| L(s) = 1 | − 9i·3-s − 21.1i·5-s − 241.·7-s − 81·9-s − 641. i·11-s − 316. i·13-s − 190.·15-s − 901.·17-s − 2.16e3i·19-s + 2.17e3i·21-s + 397.·23-s + 2.67e3·25-s + 729i·27-s − 5.17e3i·29-s − 2.39e3·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.377i·5-s − 1.86·7-s − 0.333·9-s − 1.59i·11-s − 0.519i·13-s − 0.218·15-s − 0.756·17-s − 1.37i·19-s + 1.07i·21-s + 0.156·23-s + 0.857·25-s + 0.192i·27-s − 1.14i·29-s − 0.446·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1988931594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1988931594\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
| good | 5 | \( 1 + 21.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 241.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 641. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 316. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 901.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.16e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 397.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.17e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.35e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.69e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.71e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.86e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.07e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.59e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.79e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.07e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.59e4T + 8.58e9T^{2} \) |
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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
−9.678398669546732914720191085639, −8.915197941932102957363107425085, −8.069246904934994531646242965736, −6.59876568665710594119033735599, −6.35307734919723049695251859019, −5.04209922901795166158558735483, −3.40528935894169322581086340310, −2.74033075021766116724948194045, −0.809355370337969464397926778370, −0.06496295618053587088206836800,
2.04069676201246267082796439914, 3.29929373435672725286759717672, 4.13694399534161114059354671995, 5.44280884770481673324080173268, 6.69608017757676662359784185275, 7.09737154723162837541034569346, 8.735786473460331042017355136061, 9.602073314027207763044335856890, 10.09558546710336610675383233089, 10.95496764081577046948945265303
