L(s) = 1 | + 3.80i·3-s − 2.23·5-s + 8.50i·7-s − 5.47·9-s + 1.79i·11-s − 0.472·13-s − 8.50i·15-s − 23.8·17-s − 9.40i·19-s − 32.3·21-s + 16.1i·23-s + 5.00·25-s + 13.4i·27-s − 6.94·29-s − 47.4i·31-s + ⋯ |
L(s) = 1 | + 1.26i·3-s − 0.447·5-s + 1.21i·7-s − 0.608·9-s + 0.163i·11-s − 0.0363·13-s − 0.567i·15-s − 1.40·17-s − 0.494i·19-s − 1.54·21-s + 0.700i·23-s + 0.200·25-s + 0.497i·27-s − 0.239·29-s − 1.53i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.969692i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.969692i\) |
\(L(2)\) |
|
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 | \( 1 - 3.80iT - 9T^{2} \) |
| 7 | \( 1 - 8.50iT - 49T^{2} \) |
| 11 | \( 1 - 1.79iT - 121T^{2} \) |
| 13 | \( 1 + 0.472T + 169T^{2} \) |
| 17 | \( 1 + 23.8T + 289T^{2} \) |
| 19 | \( 1 + 9.40iT - 361T^{2} \) |
| 23 | \( 1 - 16.1iT - 529T^{2} \) |
| 29 | \( 1 + 6.94T + 841T^{2} \) |
| 31 | \( 1 + 47.4iT - 961T^{2} \) |
| 37 | \( 1 + 26.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 41.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 2.00iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 21.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 73.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 88.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 39.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 137.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 21.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 67.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 39.1T + 9.40e3T^{2} \) |
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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
−11.59151071566430929221383156025, −11.04900608914055974496259867117, −9.895683943238428710400436239266, −9.147279943720264490584316953772, −8.451578601633214421155859003268, −7.05420268709369504086731114474, −5.72102826704691937217778603235, −4.76683187146510693734183114139, −3.80240239140930554536661662757, −2.41804115093982427079582370547,
0.44881164330531768164790096116, 1.87010954949121341654638922819, 3.57646773042693015747548752794, 4.80447769971993259211372658725, 6.51963253962385560731232422003, 6.98799499083442646050864203757, 7.917032670545448728508925336146, 8.762734025003559397603287348353, 10.25262410122465889351417669922, 11.00561450322235681643308199020