| L(s) = 1 | + (26.0 + 15.0i)2-s + (−77.6 + 23.0i)3-s + (324. + 562. i)4-s + (189. − 109. i)5-s + (−2.37e3 − 566. i)6-s + (1.40e3 − 2.43e3i)7-s + 1.18e4i·8-s + (5.49e3 − 3.58e3i)9-s + 6.57e3·10-s + (−4.50e3 − 2.59e3i)11-s + (−3.81e4 − 3.61e4i)12-s + (−5.26e3 − 9.11e3i)13-s + (7.32e4 − 4.22e4i)14-s + (−1.21e4 + 1.28e4i)15-s + (−9.49e4 + 1.64e5i)16-s − 1.52e4i·17-s + ⋯ |
| L(s) = 1 | + (1.62 + 0.940i)2-s + (−0.958 + 0.284i)3-s + (1.26 + 2.19i)4-s + (0.302 − 0.174i)5-s + (−1.82 − 0.437i)6-s + (0.585 − 1.01i)7-s + 2.88i·8-s + (0.837 − 0.546i)9-s + 0.657·10-s + (−0.307 − 0.177i)11-s + (−1.84 − 1.74i)12-s + (−0.184 − 0.319i)13-s + (1.90 − 1.10i)14-s + (−0.240 + 0.253i)15-s + (−1.44 + 2.50i)16-s − 0.182i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.98411 + 1.75860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98411 + 1.75860i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (77.6 - 23.0i)T \) |
| good | 2 | \( 1 + (-26.0 - 15.0i)T + (128 + 221. i)T^{2} \) |
| 5 | \( 1 + (-189. + 109. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 7 | \( 1 + (-1.40e3 + 2.43e3i)T + (-2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (4.50e3 + 2.59e3i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 + (5.26e3 + 9.11e3i)T + (-4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + 1.52e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 9.26e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + (5.31e4 - 3.06e4i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-7.59e5 - 4.38e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + (8.14e5 + 1.41e6i)T + (-4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + 1.66e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + (-4.15e6 + 2.39e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (-2.56e5 + 4.44e5i)T + (-5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-1.08e6 - 6.25e5i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 - 8.87e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (8.34e6 - 4.81e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (4.51e6 - 7.81e6i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (5.32e6 + 9.22e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 2.16e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.00e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + (2.83e7 - 4.90e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (1.29e6 + 7.45e5i)T + (1.12e15 + 1.95e15i)T^{2} \) |
| 89 | \( 1 - 5.42e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (3.06e6 - 5.31e6i)T + (-3.91e15 - 6.78e15i)T^{2} \) |
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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
−20.66251918101507233916893904418, −17.50982104924214411640058155553, −16.66069185507380313789946519107, −15.34829528871942044183135934949, −13.86867990460378759459106758844, −12.56972981868232179493212238485, −10.96911345731144013556723545216, −7.35448750677955140379954298810, −5.67150760744565750228719329806, −4.25711382585556720638867140868,
2.04319763270510197248861529935, 4.84297270831759192925185483629, 6.20235538104929713905491890132, 10.47252406693046894189030953528, 11.77742855449427781081124139164, 12.71272130850156512659576106833, 14.29829170150225354321997224954, 15.73270224488315801460627351259, 18.02066988961104564268508049708, 19.40032147254588016414843588520
