| L(s) = 1 | + 0.603i·2-s + 3.63·4-s − 2.92i·5-s + 7.63·7-s + 4.60i·8-s + 1.76·10-s − 3.31i·11-s − 2.58·13-s + 4.60i·14-s + 11.7·16-s − 14.2i·17-s − 0.921·19-s − 10.6i·20-s + 2.00·22-s + 0.528i·23-s + ⋯ |
| L(s) = 1 | + 0.301i·2-s + 0.908·4-s − 0.584i·5-s + 1.09·7-s + 0.576i·8-s + 0.176·10-s − 0.301i·11-s − 0.198·13-s + 0.328i·14-s + 0.735·16-s − 0.835i·17-s − 0.0485·19-s − 0.531i·20-s + 0.0909·22-s + 0.0229i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.24362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.24362\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + 3.31iT \) |
| good | 2 | \( 1 - 0.603iT - 4T^{2} \) |
| 5 | \( 1 + 2.92iT - 25T^{2} \) |
| 7 | \( 1 - 7.63T + 49T^{2} \) |
| 13 | \( 1 + 2.58T + 169T^{2} \) |
| 17 | \( 1 + 14.2iT - 289T^{2} \) |
| 19 | \( 1 + 0.921T + 361T^{2} \) |
| 23 | \( 1 - 0.528iT - 529T^{2} \) |
| 29 | \( 1 - 3.87iT - 841T^{2} \) |
| 31 | \( 1 - 0.158T + 961T^{2} \) |
| 37 | \( 1 - 41.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 24.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 77.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 82.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 17.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 26.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + 5.98T + 4.48e3T^{2} \) |
| 71 | \( 1 + 108. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 128.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 98.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 103. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 46.6iT - 7.92e3T^{2} \) |
| 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
−11.46164132160489482241815886098, −10.88484154932863619660759818328, −9.606969972350876660652272185098, −8.423975792197971061348907899106, −7.74577618385731950523715346326, −6.69363295289795894212963368111, −5.49405891187167360433473789634, −4.60486880391486857773223335089, −2.79693686322240657088390666061, −1.35132808353482693056060210140,
1.58784044939769376815178730613, 2.75251711137345244052714570382, 4.18261380385458605947058545771, 5.60390065300758536736404158042, 6.75689978639202264067394636065, 7.57193447655609508805889697199, 8.567685375397449630727450703603, 10.02521129960714759052886860763, 10.72682872425880978995243855598, 11.43937736444950984370878194562
