| L(s) = 1 | − 15.9i·2-s + 1.18·4-s + 230. i·5-s + 3.84e3·7-s − 4.10e3i·8-s + 3.68e3·10-s − 1.95e3i·11-s + 9.93e3·13-s − 6.13e4i·14-s − 6.52e4·16-s − 1.12e5i·17-s − 7.54e4·19-s + 273. i·20-s − 3.12e4·22-s − 3.45e5i·23-s + ⋯ |
| L(s) = 1 | − 0.997i·2-s + 0.00463·4-s + 0.369i·5-s + 1.60·7-s − 1.00i·8-s + 0.368·10-s − 0.133i·11-s + 0.347·13-s − 1.59i·14-s − 0.995·16-s − 1.34i·17-s − 0.579·19-s + 0.00171i·20-s − 0.133·22-s − 1.23i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.58663 - 1.58663i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58663 - 1.58663i\) |
| \(L(5)\) |
|
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 | \( 1 + 15.9iT - 256T^{2} \) |
| 5 | \( 1 - 230. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 3.84e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.95e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 9.93e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.12e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 7.54e4T + 1.69e10T^{2} \) |
| 23 | \( 1 + 3.45e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.34e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.51e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.77e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.33e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 7.13e4T + 1.16e13T^{2} \) |
| 47 | \( 1 + 5.72e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 4.25e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 7.30e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.95e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 8.03e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.86e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.24e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.63e6T + 1.51e15T^{2} \) |
| 83 | \( 1 - 7.31e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 4.03e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 2.17e7T + 7.83e15T^{2} \) |
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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
−15.00981000103508170601843333331, −13.88812462981093458242848470475, −12.26997544296124190765595385574, −11.24154110953913366796557071051, −10.45436521526607730504937236805, −8.606960984924890374716389004745, −6.92194473186964562869823690657, −4.69885965169273100444113555891, −2.74723674656716822079625196628, −1.20427150293273642966561904755,
1.74061209506399329843281085314, 4.63642831290741279674521787922, 6.03538018105473245959179410938, 7.74753875202102207734839826653, 8.569355832850968789060778290433, 10.76605407640637222447807720565, 11.91636279678670281144586900196, 13.71898381867028493547858983937, 14.86944563730547847426177366161, 15.62769336988343172902926635354
