| L(s) = 1 | + (−1.37 + 1.64i)2-s + (−231. + 72.3i)3-s + (177. + 1.00e3i)4-s + (−1.39e3 + 3.82e3i)5-s + (200. − 480. i)6-s + (−3.23e3 + 1.83e4i)7-s + (−3.79e3 − 2.18e3i)8-s + (4.85e4 − 3.35e4i)9-s + (−4.36e3 − 7.55e3i)10-s + (7.03e4 + 1.93e5i)11-s + (−1.13e5 − 2.20e5i)12-s + (2.65e4 − 2.22e4i)13-s + (−2.56e4 − 3.05e4i)14-s + (4.62e4 − 9.88e5i)15-s + (−9.72e5 + 3.53e5i)16-s + (−1.17e6 + 6.78e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0430 + 0.0512i)2-s + (−0.954 + 0.297i)3-s + (0.172 + 0.980i)4-s + (−0.445 + 1.22i)5-s + (0.0258 − 0.0617i)6-s + (−0.192 + 1.09i)7-s + (−0.115 − 0.0667i)8-s + (0.822 − 0.568i)9-s + (−0.0436 − 0.0755i)10-s + (0.436 + 1.19i)11-s + (−0.456 − 0.884i)12-s + (0.0714 − 0.0599i)13-s + (−0.0477 − 0.0568i)14-s + (0.0608 − 1.30i)15-s + (−0.927 + 0.337i)16-s + (−0.827 + 0.477i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.307142 - 0.876193i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.307142 - 0.876193i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (231. - 72.3i)T \) |
| good | 2 | \( 1 + (1.37 - 1.64i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (1.39e3 - 3.82e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (3.23e3 - 1.83e4i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (-7.03e4 - 1.93e5i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (-2.65e4 + 2.22e4i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (1.17e6 - 6.78e5i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-5.96e5 + 1.03e6i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-5.53e6 + 9.75e5i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (-3.19e6 + 3.80e6i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (6.29e6 + 3.57e7i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (-4.53e7 - 7.85e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (-5.44e7 - 6.49e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (-2.34e8 + 8.52e7i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (-1.05e8 - 1.86e7i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 - 6.29e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (4.90e6 - 1.34e7i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (-2.28e8 + 1.29e9i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (1.56e8 - 1.31e8i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (2.93e9 - 1.69e9i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.16e9 + 2.02e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (1.59e9 + 1.33e9i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (3.27e9 - 3.90e9i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (-5.78e9 - 3.34e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (1.20e10 - 4.36e9i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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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
−15.56649279628083764381596047318, −15.10080994437008384391532144178, −12.82025812996706406225297971650, −11.84107528125578195192671464277, −10.95714400169759231025785118775, −9.286529748205098988650514637352, −7.35581454296486632795159417241, −6.33563948671574733636244920934, −4.25890679842756904746333633642, −2.61474265411163002300299897189,
0.49287142637540022043354929310, 1.12289633770849093140390527361, 4.38044462810140327779716532069, 5.67247103250694920864082367713, 7.09650539564096893783281753086, 9.003167834409399534086379709054, 10.61219741101645328491239873995, 11.50074809965435828210270311929, 12.93794211837214170596382906869, 14.04718582775357044570325038679
