| L(s) = 1 | + 3i·3-s − 17.4i·5-s − 2.99·7-s − 9·9-s + 10.6i·11-s − 43.3i·13-s + 52.2·15-s − 37.8·17-s + 79.8i·19-s − 8.97i·21-s − 191.·23-s − 178.·25-s − 27i·27-s + 138. i·29-s + 212.·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.55i·5-s − 0.161·7-s − 0.333·9-s + 0.291i·11-s − 0.924i·13-s + 0.900·15-s − 0.540·17-s + 0.964i·19-s − 0.0932i·21-s − 1.73·23-s − 1.43·25-s − 0.192i·27-s + 0.889i·29-s + 1.22·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2413127317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2413127317\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| good | 5 | \( 1 + 17.4iT - 125T^{2} \) |
| 7 | \( 1 + 2.99T + 343T^{2} \) |
| 11 | \( 1 - 10.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 43.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 37.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 79.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 138. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 441.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 64.1iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 278. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 830. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 724. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 859. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 467. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 510.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 234.T + 9.12e5T^{2} \) |
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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
−10.07242244999140071521438145037, −9.773627085119473905221594001613, −8.330961838876766229428879229891, −8.247045087014065518773063454614, −6.43899837407399307510722675689, −5.31472131204541080912281350703, −4.61374689467733315313325985410, −3.46142582503955074894277109668, −1.64643899044992633638583610930, −0.07742737098673125996789373351,
2.03346861200024048402022574257, 3.00951455472590295018882837129, 4.30149400629757674648412390117, 6.04712556058499363767824562072, 6.62875123709242923596242867017, 7.42726144599253640288007055834, 8.482614241651220348511013378721, 9.680560185763102442005025556567, 10.54148952129158656794922363034, 11.47089506776329176799900712717
