| L(s) = 1 | + 5.29i·3-s + 8·5-s − 2.64i·7-s − 19.0·9-s + 10.5i·11-s − 4·13-s + 42.3i·15-s − 2·17-s − 26.4i·19-s + 14.0·21-s − 21.1i·23-s + 39·25-s − 52.9i·27-s + 14·29-s + 31.7i·31-s + ⋯ |
| L(s) = 1 | + 1.76i·3-s + 1.60·5-s − 0.377i·7-s − 2.11·9-s + 0.962i·11-s − 0.307·13-s + 2.82i·15-s − 0.117·17-s − 1.39i·19-s + 0.666·21-s − 0.920i·23-s + 1.56·25-s − 1.95i·27-s + 0.482·29-s + 1.02i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.14289 + 1.14289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14289 + 1.14289i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 2.64iT \) |
| good | 3 | \( 1 - 5.29iT - 9T^{2} \) |
| 5 | \( 1 - 8T + 25T^{2} \) |
| 11 | \( 1 - 10.5iT - 121T^{2} \) |
| 13 | \( 1 + 4T + 169T^{2} \) |
| 17 | \( 1 + 2T + 289T^{2} \) |
| 19 | \( 1 + 26.4iT - 361T^{2} \) |
| 23 | \( 1 + 21.1iT - 529T^{2} \) |
| 29 | \( 1 - 14T + 841T^{2} \) |
| 31 | \( 1 - 31.7iT - 961T^{2} \) |
| 37 | \( 1 - 14T + 1.36e3T^{2} \) |
| 41 | \( 1 - 46T + 1.68e3T^{2} \) |
| 43 | \( 1 + 10.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 22T + 2.80e3T^{2} \) |
| 59 | \( 1 + 89.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 84.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 110T + 5.32e3T^{2} \) |
| 79 | \( 1 - 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 37.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 134T + 7.92e3T^{2} \) |
| 97 | \( 1 + 178T + 9.40e3T^{2} \) |
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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
−13.94685507212963549163486608800, −12.73452966790673885555518562537, −11.09040867852138910565025180795, −10.21336811093121172273189282744, −9.654019853187693809039692158927, −8.801801340347135863674983002998, −6.70667778550050906414932982390, −5.29344526219146188010973232090, −4.46755723642027647351788651704, −2.61737207192282527323600793105,
1.42992247741257885403348369265, 2.64391421228244465049837202373, 5.78857047819066157170620777864, 6.09186956031030647764937222976, 7.52169407207388329708972029313, 8.645744602984799093649647491056, 9.818856830119417266232557690101, 11.28358800586661379073193097480, 12.38001279636780710645655746634, 13.21432697895065728119013912910
