| L(s) = 1 | + 8i·5-s − 9·9-s + 24i·13-s + 30·17-s − 39·25-s − 40i·29-s − 24i·37-s + 18·41-s − 72i·45-s + 49·49-s − 56i·53-s + 120i·61-s − 192·65-s + 110·73-s + 81·81-s + ⋯ |
| L(s) = 1 | + 1.60i·5-s − 9-s + 1.84i·13-s + 1.76·17-s − 1.56·25-s − 1.37i·29-s − 0.648i·37-s + 0.439·41-s − 1.60i·45-s + 0.999·49-s − 1.05i·53-s + 1.96i·61-s − 2.95·65-s + 1.50·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.866553 + 0.866553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.866553 + 0.866553i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 9T^{2} \) |
| 5 | \( 1 - 8iT - 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 - 24iT - 169T^{2} \) |
| 17 | \( 1 - 30T + 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 40iT - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 + 24iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 18T + 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 56iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 3.48e3T^{2} \) |
| 61 | \( 1 - 120iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 110T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 - 78T + 7.92e3T^{2} \) |
| 97 | \( 1 + 130T + 9.40e3T^{2} \) |
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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
−13.77676078865330230683853503155, −11.96485602203701156075599325867, −11.39787022735936419902063736354, −10.33729263689225348720288237108, −9.294466469909227778540559662291, −7.83655585834435200093201307417, −6.77517785117455227179730627952, −5.77243802406384235237393007373, −3.79571626667575362714368958340, −2.47511284011557875413663458057,
0.885053195404485168075259896355, 3.26350088861955763918854450182, 5.11864648982598716729457242297, 5.71263577535222720521205607235, 7.82339865808919886715035214529, 8.481061015318365249472335382929, 9.584591927863906772345047956544, 10.76492532689609751239162206837, 12.20865951635231157041891798894, 12.60696087425977997847109937278
