| L(s) = 1 | − 4.03i·3-s + 11.1·7-s − 7.27·9-s − 17.3·11-s − 6.70·13-s − 3.45i·17-s + 0.787·19-s − 44.8i·21-s − 38.1·23-s − 6.96i·27-s − 37.7i·29-s − 42.7i·31-s + 69.8i·33-s − 0.378·37-s + 27.0i·39-s + ⋯ |
| L(s) = 1 | − 1.34i·3-s + 1.58·7-s − 0.808·9-s − 1.57·11-s − 0.515·13-s − 0.203i·17-s + 0.0414·19-s − 2.13i·21-s − 1.65·23-s − 0.257i·27-s − 1.30i·29-s − 1.37i·31-s + 2.11i·33-s − 0.0102·37-s + 0.693i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0513i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 + 0.0513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.201914204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.201914204\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 4.03iT - 9T^{2} \) |
| 7 | \( 1 - 11.1T + 49T^{2} \) |
| 11 | \( 1 + 17.3T + 121T^{2} \) |
| 13 | \( 1 + 6.70T + 169T^{2} \) |
| 17 | \( 1 + 3.45iT - 289T^{2} \) |
| 19 | \( 1 - 0.787T + 361T^{2} \) |
| 23 | \( 1 + 38.1T + 529T^{2} \) |
| 29 | \( 1 + 37.7iT - 841T^{2} \) |
| 31 | \( 1 + 42.7iT - 961T^{2} \) |
| 37 | \( 1 + 0.378T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.91T + 1.68e3T^{2} \) |
| 43 | \( 1 + 9.21iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 31.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 23.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 47.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 59.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 96.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 41.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 112. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 69.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 58.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 44.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 126. iT - 9.40e3T^{2} \) |
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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
−9.721591208029243809829339364301, −8.199010389971907246681766343630, −7.950723041488229804516571116292, −7.38650491563563223422654439895, −6.13927083081579625629656723984, −5.29136762653996443847093467244, −4.32037389013295108597481200112, −2.45456339445501710358834627014, −1.87283692086784493068025341903, −0.37857488744364189559154595597,
1.83114705863270812309112205757, 3.15004807360075445691456545165, 4.40941638046840412999542385419, 4.98100182001345941866585362871, 5.62773378814655545148334116719, 7.27366611047215224911118863945, 8.112907060572653753597961967955, 8.726603901531763404235570657078, 9.897138788676511227298268656431, 10.48122378253908482664914223560
