 - 554.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (574. - 574. i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 - 4.12e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-1.67e3 + 1.67e3i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 - 1.02e4T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-235. - 235. i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 542. iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.04e4 - 1.04e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + 4.00e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (1.99e4 + 1.99e4i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (2.99e4 + 2.99e4i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (-1.53e4 + 1.53e4i)T - 8.44e8iT^{2} \) |
| 67 | \( 1 + (-5.10e3 + 5.10e3i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 5.42e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 8.30e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (2.87e4 - 2.87e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 4.86e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.32e5T + 8.58e9T^{2} \) |
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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
−14.05357000665378140706848014164, −13.06872318440315469676700970618, −11.91130488970016499036123161727, −10.90959616373565160554476510559, −9.891981302236983632374756636202, −7.975149235450878431556552366301, −6.08086427874158911786369683383, −4.73400591384988677579702104787, −3.45215106783899589106295198414, −0.788945713611762802869316305200,
2.65624388151735211179912095070, 4.58651872781698825836834800527, 6.11426273292531588363978451585, 7.22380349315093673651238989815, 8.490802270660028260038739228218, 10.70771675529755417284446473075, 11.89270269996848716676164008040, 12.65360573584584707920629146766, 13.95221870123077107927663463506, 15.10789424740079795860702258609