L(s) = 1 | + (−15.9 + 1.67i)2-s + (−93.3 − 93.3i)3-s + (250. − 53.3i)4-s + (−695. − 695. i)5-s + (1.64e3 + 1.32e3i)6-s + 805.·7-s + (−3.89e3 + 1.26e3i)8-s + 1.08e4i·9-s + (1.22e4 + 9.89e3i)10-s + (−1.19e4 + 1.19e4i)11-s + (−2.83e4 − 1.83e4i)12-s + (2.45e4 − 2.45e4i)13-s + (−1.28e4 + 1.35e3i)14-s + 1.29e5i·15-s + (5.98e4 − 2.67e4i)16-s + 8.61e4·17-s + ⋯ |
L(s) = 1 | + (−0.994 + 0.104i)2-s + (−1.15 − 1.15i)3-s + (0.978 − 0.208i)4-s + (−1.11 − 1.11i)5-s + (1.26 + 1.02i)6-s + 0.335·7-s + (−0.950 + 0.309i)8-s + 1.65i·9-s + (1.22 + 0.989i)10-s + (−0.817 + 0.817i)11-s + (−1.36 − 0.887i)12-s + (0.861 − 0.861i)13-s + (−0.333 + 0.0351i)14-s + 2.56i·15-s + (0.913 − 0.407i)16-s + 1.03·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0274 - 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0274 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.00371759 + 0.00382122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00371759 + 0.00382122i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (15.9 - 1.67i)T \) |
good | 3 | \( 1 + (93.3 + 93.3i)T + 6.56e3iT^{2} \) |
| 5 | \( 1 + (695. + 695. i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 - 805.T + 5.76e6T^{2} \) |
| 11 | \( 1 + (1.19e4 - 1.19e4i)T - 2.14e8iT^{2} \) |
| 13 | \( 1 + (-2.45e4 + 2.45e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 - 8.61e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + (6.42e4 + 6.42e4i)T + 1.69e10iT^{2} \) |
| 23 | \( 1 + 1.93e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (3.52e5 - 3.52e5i)T - 5.00e11iT^{2} \) |
| 31 | \( 1 - 9.10e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (5.12e4 + 5.12e4i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 + 4.16e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (1.53e6 - 1.53e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + 1.86e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 + (-8.08e6 - 8.08e6i)T + 6.22e13iT^{2} \) |
| 59 | \( 1 + (1.04e7 - 1.04e7i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (-2.57e6 + 2.57e6i)T - 1.91e14iT^{2} \) |
| 67 | \( 1 + (-7.07e6 - 7.07e6i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 + 1.58e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 4.16e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 5.49e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (2.38e7 + 2.38e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 - 2.35e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.07e8T + 7.83e15T^{2} \) |
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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
−16.55618650347257246890160330285, −15.53558658159476936873557705243, −12.74280414759160725136871639525, −12.00393020193686414985656745666, −10.72193506729124553554673914736, −8.309046477810237314049551985767, −7.36587338111530083886396598772, −5.44778191978982871525112141478, −1.24475541790312902368950344304, −0.00533596407071888730027963976,
3.66078319341679115371052936813, 6.14743848855659397268490065651, 8.011125760811945290692849009005, 10.09283744169955285911465813042, 11.12162941363498226214051014713, 11.65560378038933904228625395302, 14.89946248952367090727523591030, 15.96125747204176108325459272835, 16.66729319587409656620972395167, 18.28050693361324558702331115542