| L(s) = 1 | − 8.36i·5-s + 8.08·7-s − 3.31i·11-s − 19.8·13-s + 30.8i·17-s − 13.9·19-s − 24.1i·23-s − 44.9·25-s − 8.62i·29-s − 46.9·31-s − 67.6i·35-s − 58.1·37-s + 12.6i·41-s + 65.2·43-s + 57.8i·47-s + ⋯ |
| L(s) = 1 | − 1.67i·5-s + 1.15·7-s − 0.301i·11-s − 1.52·13-s + 1.81i·17-s − 0.732·19-s − 1.05i·23-s − 1.79·25-s − 0.297i·29-s − 1.51·31-s − 1.93i·35-s − 1.57·37-s + 0.307i·41-s + 1.51·43-s + 1.23i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2862531796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2862531796\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + 3.31iT \) |
| good | 5 | \( 1 + 8.36iT - 25T^{2} \) |
| 7 | \( 1 - 8.08T + 49T^{2} \) |
| 13 | \( 1 + 19.8T + 169T^{2} \) |
| 17 | \( 1 - 30.8iT - 289T^{2} \) |
| 19 | \( 1 + 13.9T + 361T^{2} \) |
| 23 | \( 1 + 24.1iT - 529T^{2} \) |
| 29 | \( 1 + 8.62iT - 841T^{2} \) |
| 31 | \( 1 + 46.9T + 961T^{2} \) |
| 37 | \( 1 + 58.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 12.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 65.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 57.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 58.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 77.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 71.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 100.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 24.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 38.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 10.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 50.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 59.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 44.6T + 9.40e3T^{2} \) |
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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
−8.639446395835449502467223304607, −8.169941739216434029551738725428, −7.41434792437559033439516764973, −6.10610603362137441443176689532, −5.26520892137237442770615054497, −4.63144119857648169260368456499, −3.97287444408732761600126226445, −2.19440422033851195018613129364, −1.41989566709106857302498894043, −0.07181861685022917191308764667,
1.94570206889205319000528503092, 2.63051336608864541904044867873, 3.67716328609056512851595323142, 4.87088395015219789474029820783, 5.48820162894439694963195511692, 6.83342139941396150818588549946, 7.30178047825303124088329039167, 7.73995822025816628964997102237, 9.062089030430516427346281204995, 9.755625951742837484264493675024
