| L(s) = 1 | + (9.99 − 1.76i)2-s + (58.8 + 235. i)3-s + (−865. + 315. i)4-s + (2.61e3 − 3.12e3i)5-s + (1.00e3 + 2.25e3i)6-s + (−1.51e4 − 5.50e3i)7-s + (−1.70e4 + 9.86e3i)8-s + (−5.21e4 + 2.77e4i)9-s + (2.06e4 − 3.58e4i)10-s + (4.56e4 + 5.43e4i)11-s + (−1.25e5 − 1.85e5i)12-s + (1.21e5 − 6.91e5i)13-s + (−1.60e5 − 2.83e4i)14-s + (8.90e5 + 4.33e5i)15-s + (5.69e5 − 4.77e5i)16-s + (1.09e5 + 6.30e4i)17-s + ⋯ |
| L(s) = 1 | + (0.312 − 0.0550i)2-s + (0.242 + 0.970i)3-s + (−0.845 + 0.307i)4-s + (0.838 − 0.998i)5-s + (0.128 + 0.289i)6-s + (−0.900 − 0.327i)7-s + (−0.521 + 0.301i)8-s + (−0.882 + 0.469i)9-s + (0.206 − 0.358i)10-s + (0.283 + 0.337i)11-s + (−0.503 − 0.745i)12-s + (0.328 − 1.86i)13-s + (−0.299 − 0.0527i)14-s + (1.17 + 0.571i)15-s + (0.542 − 0.455i)16-s + (0.0769 + 0.0444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0808 + 0.996i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0808 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.727843 - 0.789304i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.727843 - 0.789304i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-58.8 - 235. i)T \) |
| good | 2 | \( 1 + (-9.99 + 1.76i)T + (962. - 350. i)T^{2} \) |
| 5 | \( 1 + (-2.61e3 + 3.12e3i)T + (-1.69e6 - 9.61e6i)T^{2} \) |
| 7 | \( 1 + (1.51e4 + 5.50e3i)T + (2.16e8 + 1.81e8i)T^{2} \) |
| 11 | \( 1 + (-4.56e4 - 5.43e4i)T + (-4.50e9 + 2.55e10i)T^{2} \) |
| 13 | \( 1 + (-1.21e5 + 6.91e5i)T + (-1.29e11 - 4.71e10i)T^{2} \) |
| 17 | \( 1 + (-1.09e5 - 6.30e4i)T + (1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (1.20e6 + 2.07e6i)T + (-3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (2.26e6 + 6.21e6i)T + (-3.17e13 + 2.66e13i)T^{2} \) |
| 29 | \( 1 + (-2.11e7 + 3.72e6i)T + (3.95e14 - 1.43e14i)T^{2} \) |
| 31 | \( 1 + (4.32e7 - 1.57e7i)T + (6.27e14 - 5.26e14i)T^{2} \) |
| 37 | \( 1 + (-3.39e7 + 5.87e7i)T + (-2.40e15 - 4.16e15i)T^{2} \) |
| 41 | \( 1 + (8.50e7 + 1.49e7i)T + (1.26e16 + 4.59e15i)T^{2} \) |
| 43 | \( 1 + (8.62e7 - 7.23e7i)T + (3.75e15 - 2.12e16i)T^{2} \) |
| 47 | \( 1 + (8.10e7 - 2.22e8i)T + (-4.02e16 - 3.38e16i)T^{2} \) |
| 53 | \( 1 - 5.12e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-7.08e8 + 8.44e8i)T + (-8.87e16 - 5.03e17i)T^{2} \) |
| 61 | \( 1 + (6.03e8 + 2.19e8i)T + (5.46e17 + 4.58e17i)T^{2} \) |
| 67 | \( 1 + (3.30e8 - 1.87e9i)T + (-1.71e18 - 6.23e17i)T^{2} \) |
| 71 | \( 1 + (1.80e9 + 1.04e9i)T + (1.62e18 + 2.81e18i)T^{2} \) |
| 73 | \( 1 + (1.61e7 + 2.80e7i)T + (-2.14e18 + 3.72e18i)T^{2} \) |
| 79 | \( 1 + (5.52e7 + 3.13e8i)T + (-8.89e18 + 3.23e18i)T^{2} \) |
| 83 | \( 1 + (5.26e9 - 9.28e8i)T + (1.45e19 - 5.30e18i)T^{2} \) |
| 89 | \( 1 + (-5.25e9 + 3.03e9i)T + (1.55e19 - 2.70e19i)T^{2} \) |
| 97 | \( 1 + (-4.82e9 + 4.05e9i)T + (1.28e19 - 7.26e19i)T^{2} \) |
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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
−14.57238214385555249702976343169, −13.26521756224861015714607771797, −12.67534483704098219232156263795, −10.38408715112010283346026921544, −9.394672446691123349530483483774, −8.427157419681144059001438492550, −5.71392112018272444481804128898, −4.56857080254084318314706973047, −3.11450849437118277545595077083, −0.35923330394016826086255244288,
1.79169605940887495760654291888, 3.49296515246502287225698854865, 5.96582774534385220664913119110, 6.69759297276090691210129703371, 8.833893325157400510016195838549, 9.894276467093437751897745224057, 11.78570794699522046148995213039, 13.22684672621351615268853605844, 13.95697787261189730789329001123, 14.73973478780006501431757668819
