| L(s) = 1 | + 3i·2-s + 7·4-s + 33i·5-s − 19·7-s + 69i·8-s − 99·10-s − 123i·11-s + 302·13-s − 57i·14-s − 95·16-s − 414i·17-s − 304·19-s + 231i·20-s + 369·22-s − 300i·23-s + ⋯ |
| L(s) = 1 | + 0.750i·2-s + 0.437·4-s + 1.32i·5-s − 0.387·7-s + 1.07i·8-s − 0.989·10-s − 1.01i·11-s + 1.78·13-s − 0.290i·14-s − 0.371·16-s − 1.43i·17-s − 0.842·19-s + 0.577i·20-s + 0.762·22-s − 0.567i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.05829 + 1.05829i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05829 + 1.05829i\) |
| \(L(3)\) |
|
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 | \( 1 - 3iT - 16T^{2} \) |
| 5 | \( 1 - 33iT - 625T^{2} \) |
| 7 | \( 1 + 19T + 2.40e3T^{2} \) |
| 11 | \( 1 + 123iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 302T + 2.85e4T^{2} \) |
| 17 | \( 1 + 414iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 304T + 1.30e5T^{2} \) |
| 23 | \( 1 + 300iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 678iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 239T + 9.23e5T^{2} \) |
| 37 | \( 1 - 740T + 1.87e6T^{2} \) |
| 41 | \( 1 + 228iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 982T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.16e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.59e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 2.92e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 316T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.62e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.81e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.03e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.04e4T + 3.89e7T^{2} \) |
| 83 | \( 1 + 1.26e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.00e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 6.51e3T + 8.85e7T^{2} \) |
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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
−16.50259104642843069262332811953, −15.76107984422775803842044476115, −14.57240935625533123466617305303, −13.57193457608714596607695177070, −11.42635878253882303189727707945, −10.66317134386680375165190413963, −8.537052936300178784252267315297, −6.92663286302957008782946200185, −6.04368205348361137920981572860, −3.05553397015761697482722952315,
1.50186504539961395047944735848, 4.03088518790966787447861190872, 6.25477143298326408900515905722, 8.356504099295465831842273148100, 9.810941832294436441718305423684, 11.18386426520262237837112012986, 12.58931403462131941534971713157, 13.11924741795611269714341046009, 15.31650150181296386185682261607, 16.23279407038076774837417964119
