| L(s) = 1 | + 1.41i·2-s + 1.73·3-s − 2.00·4-s + 4.92·5-s + 2.44i·6-s + 6.03i·7-s − 2.82i·8-s + 2.99·9-s + 6.96i·10-s + (−9.19 − 6.03i)11-s − 3.46·12-s + 8.48i·13-s − 8.53·14-s + 8.53·15-s + 4.00·16-s − 28.7i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.500·4-s + 0.985·5-s + 0.408i·6-s + 0.862i·7-s − 0.353i·8-s + 0.333·9-s + 0.696i·10-s + (−0.836 − 0.548i)11-s − 0.288·12-s + 0.652i·13-s − 0.609·14-s + 0.569·15-s + 0.250·16-s − 1.69i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.30338 + 0.703586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30338 + 0.703586i\) |
| \(L(2)\) |
|
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 | \( 1 - 1.73T \) |
| 11 | \( 1 + (9.19 + 6.03i)T \) |
| good | 5 | \( 1 - 4.92T + 25T^{2} \) |
| 7 | \( 1 - 6.03iT - 49T^{2} \) |
| 13 | \( 1 - 8.48iT - 169T^{2} \) |
| 17 | \( 1 + 28.7iT - 289T^{2} \) |
| 19 | \( 1 + 10.9iT - 361T^{2} \) |
| 23 | \( 1 - 2.39T + 529T^{2} \) |
| 29 | \( 1 + 30.0iT - 841T^{2} \) |
| 31 | \( 1 + 41.8T + 961T^{2} \) |
| 37 | \( 1 - 23.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 81.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 28.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 66.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 28.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 26.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 82.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 81.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 89.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 88.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 24.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 50T + 7.92e3T^{2} \) |
| 97 | \( 1 - 173.T + 9.40e3T^{2} \) |
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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
−14.74811649453002791283937019221, −13.74087385489881646149108516208, −13.08861818202419915042467088667, −11.45589026050027880772320235710, −9.699168243700645667884356334903, −9.059136718923635700165173662063, −7.70742839641632382960733601866, −6.21799676482337282905895416533, −5.00995349875738341976094148202, −2.61671934473076616789477305235,
1.93585249717617379669126970953, 3.76775402893384780021643772293, 5.56635617124005436861226534719, 7.46000877935938423762843578143, 8.802703402663199491862780185992, 10.24468533543437870043932893765, 10.51982209190704573488687102497, 12.57577716123079824456795788623, 13.21832494129263500687547235089, 14.18876692858429629884136997116
