| L(s) = 1 | + 2.76i·3-s + 0.377·5-s + 10.4i·7-s + 1.38·9-s + 5.53i·11-s + 10.3·13-s + 1.04i·15-s − 32.8·17-s + 4.35i·19-s − 28.8·21-s + 28.0i·23-s − 24.8·25-s + 28.6i·27-s + 35.5·29-s − 35.5i·31-s + ⋯ |
| L(s) = 1 | + 0.920i·3-s + 0.0754·5-s + 1.49i·7-s + 0.153·9-s + 0.503i·11-s + 0.794·13-s + 0.0694i·15-s − 1.93·17-s + 0.229i·19-s − 1.37·21-s + 1.21i·23-s − 0.994·25-s + 1.06i·27-s + 1.22·29-s − 1.14i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.482002018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482002018\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 - 2.76iT - 9T^{2} \) |
| 5 | \( 1 - 0.377T + 25T^{2} \) |
| 7 | \( 1 - 10.4iT - 49T^{2} \) |
| 11 | \( 1 - 5.53iT - 121T^{2} \) |
| 13 | \( 1 - 10.3T + 169T^{2} \) |
| 17 | \( 1 + 32.8T + 289T^{2} \) |
| 23 | \( 1 - 28.0iT - 529T^{2} \) |
| 29 | \( 1 - 35.5T + 841T^{2} \) |
| 31 | \( 1 + 35.5iT - 961T^{2} \) |
| 37 | \( 1 - 40.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 32.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + 26.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 92.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.54T + 2.80e3T^{2} \) |
| 59 | \( 1 + 88.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 98.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 57.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 78.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 88.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 56.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 44.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 70.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 168.T + 9.40e3T^{2} \) |
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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
−9.592563418956155472242739764243, −9.384458633491643081998474895793, −8.554932650983712015620087408814, −7.61663625907609426719444924744, −6.35630903461337388233999910542, −5.77613997055904160177683388945, −4.68790144688492603635396542257, −4.06253767866056677373203629644, −2.78491175453027390294772060978, −1.77944604345698358578445737117,
0.44305665912775991145229007818, 1.37883999398467750342833322858, 2.62561548202161741177818507240, 4.02591261539007292717805827690, 4.57785336677869777785531650713, 6.19524998391717668566503694064, 6.66098620988256308406105463483, 7.35978937530115056386063330641, 8.245658521460713075397637064562, 8.924205451111232485133471981358
