| L(s) = 1 | + (−25.8 + 14.9i)2-s + (316. − 547. i)4-s + (−153. − 88.8i)5-s + (−1.69e3 − 2.93e3i)7-s + 1.12e4i·8-s + 5.29e3·10-s + (−1.96e4 + 1.13e4i)11-s + (−2.43e4 + 4.22e4i)13-s + (8.75e4 + 5.05e4i)14-s + (−8.60e4 − 1.49e5i)16-s + 3.66e4i·17-s − 8.15e4·19-s + (−9.73e4 + 5.61e4i)20-s + (3.38e5 − 5.86e5i)22-s + (3.86e4 + 2.23e4i)23-s + ⋯ |
| L(s) = 1 | + (−1.61 + 0.931i)2-s + (1.23 − 2.13i)4-s + (−0.246 − 0.142i)5-s + (−0.706 − 1.22i)7-s + 2.73i·8-s + 0.529·10-s + (−1.34 + 0.775i)11-s + (−0.853 + 1.47i)13-s + (2.27 + 1.31i)14-s + (−1.31 − 2.27i)16-s + 0.438i·17-s − 0.625·19-s + (−0.608 + 0.351i)20-s + (1.44 − 2.50i)22-s + (0.138 + 0.0798i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.317459 + 0.0703790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.317459 + 0.0703790i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (25.8 - 14.9i)T + (128 - 221. i)T^{2} \) |
| 5 | \( 1 + (153. + 88.8i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + (1.69e3 + 2.93e3i)T + (-2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.96e4 - 1.13e4i)T + (1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 + (2.43e4 - 4.22e4i)T + (-4.07e8 - 7.06e8i)T^{2} \) |
| 17 | \( 1 - 3.66e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 8.15e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (-3.86e4 - 2.23e4i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (2.65e5 - 1.53e5i)T + (2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-1.88e5 + 3.27e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 - 1.82e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-2.22e6 - 1.28e6i)T + (3.99e12 + 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-4.29e5 - 7.44e5i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (2.71e6 - 1.57e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + 1.04e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.74e7 + 1.00e7i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-5.33e6 - 9.23e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.68e7 + 2.92e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.87e5iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.74e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (6.52e6 + 1.13e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (5.41e7 - 3.12e7i)T + (1.12e15 - 1.95e15i)T^{2} \) |
| 89 | \( 1 + 6.33e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (-3.26e7 - 5.66e7i)T + (-3.91e15 + 6.78e15i)T^{2} \) |
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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
−12.81817704438544524095212373722, −11.13304192979815155133890275070, −10.10027558116494140011476720231, −9.516647699083911561643509536038, −8.042892722902056112558391670873, −7.27372034743002179172510162782, −6.36920105457835952443831223375, −4.55709635921481043807190696587, −2.05255904451766560073332755962, −0.38196645667676087096861326009,
0.46284985681833295037441073667, 2.49199800729044545302689965462, 3.05603150160676022861650781337, 5.67785593362325059994310975805, 7.49067035476941408301655170841, 8.329489159953913335754830096217, 9.388303540837289385709951903730, 10.33420292142943438558509603553, 11.23350243781549141664544531821, 12.39876019507074165775304164107
