| L(s) = 1 | + (0.868 + 1.03i)2-s + (−7.41 − 5.10i)3-s + (2.46 − 13.9i)4-s + (−3.20 − 8.80i)5-s + (−1.14 − 12.1i)6-s + (−7.41 − 42.0i)7-s + (35.2 − 20.3i)8-s + (28.8 + 75.6i)9-s + (6.32 − 10.9i)10-s + (−6.44 + 17.7i)11-s + (−89.5 + 90.8i)12-s + (−37.7 − 31.7i)13-s + (37.0 − 44.1i)14-s + (−21.2 + 81.5i)15-s + (−161. − 58.7i)16-s + (390. + 225. i)17-s + ⋯ |
| L(s) = 1 | + (0.217 + 0.258i)2-s + (−0.823 − 0.567i)3-s + (0.153 − 0.872i)4-s + (−0.128 − 0.352i)5-s + (−0.0319 − 0.336i)6-s + (−0.151 − 0.858i)7-s + (0.551 − 0.318i)8-s + (0.355 + 0.934i)9-s + (0.0632 − 0.109i)10-s + (−0.0532 + 0.146i)11-s + (−0.621 + 0.631i)12-s + (−0.223 − 0.187i)13-s + (0.189 − 0.225i)14-s + (−0.0942 + 0.362i)15-s + (−0.630 − 0.229i)16-s + (1.35 + 0.780i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00699 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.00699 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.790867 - 0.796419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.790867 - 0.796419i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.41 + 5.10i)T \) |
| good | 2 | \( 1 + (-0.868 - 1.03i)T + (-2.77 + 15.7i)T^{2} \) |
| 5 | \( 1 + (3.20 + 8.80i)T + (-478. + 401. i)T^{2} \) |
| 7 | \( 1 + (7.41 + 42.0i)T + (-2.25e3 + 821. i)T^{2} \) |
| 11 | \( 1 + (6.44 - 17.7i)T + (-1.12e4 - 9.41e3i)T^{2} \) |
| 13 | \( 1 + (37.7 + 31.7i)T + (4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (-390. - 225. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (57.0 + 98.7i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-608. - 107. i)T + (2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (179. + 213. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-211. + 1.20e3i)T + (-8.67e5 - 3.15e5i)T^{2} \) |
| 37 | \( 1 + (1.10e3 - 1.91e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (348. - 415. i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-3.15e3 - 1.14e3i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (3.96e3 - 698. i)T + (4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 - 3.91e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (1.53e3 + 4.23e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-208. - 1.18e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (2.41e3 + 2.02e3i)T + (3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-2.47e3 - 1.43e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (1.17e3 + 2.03e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-3.85e3 + 3.23e3i)T + (6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (-7.26e3 - 8.65e3i)T + (-8.24e6 + 4.67e7i)T^{2} \) |
| 89 | \( 1 + (982. - 567. i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-1.75e3 - 637. i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−16.45730876029509750491091480147, −15.04908879741380942262449630466, −13.69226684317514184840743104069, −12.58435668130756855023643303370, −11.04580630407163766081895926572, −9.999637877555153716451750481899, −7.61500150897246179611455316996, −6.28037671920092307064392495954, −4.83509074586545065354220037378, −1.00532107268942254850921903617,
3.26880615808587728649306418159, 5.22560661778283111677799332472, 7.09810122523671597666175785741, 9.028220608884948307604448694171, 10.70680880114789564832973888545, 11.86566373921229954015235635800, 12.64347976853300002863002436755, 14.54279759666025728824456456177, 15.91236688637734457433253049769, 16.72790541342365176827079722633
