| L(s) = 1 | − 1.23i·3-s − 2.23·5-s + 1.23i·7-s + 7.47·9-s − 11.4i·11-s − 5.41·13-s + 2.76i·15-s + 6.94·17-s − 29.8i·19-s + 1.52·21-s − 19.1i·23-s + 5.00·25-s − 20.3i·27-s − 21.0·29-s − 34.4i·31-s + ⋯ |
| L(s) = 1 | − 0.412i·3-s − 0.447·5-s + 0.176i·7-s + 0.830·9-s − 1.03i·11-s − 0.416·13-s + 0.184i·15-s + 0.408·17-s − 1.57i·19-s + 0.0727·21-s − 0.831i·23-s + 0.200·25-s − 0.754i·27-s − 0.726·29-s − 1.11i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.969508 - 0.969508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.969508 - 0.969508i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 3 | \( 1 + 1.23iT - 9T^{2} \) |
| 7 | \( 1 - 1.23iT - 49T^{2} \) |
| 11 | \( 1 + 11.4iT - 121T^{2} \) |
| 13 | \( 1 + 5.41T + 169T^{2} \) |
| 17 | \( 1 - 6.94T + 289T^{2} \) |
| 19 | \( 1 + 29.8iT - 361T^{2} \) |
| 23 | \( 1 + 19.1iT - 529T^{2} \) |
| 29 | \( 1 + 21.0T + 841T^{2} \) |
| 31 | \( 1 + 34.4iT - 961T^{2} \) |
| 37 | \( 1 - 19.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 58.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 62.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 63.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 98.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 1.19T + 3.72e3T^{2} \) |
| 67 | \( 1 - 5.01iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 84.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 70.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 124. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 160. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 46.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 133.T + 9.40e3T^{2} \) |
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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
−11.32237195693699629465774986548, −10.31689349930658523429344761664, −9.238014901867500248203168469470, −8.259722392051670011717886954926, −7.32334930661057567819616074442, −6.44569036324834749405014746856, −5.14485394367414349172233869116, −3.93545325401174367291680205090, −2.51379280919938581893936824386, −0.68563582088384171927285637133,
1.66174953857410579627811185639, 3.54374363148245652089410612753, 4.44047934880012673032000338112, 5.56157293953218821848366587487, 7.10265297184659652360251962424, 7.62796908337782289562339813750, 8.938857382703669926362716962060, 10.05042935634693216594435135845, 10.37897318220514519951693695996, 11.81836748964129258053455630958
