L(s) = 1 | − 17.8i·2-s + (18.7 − 78.8i)3-s − 61.9·4-s − 794. i·5-s + (−1.40e3 − 334. i)6-s + 2.43e3·7-s − 3.45e3i·8-s + (−5.85e3 − 2.95e3i)9-s − 1.41e4·10-s − 4.41e3i·11-s + (−1.16e3 + 4.88e3i)12-s + 4.37e4·13-s − 4.34e4i·14-s + (−6.26e4 − 1.48e4i)15-s − 7.75e4·16-s + 1.49e5i·17-s + ⋯ |
L(s) = 1 | − 1.11i·2-s + (0.231 − 0.972i)3-s − 0.242·4-s − 1.27i·5-s + (−1.08 − 0.257i)6-s + 1.01·7-s − 0.844i·8-s + (−0.893 − 0.449i)9-s − 1.41·10-s − 0.301i·11-s + (−0.0559 + 0.235i)12-s + 1.53·13-s − 1.13i·14-s + (−1.23 − 0.293i)15-s − 1.18·16-s + 1.79i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.972 - 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.267769 + 2.28446i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.267769 + 2.28446i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-18.7 + 78.8i)T \) |
| 11 | \( 1 + 4.41e3iT \) |
good | 2 | \( 1 + 17.8iT - 256T^{2} \) |
| 5 | \( 1 + 794. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 2.43e3T + 5.76e6T^{2} \) |
| 13 | \( 1 - 4.37e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.49e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.74e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.63e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.99e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.37e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 5.64e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.76e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.67e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 9.56e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 4.43e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 2.31e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.29e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 7.17e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 3.78e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 9.80e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.13e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 2.95e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 5.92e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 7.89e7T + 7.83e15T^{2} \) |
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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
−13.63685339250957481447173517625, −12.89651926570944800168976073100, −11.84699962335441185136319990659, −10.91372619410166541631814624937, −8.947773934151861401803540756043, −8.002049013628538307475870692737, −5.93042654138818907570867628387, −3.84228340758816966863074175690, −1.72445895134968769410061466573, −1.06113454910238854219430288537,
2.72483705569882979905979496432, 4.70063842732658880954119541833, 6.23754130988543656672087118944, 7.61467325195136549456370439151, 8.876394145881647005532139055439, 10.73176950512624799364520146670, 11.32063334089436037209164169286, 14.13618916298088666562317268816, 14.44679875184770275183613750452, 15.57582469813624888767751600726