| L(s) = 1 | + 9.48i·2-s − 26·4-s + 132. i·5-s − 403·7-s + 360. i·8-s − 1.26e3·10-s − 1.49e3i·11-s − 961·13-s − 3.82e3i·14-s − 5.08e3·16-s + 9.61e3i·17-s + 8.02e3·19-s − 3.45e3i·20-s + 1.42e4·22-s + 1.06e4i·23-s + ⋯ |
| L(s) = 1 | + 1.18i·2-s − 0.406·4-s + 1.06i·5-s − 1.17·7-s + 0.704i·8-s − 1.26·10-s − 1.12i·11-s − 0.437·13-s − 1.39i·14-s − 1.24·16-s + 1.95i·17-s + 1.16·19-s − 0.431i·20-s + 1.33·22-s + 0.871i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.23619i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23619i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 9.48iT - 64T^{2} \) |
| 5 | \( 1 - 132. iT - 1.56e4T^{2} \) |
| 7 | \( 1 + 403T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.49e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 961T + 4.82e6T^{2} \) |
| 17 | \( 1 - 9.61e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.02e3T + 4.70e7T^{2} \) |
| 23 | \( 1 - 1.06e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.46e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 4.88e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.41e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 7.07e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.08e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 2.69e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.37e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 9.75e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.72e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 8.55e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.41e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.52e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 7.40e4T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.65e4iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.19e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.19e6T + 8.32e11T^{2} \) |
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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
−16.43328849811624530887720060325, −15.48528904324243096361972788896, −14.48987751077665638330503509048, −13.31608998005064072822578027533, −11.43634296735836123638826621455, −10.01190516188246852502575111419, −8.203161728507335186059484202552, −6.76987967388284828238587835056, −5.90714014983130655915811795132, −3.17000916963960605459453982421,
0.70185214480232132091393061711, 2.76805068737731192224677330883, 4.71137393850273725566382844012, 7.03267054245268434675094644345, 9.320413258663293903084936825604, 9.986836991501613221822444285694, 11.87650182552932993478558746076, 12.51230793523069452017677542745, 13.61115121788217362920383645640, 15.70354160908033361893694124742
