| L(s) = 1 | + (−27.9 + 33.3i)2-s + (−113. + 214. i)3-s + (−151. − 857. i)4-s + (−874. + 2.40e3i)5-s + (−3.98e3 − 9.79e3i)6-s + (−1.72e3 + 9.75e3i)7-s + (−5.79e3 − 3.34e3i)8-s + (−3.32e4 − 4.87e4i)9-s + (−5.56e4 − 9.63e4i)10-s + (−6.49e4 − 1.78e5i)11-s + (2.01e5 + 6.48e4i)12-s + (−3.17e5 + 2.66e5i)13-s + (−2.77e5 − 3.30e5i)14-s + (−4.16e5 − 4.60e5i)15-s + (1.11e6 − 4.04e5i)16-s + (7.36e5 − 4.25e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.874 + 1.04i)2-s + (−0.467 + 0.884i)3-s + (−0.147 − 0.836i)4-s + (−0.279 + 0.768i)5-s + (−0.512 − 1.25i)6-s + (−0.102 + 0.580i)7-s + (−0.176 − 0.102i)8-s + (−0.563 − 0.826i)9-s + (−0.556 − 0.963i)10-s + (−0.403 − 1.10i)11-s + (0.808 + 0.260i)12-s + (−0.854 + 0.717i)13-s + (−0.515 − 0.614i)14-s + (−0.548 − 0.606i)15-s + (1.05 − 0.385i)16-s + (0.518 − 0.299i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0170096 - 0.00103014i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0170096 - 0.00103014i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (113. - 214. i)T \) |
| good | 2 | \( 1 + (27.9 - 33.3i)T + (-177. - 1.00e3i)T^{2} \) |
| 5 | \( 1 + (874. - 2.40e3i)T + (-7.48e6 - 6.27e6i)T^{2} \) |
| 7 | \( 1 + (1.72e3 - 9.75e3i)T + (-2.65e8 - 9.66e7i)T^{2} \) |
| 11 | \( 1 + (6.49e4 + 1.78e5i)T + (-1.98e10 + 1.66e10i)T^{2} \) |
| 13 | \( 1 + (3.17e5 - 2.66e5i)T + (2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (-7.36e5 + 4.25e5i)T + (1.00e12 - 1.74e12i)T^{2} \) |
| 19 | \( 1 + (2.37e6 - 4.10e6i)T + (-3.06e12 - 5.30e12i)T^{2} \) |
| 23 | \( 1 + (-6.27e6 + 1.10e6i)T + (3.89e13 - 1.41e13i)T^{2} \) |
| 29 | \( 1 + (9.44e6 - 1.12e7i)T + (-7.30e13 - 4.14e14i)T^{2} \) |
| 31 | \( 1 + (2.66e6 + 1.50e7i)T + (-7.70e14 + 2.80e14i)T^{2} \) |
| 37 | \( 1 + (-1.62e7 - 2.81e7i)T + (-2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 + (8.58e7 + 1.02e8i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (1.48e8 - 5.40e7i)T + (1.65e16 - 1.38e16i)T^{2} \) |
| 47 | \( 1 + (3.52e7 + 6.22e6i)T + (4.94e16 + 1.79e16i)T^{2} \) |
| 53 | \( 1 + 3.23e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + (6.79e7 - 1.86e8i)T + (-3.91e17 - 3.28e17i)T^{2} \) |
| 61 | \( 1 + (-4.03e6 + 2.28e7i)T + (-6.70e17 - 2.43e17i)T^{2} \) |
| 67 | \( 1 + (-1.41e9 + 1.18e9i)T + (3.16e17 - 1.79e18i)T^{2} \) |
| 71 | \( 1 + (-9.22e8 + 5.32e8i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (-1.44e9 + 2.50e9i)T + (-2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (2.06e9 + 1.72e9i)T + (1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (2.41e9 - 2.87e9i)T + (-2.69e18 - 1.52e19i)T^{2} \) |
| 89 | \( 1 + (-1.77e9 - 1.02e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 + (1.22e10 - 4.46e9i)T + (5.64e19 - 4.74e19i)T^{2} \) |
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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
−15.19324848781062268289116805018, −14.56094744840916302933409784289, −12.13793808335442020992183263330, −10.76860526537999756495201238278, −9.531736091208371461685014858666, −8.313223008281766642029860097340, −6.73176715877939401754061537931, −5.52293136821065316292890068525, −3.30619205871035912751150281061, −0.01158566418847023419698374542,
0.988497174700748447822827283632, 2.46664684518371371196605420318, 4.99506854914211004816536433207, 7.16784293044998119619108486678, 8.474396793262202673191348882661, 10.00431146440756585977193093544, 11.14928214613277281242241673995, 12.42681670319917023211728250299, 13.01901404673805042788065634769, 15.07944180179453008765814084820
