| L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 2.23i·5-s − 6.63·7-s + 2.82i·8-s − 3.16·10-s − 21.0i·11-s − 9.78·13-s + 9.37i·14-s + 4.00·16-s + 12.9i·17-s + 16.1·19-s + 4.47i·20-s − 29.7·22-s − 16.9i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 0.447i·5-s − 0.947·7-s + 0.353i·8-s − 0.316·10-s − 1.91i·11-s − 0.752·13-s + 0.669i·14-s + 0.250·16-s + 0.758i·17-s + 0.851·19-s + 0.223i·20-s − 1.35·22-s − 0.736i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06008893899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06008893899\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| good | 7 | \( 1 + 6.63T + 49T^{2} \) |
| 11 | \( 1 + 21.0iT - 121T^{2} \) |
| 13 | \( 1 + 9.78T + 169T^{2} \) |
| 17 | \( 1 - 12.9iT - 289T^{2} \) |
| 19 | \( 1 - 16.1T + 361T^{2} \) |
| 23 | \( 1 + 16.9iT - 529T^{2} \) |
| 29 | \( 1 - 50.0iT - 841T^{2} \) |
| 31 | \( 1 + 10.0T + 961T^{2} \) |
| 37 | \( 1 - 2.54T + 1.36e3T^{2} \) |
| 41 | \( 1 - 53.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 74.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 47.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 1.08iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 9.63iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 44.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 3.34T + 4.48e3T^{2} \) |
| 71 | \( 1 - 38.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 88.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 117.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 147. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 47.0T + 9.40e3T^{2} \) |
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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
−10.29934370827645264952336352228, −9.456847157631039631648169438432, −8.734776559815129385516698735191, −7.987470650421586708061476461830, −6.65999496665842286102133541080, −5.78150112559132665209057697044, −4.84109243719433718920926017316, −3.50371262319132032080475605477, −2.94708302278893956528990560940, −1.24962966894369835665250367360,
0.02136740749624916709571698247, 2.14398321098793545596934907692, 3.39633208090095022338377105515, 4.55951497194280790894250028982, 5.43691379578690213691349928017, 6.58273476062518266769800972491, 7.22830425343121532946354084956, 7.74268342016679777573631978756, 9.232121872446925350665060829736, 9.760200551208934023990053063249
