| L(s) = 1 | + 1.73i·3-s − 4.58·5-s − 2.99·9-s − 0.594i·11-s − 2.10·13-s − 7.94i·15-s + 1.07·17-s − 14.6i·19-s − 21.3i·23-s − 3.97·25-s − 5.19i·27-s + 15.5·29-s + 37.1i·31-s + 1.02·33-s + 33.9·37-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.917·5-s − 0.333·9-s − 0.0540i·11-s − 0.161·13-s − 0.529i·15-s + 0.0630·17-s − 0.773i·19-s − 0.929i·23-s − 0.158·25-s − 0.192i·27-s + 0.534·29-s + 1.19i·31-s + 0.0311·33-s + 0.917·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.133796491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133796491\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.58T + 25T^{2} \) |
| 11 | \( 1 + 0.594iT - 121T^{2} \) |
| 13 | \( 1 + 2.10T + 169T^{2} \) |
| 17 | \( 1 - 1.07T + 289T^{2} \) |
| 19 | \( 1 + 14.6iT - 361T^{2} \) |
| 23 | \( 1 + 21.3iT - 529T^{2} \) |
| 29 | \( 1 - 15.5T + 841T^{2} \) |
| 31 | \( 1 - 37.1iT - 961T^{2} \) |
| 37 | \( 1 - 33.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 27.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 28.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 0.840iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 28.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 92.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 81.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 10.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 89.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 37.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 7.42iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 64.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 3.12T + 7.92e3T^{2} \) |
| 97 | \( 1 + 92.4T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.935156089415645380575062423667, −8.308364533452295560904525707054, −7.52048083785411364455931716878, −6.75343843408448890792014308528, −5.81746045916065285190528142100, −4.79802682659762779531522824768, −4.26194134248766390492267804444, −3.33153689831589355583621366675, −2.45259108816678130963454721727, −0.807101003901574978472195774843,
0.37946949340947704206905592052, 1.59446125986311710729998264318, 2.75305071961008180343236585240, 3.73411003909499307587379950962, 4.48294631286003917358341532065, 5.60363968639035704168883596769, 6.29369475195929868833100784191, 7.28817095428368992093868565909, 7.81815672967214012346585485590, 8.323773226133496657015867354496
