| L(s) = 1 | − 1.73i·3-s + 8.29·5-s − 8.55i·7-s − 2.99·9-s − 13.7i·11-s − 17.0·13-s − 14.3i·15-s + 20.3·17-s + 20.4i·19-s − 14.8·21-s + 5.51i·23-s + 43.7·25-s + 5.19i·27-s − 41.0·29-s − 22.2i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.65·5-s − 1.22i·7-s − 0.333·9-s − 1.25i·11-s − 1.31·13-s − 0.957i·15-s + 1.19·17-s + 1.07i·19-s − 0.705·21-s + 0.239i·23-s + 1.75·25-s + 0.192i·27-s − 1.41·29-s − 0.719i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.45468 - 1.45468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45468 - 1.45468i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 - 8.29T + 25T^{2} \) |
| 7 | \( 1 + 8.55iT - 49T^{2} \) |
| 11 | \( 1 + 13.7iT - 121T^{2} \) |
| 13 | \( 1 + 17.0T + 169T^{2} \) |
| 17 | \( 1 - 20.3T + 289T^{2} \) |
| 19 | \( 1 - 20.4iT - 361T^{2} \) |
| 23 | \( 1 - 5.51iT - 529T^{2} \) |
| 29 | \( 1 + 41.0T + 841T^{2} \) |
| 31 | \( 1 + 22.2iT - 961T^{2} \) |
| 37 | \( 1 - 11.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 66.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 19.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 17.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 47.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 74.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 16.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 101.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.879iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 23.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 16.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 188.T + 9.40e3T^{2} \) |
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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
−10.66852437953348563423893095756, −10.02974356984481768541471419871, −9.256626137116050560687090586298, −7.913340647450998361230132475407, −7.14147090353369775982866389610, −5.95521475305667273569695891775, −5.40313510276454460128871867066, −3.64595969747790001748624535247, −2.22326937426759364830426984407, −0.927063185339134071842689922414,
1.98467645101252774206492331805, 2.79956219368289020129756411452, 4.83861796647265328502114287882, 5.37383342987255703696372286089, 6.37358823414101723612343996625, 7.59103831501446531156444611562, 9.081444877562347572002139876875, 9.584078595583123259775453122713, 10.04758419688843113575351535352, 11.27355697721163013701928361365
