| L(s) = 1 | + (−59.6 + 103. i)5-s + (2.05e3 − 1.18e3i)7-s + (1.64e4 − 9.48e3i)11-s + (1.77e4 − 3.07e4i)13-s − 9.49e4·17-s − 2.27e5i·19-s + (−3.06e5 − 1.76e5i)23-s + (1.88e5 + 3.25e5i)25-s + (−4.44e5 − 7.70e5i)29-s + (−8.98e5 − 5.18e5i)31-s + 2.82e5i·35-s − 1.51e6·37-s + (−2.10e6 + 3.65e6i)41-s + (5.17e6 − 2.98e6i)43-s + (−6.84e5 + 3.94e5i)47-s + ⋯ |
| L(s) = 1 | + (−0.0953 + 0.165i)5-s + (0.855 − 0.493i)7-s + (1.12 − 0.647i)11-s + (0.620 − 1.07i)13-s − 1.13·17-s − 1.74i·19-s + (−1.09 − 0.631i)23-s + (0.481 + 0.834i)25-s + (−0.628 − 1.08i)29-s + (−0.972 − 0.561i)31-s + 0.188i·35-s − 0.806·37-s + (−0.746 + 1.29i)41-s + (1.51 − 0.874i)43-s + (−0.140 + 0.0809i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.377180467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.377180467\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (59.6 - 103. i)T + (-1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-2.05e3 + 1.18e3i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-1.64e4 + 9.48e3i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (-1.77e4 + 3.07e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 + 9.49e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 2.27e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (3.06e5 + 1.76e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (4.44e5 + 7.70e5i)T + (-2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (8.98e5 + 5.18e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + 1.51e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (2.10e6 - 3.65e6i)T + (-3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-5.17e6 + 2.98e6i)T + (5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (6.84e5 - 3.94e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 - 9.03e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.10e6 - 6.38e5i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (7.13e5 + 1.23e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.96e7 + 1.13e7i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 4.77e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.77e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (4.32e6 - 2.49e6i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-3.46e7 + 2.00e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 3.96e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (-3.77e7 - 6.53e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
| 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
−9.211513783252296510271138877537, −8.566829523095772214452612984266, −7.57276555528214660855911258176, −6.65628457959546353638603445353, −5.67279894634245338575304973856, −4.48929251614356349175838047761, −3.68115786272104772881491816323, −2.39858031521749300377255265255, −1.16138211724600169227847744978, −0.25021111424738958099978735985,
1.60577830594164149158577692580, 1.85408880947648908397695500406, 3.73694608077095583711032966820, 4.37902988946495560166829679369, 5.56882266350032509070432059593, 6.53478890262571949576003888814, 7.47826670317078203636749619490, 8.684342685188959719476748579473, 9.032406528031134237125213557910, 10.25450349192178919168746978888
