| L(s) = 1 | + (7.56 + 13.1i)2-s + (−2.65 + 4.59i)3-s + (−50.5 + 87.5i)4-s + (207. + 359. i)5-s − 80.2·6-s + 407.·8-s + (1.07e3 + 1.86e3i)9-s + (−3.14e3 + 5.44e3i)10-s + (2.13e3 − 3.69e3i)11-s + (−268. − 464. i)12-s − 1.23e4·13-s − 2.20e3·15-s + (9.55e3 + 1.65e4i)16-s + (−9.86e3 + 1.70e4i)17-s + (−1.63e4 + 2.82e4i)18-s + (5.56e3 + 9.64e3i)19-s + ⋯ |
| L(s) = 1 | + (0.668 + 1.15i)2-s + (−0.0567 + 0.0982i)3-s + (−0.394 + 0.683i)4-s + (0.742 + 1.28i)5-s − 0.151·6-s + 0.281·8-s + (0.493 + 0.854i)9-s + (−0.993 + 1.72i)10-s + (0.482 − 0.836i)11-s + (−0.0447 − 0.0775i)12-s − 1.55·13-s − 0.168·15-s + (0.583 + 1.00i)16-s + (−0.486 + 0.843i)17-s + (−0.660 + 1.14i)18-s + (0.186 + 0.322i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.647614 + 2.82324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.647614 + 2.82324i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-7.56 - 13.1i)T + (-64 + 110. i)T^{2} \) |
| 3 | \( 1 + (2.65 - 4.59i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-207. - 359. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-2.13e3 + 3.69e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + 1.23e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + (9.86e3 - 1.70e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-5.56e3 - 9.64e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (5.32e4 + 9.22e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.03e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-4.40e4 + 7.63e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-4.80e4 - 8.31e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 3.62e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.92e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.74e5 - 6.49e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (2.93e5 - 5.08e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.28e6 + 2.22e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.79e5 - 4.83e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-5.54e5 + 9.60e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 3.81e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-6.78e5 + 1.17e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-8.22e5 - 1.42e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + 8.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (3.01e6 + 5.21e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.33e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.26867304840634101209136647400, −14.18959207909434058908872586756, −12.72453193619647112904530893513, −10.85402075841345633477471252896, −10.02556562659837156763704016456, −7.969118606897678174441605444536, −6.78016776895180825320585050830, −5.92074568230731513470635002679, −4.41061953258157037692539029185, −2.34353580152147283809412058845,
0.990783352070476026118636534527, 2.25026926285255093851313671280, 4.22421337486895595956386554560, 5.22527771088380045117138216864, 7.23952140225082154495459338862, 9.367620973972621435178475878217, 9.895485427449606876847572174116, 11.83758944866762317984729380738, 12.33808999333723692373900560295, 13.23928143567607052791074784116
