| L(s) = 1 | + 27i·3-s + 218. i·5-s − 904.·7-s − 729·9-s − 1.95e3i·11-s + 1.49e3i·13-s − 5.90e3·15-s + 1.46e4·17-s − 3.25e4i·19-s − 2.44e4i·21-s + 2.42e4·23-s + 3.02e4·25-s − 1.96e4i·27-s − 4.77e4i·29-s + 3.84e4·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.783i·5-s − 0.996·7-s − 0.333·9-s − 0.443i·11-s + 0.188i·13-s − 0.452·15-s + 0.723·17-s − 1.08i·19-s − 0.575i·21-s + 0.415·23-s + 0.386·25-s − 0.192i·27-s − 0.363i·29-s + 0.231·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.670787101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.670787101\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
| good | 5 | \( 1 - 218. iT - 7.81e4T^{2} \) |
| 7 | \( 1 + 904.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.95e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 1.49e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.46e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.25e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 2.42e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.77e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 3.84e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.30e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 5.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.41e4iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 8.87e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.01e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 9.56e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.51e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 - 2.12e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.47e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.76e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.45e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 3.18e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.36e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.39446067901277836222119953015, −9.538676025195641900350167230337, −8.774707380405621851672040337961, −7.45880678669005185509960488058, −6.57201105778222973922238573083, −5.71166770962154565272330026814, −4.43438251698989532882751142980, −3.26082175264149578761339775151, −2.67257397961257434725082390762, −0.77717204180599831872126024118,
0.48826585568928068247654790016, 1.46194695971321191271856057459, 2.80213823021023906288228915920, 3.93043242653202070342975420841, 5.24169116059229128167319933894, 6.11300421642623834953208755374, 7.14229438293060879444956648006, 8.033155870704921851291341434215, 9.031473985826429286776754816992, 9.802873191808191693333488930017
