| L(s) = 1 | + (−1.22 + 1.22i)3-s + (5.27 − 5.27i)5-s + 12.7·7-s − 2.99i·9-s + (7.46 + 7.46i)11-s + (11.9 + 11.9i)13-s + 12.9i·15-s − 4.55·17-s + (−14.6 + 14.6i)19-s + (−15.6 + 15.6i)21-s + 31.6·23-s − 30.7i·25-s + (3.67 + 3.67i)27-s + (−38.3 − 38.3i)29-s − 13.8i·31-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.408i)3-s + (1.05 − 1.05i)5-s + 1.82·7-s − 0.333i·9-s + (0.678 + 0.678i)11-s + (0.919 + 0.919i)13-s + 0.862i·15-s − 0.268·17-s + (−0.769 + 0.769i)19-s + (−0.743 + 0.743i)21-s + 1.37·23-s − 1.22i·25-s + (0.136 + 0.136i)27-s + (−1.32 − 1.32i)29-s − 0.447i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.617033004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.617033004\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| good | 5 | \( 1 + (-5.27 + 5.27i)T - 25iT^{2} \) |
| 7 | \( 1 - 12.7T + 49T^{2} \) |
| 11 | \( 1 + (-7.46 - 7.46i)T + 121iT^{2} \) |
| 13 | \( 1 + (-11.9 - 11.9i)T + 169iT^{2} \) |
| 17 | \( 1 + 4.55T + 289T^{2} \) |
| 19 | \( 1 + (14.6 - 14.6i)T - 361iT^{2} \) |
| 23 | \( 1 - 31.6T + 529T^{2} \) |
| 29 | \( 1 + (38.3 + 38.3i)T + 841iT^{2} \) |
| 31 | \( 1 + 13.8iT - 961T^{2} \) |
| 37 | \( 1 + (28.7 - 28.7i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 3.98iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-10.7 - 10.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 27.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (43.3 - 43.3i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (6.84 + 6.84i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (8.18 + 8.18i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (26.6 - 26.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 16.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 50.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 75.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-0.382 + 0.382i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 165. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 63.7T + 9.40e3T^{2} \) |
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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
−10.09557692903785501160148503515, −9.113949750587792110539439330204, −8.760822279166569850635229549287, −7.67562135572929586647577867184, −6.38312216282412026105574373506, −5.56840435800281883287978960755, −4.65974861690693546063323406960, −4.17992598777688732689804881971, −1.88285969151171046495413167450, −1.36083551177570459599169328208,
1.19651737205926045643933918148, 2.14410935715367295814750711230, 3.45204243317300901380902936286, 4.99624398027126096919651536029, 5.66832986226123505095033654884, 6.60699368278224691261278381961, 7.33582255476356584168202796305, 8.468460256789984519096239342755, 9.073053043123980087484232913443, 10.57997960412308751009026842656
