| L(s) = 1 | − 9.23·5-s + 6.15i·7-s + 4.27i·11-s − 1.73·13-s + 12.3·17-s + 33.9i·19-s − 3.86i·23-s + 60.2·25-s − 35.6·29-s + 44.8i·31-s − 56.8i·35-s − 32.7·37-s − 43.7·41-s − 39.1i·43-s − 46.0i·47-s + ⋯ |
| L(s) = 1 | − 1.84·5-s + 0.879i·7-s + 0.388i·11-s − 0.133·13-s + 0.726·17-s + 1.78i·19-s − 0.168i·23-s + 2.41·25-s − 1.23·29-s + 1.44i·31-s − 1.62i·35-s − 0.884·37-s − 1.06·41-s − 0.911i·43-s − 0.979i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06089442011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06089442011\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 9.23T + 25T^{2} \) |
| 7 | \( 1 - 6.15iT - 49T^{2} \) |
| 11 | \( 1 - 4.27iT - 121T^{2} \) |
| 13 | \( 1 + 1.73T + 169T^{2} \) |
| 17 | \( 1 - 12.3T + 289T^{2} \) |
| 19 | \( 1 - 33.9iT - 361T^{2} \) |
| 23 | \( 1 + 3.86iT - 529T^{2} \) |
| 29 | \( 1 + 35.6T + 841T^{2} \) |
| 31 | \( 1 - 44.8iT - 961T^{2} \) |
| 37 | \( 1 + 32.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 43.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 39.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 46.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 46.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 26.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 46.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 65.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 96.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 14.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 39.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 94.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 81.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 15.9T + 9.40e3T^{2} \) |
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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
−8.916873807230376191373598721428, −8.373980725307178993513684247055, −7.60799533132324933801062700553, −6.98651357917327267919230740024, −5.73419927399874841712515913999, −4.91160937044760525770887174585, −3.79625214000934517984365719932, −3.26462692626461431990261952033, −1.72265679765655755856741276803, −0.02314763948603183887222644880,
0.920870353325675996582461838134, 2.86652517275073429472548829260, 3.78393540050975617143970630675, 4.37624251828620350329980849738, 5.38740662877240498054576496822, 6.80001927832318104927059599955, 7.36446339164831779745738663208, 7.961265366084357412850810705104, 8.773449610613614542842534493985, 9.714579718639646308019197468695
