| L(s) = 1 | − 3.39i·2-s − 7.49·4-s + 3.49i·5-s − 10.8·7-s + 11.8i·8-s + 11.8·10-s + 0.0247i·11-s + 8.63·13-s + 36.8i·14-s + 10.2·16-s + 4.10i·17-s + 24.5·19-s − 26.2i·20-s + 0.0838·22-s + 7.53i·23-s + ⋯ |
| L(s) = 1 | − 1.69i·2-s − 1.87·4-s + 0.699i·5-s − 1.55·7-s + 1.48i·8-s + 1.18·10-s + 0.00224i·11-s + 0.664·13-s + 2.63i·14-s + 0.638·16-s + 0.241i·17-s + 1.29·19-s − 1.31i·20-s + 0.00380·22-s + 0.327i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.248564157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248564157\) |
| \(L(2)\) |
|
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 | \( 1 + 3.39iT - 4T^{2} \) |
| 5 | \( 1 - 3.49iT - 25T^{2} \) |
| 7 | \( 1 + 10.8T + 49T^{2} \) |
| 11 | \( 1 - 0.0247iT - 121T^{2} \) |
| 13 | \( 1 - 8.63T + 169T^{2} \) |
| 17 | \( 1 - 4.10iT - 289T^{2} \) |
| 19 | \( 1 - 24.5T + 361T^{2} \) |
| 23 | \( 1 - 7.53iT - 529T^{2} \) |
| 29 | \( 1 - 19.4iT - 841T^{2} \) |
| 31 | \( 1 + 10.9T + 961T^{2} \) |
| 37 | \( 1 - 10.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 77.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 19.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 46.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 51.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 10.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 106.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 48.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 72.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 92.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 43.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 93.8T + 9.40e3T^{2} \) |
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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
−10.19660091938388533001423362872, −9.428284219759415564132711360295, −8.812393387814581994872495929788, −7.31597043612145197560503880945, −6.43053574339897850921405317001, −5.27667359284526529956313951683, −3.67316959148081449737507531530, −3.38401535050537409025223834731, −2.32718930387982929057220484445, −0.797665740210459767478258662201,
0.68958011814167666082984899151, 3.11071952219690223898602403480, 4.30438236323949313951661056319, 5.32568609568778425475719736487, 6.13514101358400453301084734217, 6.77043074847932397612345011274, 7.70297112179200407716394016662, 8.543135453801396345357406897127, 9.394154999673396966613261480881, 9.788911890659331223409914743424
