| L(s) = 1 | + (3.24 + 0.571i)2-s + (−230. − 83.8i)4-s + (−387. − 462. i)5-s + (−3.86e3 + 1.40e3i)7-s + (−1.42e3 − 824. i)8-s + (−992. − 1.71e3i)10-s + (−1.14e4 + 1.36e4i)11-s + (−882. − 5.00e3i)13-s + (−1.33e4 + 2.34e3i)14-s + (4.39e4 + 3.68e4i)16-s + (−2.37e4 + 1.37e4i)17-s + (9.28e4 − 1.60e5i)19-s + (5.05e4 + 1.38e5i)20-s + (−4.48e4 + 3.76e4i)22-s + (8.84e4 − 2.42e5i)23-s + ⋯ |
| L(s) = 1 | + (0.202 + 0.0357i)2-s + (−0.899 − 0.327i)4-s + (−0.620 − 0.739i)5-s + (−1.60 + 0.585i)7-s + (−0.348 − 0.201i)8-s + (−0.0992 − 0.171i)10-s + (−0.780 + 0.929i)11-s + (−0.0309 − 0.175i)13-s + (−0.346 + 0.0611i)14-s + (0.670 + 0.562i)16-s + (−0.284 + 0.164i)17-s + (0.712 − 1.23i)19-s + (0.316 + 0.868i)20-s + (−0.191 + 0.160i)22-s + (0.315 − 0.868i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.599200 + 0.0628615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.599200 + 0.0628615i\) |
| \(L(5)\) |
|
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 | \( 1 + (-3.24 - 0.571i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (387. + 462. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (3.86e3 - 1.40e3i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (1.14e4 - 1.36e4i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (882. + 5.00e3i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (2.37e4 - 1.37e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-9.28e4 + 1.60e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-8.84e4 + 2.42e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (-2.34e5 - 4.13e4i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (6.18e5 + 2.25e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (-7.79e5 - 1.34e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (3.82e6 - 6.74e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (1.19e6 + 1.00e6i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (4.07e5 + 1.11e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 - 1.29e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.38e7 - 1.64e7i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (-1.39e7 + 5.08e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (1.06e6 + 6.02e6i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (3.07e7 - 1.77e7i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-6.89e6 + 1.19e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (7.76e6 - 4.40e7i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (-4.57e7 - 8.06e6i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (-3.60e6 - 2.07e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (1.08e8 + 9.11e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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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
−12.86529582147472922632565935155, −12.08543061010236590318891675109, −10.24892910900828900891821714749, −9.360054145296662078185785322180, −8.467781296136203684396413320773, −6.84341012426552291103208241176, −5.39567016504951829839695249628, −4.36855517334169756368100582257, −2.87783425602871505835491470900, −0.54516016211416707587357095627,
0.36523079750711164251316269099, 3.23320134485691458503300925474, 3.64778261318117496925711785691, 5.50883228023285455515452845743, 6.92637377102401733225730579503, 8.045936556985169827752050707896, 9.430689342625330867972938217258, 10.37459365118979037436458939827, 11.67049829577635781978101457290, 12.94890423592124720919253649885
