| 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
−14.73973478780006501431757668819, −13.95697787261189730789329001123, −13.22684672621351615268853605844, −11.78570794699522046148995213039, −9.894276467093437751897745224057, −8.833893325157400510016195838549, −6.69759297276090691210129703371, −5.96582774534385220664913119110, −3.49296515246502287225698854865, −1.79169605940887495760654291888,
0.35923330394016826086255244288, 3.11450849437118277545595077083, 4.56857080254084318314706973047, 5.71392112018272444481804128898, 8.427157419681144059001438492550, 9.394672446691123349530483483774, 10.38408715112010283346026921544, 12.67534483704098219232156263795, 13.26521756224861015714607771797, 14.57238214385555249702976343169
