| L(s) = 1 | + (12.8 + 2.27i)2-s + (−242. + 0.792i)3-s + (−801. − 291. i)4-s + (−2.00e3 − 2.38e3i)5-s + (−3.13e3 − 541. i)6-s + (−5.41e3 + 1.97e3i)7-s + (−2.12e4 − 1.22e4i)8-s + (5.90e4 − 385. i)9-s + (−2.03e4 − 3.52e4i)10-s + (5.86e3 − 6.98e3i)11-s + (1.95e5 + 7.02e4i)12-s + (5.01e4 + 2.84e5i)13-s + (−7.41e4 + 1.30e4i)14-s + (4.88e5 + 5.78e5i)15-s + (4.23e5 + 3.55e5i)16-s + (−4.83e5 + 2.79e5i)17-s + ⋯ |
| L(s) = 1 | + (0.402 + 0.0709i)2-s + (−0.999 + 0.00326i)3-s + (−0.782 − 0.284i)4-s + (−0.640 − 0.763i)5-s + (−0.402 − 0.0696i)6-s + (−0.322 + 0.117i)7-s + (−0.648 − 0.374i)8-s + (0.999 − 0.00652i)9-s + (−0.203 − 0.352i)10-s + (0.0363 − 0.0433i)11-s + (0.783 + 0.282i)12-s + (0.134 + 0.765i)13-s + (−0.137 + 0.0243i)14-s + (0.643 + 0.761i)15-s + (0.403 + 0.338i)16-s + (−0.340 + 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.734i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.679 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.622319 + 0.272061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.622319 + 0.272061i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (242. - 0.792i)T \) |
| good | 2 | \( 1 + (-12.8 - 2.27i)T + (962. + 350. i)T^{2} \) |
| 5 | \( 1 + (2.00e3 + 2.38e3i)T + (-1.69e6 + 9.61e6i)T^{2} \) |
| 7 | \( 1 + (5.41e3 - 1.97e3i)T + (2.16e8 - 1.81e8i)T^{2} \) |
| 11 | \( 1 + (-5.86e3 + 6.98e3i)T + (-4.50e9 - 2.55e10i)T^{2} \) |
| 13 | \( 1 + (-5.01e4 - 2.84e5i)T + (-1.29e11 + 4.71e10i)T^{2} \) |
| 17 | \( 1 + (4.83e5 - 2.79e5i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-4.61e5 + 7.99e5i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (5.47e5 - 1.50e6i)T + (-3.17e13 - 2.66e13i)T^{2} \) |
| 29 | \( 1 + (-1.05e7 - 1.86e6i)T + (3.95e14 + 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-3.50e7 - 1.27e7i)T + (6.27e14 + 5.26e14i)T^{2} \) |
| 37 | \( 1 + (-8.95e6 - 1.55e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (-1.02e7 + 1.80e6i)T + (1.26e16 - 4.59e15i)T^{2} \) |
| 43 | \( 1 + (9.64e7 + 8.09e7i)T + (3.75e15 + 2.12e16i)T^{2} \) |
| 47 | \( 1 + (1.34e8 + 3.69e8i)T + (-4.02e16 + 3.38e16i)T^{2} \) |
| 53 | \( 1 - 2.30e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (-4.14e8 - 4.93e8i)T + (-8.87e16 + 5.03e17i)T^{2} \) |
| 61 | \( 1 + (1.44e8 - 5.25e7i)T + (5.46e17 - 4.58e17i)T^{2} \) |
| 67 | \( 1 + (-3.88e8 - 2.20e9i)T + (-1.71e18 + 6.23e17i)T^{2} \) |
| 71 | \( 1 + (-2.85e9 + 1.64e9i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (1.88e9 - 3.25e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (8.39e8 - 4.76e9i)T + (-8.89e18 - 3.23e18i)T^{2} \) |
| 83 | \( 1 + (-6.01e9 - 1.06e9i)T + (1.45e19 + 5.30e18i)T^{2} \) |
| 89 | \( 1 + (5.85e7 + 3.38e7i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (4.70e9 + 3.94e9i)T + (1.28e19 + 7.26e19i)T^{2} \) |
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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
−15.36891404014526462804414376367, −13.69129954028131177842086880452, −12.58953133373843378261117186567, −11.65546189890741969173370264573, −9.974006476686371381371512081803, −8.603314816899617510617953577950, −6.55527186401810307906852593205, −5.09214748669445270615614163729, −4.08673836328845161039814140112, −0.894181968361369492419867506751,
0.38507068184737270446479731833, 3.34208971478997059535986994950, 4.75179452450325577962325941872, 6.32542666508051203052794092226, 7.888966901448203217064707576900, 9.848519053536004271301124239849, 11.18315088187924644958510376971, 12.30596209530384304676201490975, 13.38774022936963975000588122859, 14.86884567368952121253813254611
