| L(s) = 1 | + 1.41i·2-s − 1.73·3-s − 2.00·4-s − 2.44i·6-s + (−4.78 − 5.11i)7-s − 2.82i·8-s + 2.99·9-s + 10.3·11-s + 3.46·12-s + 0.605·13-s + (7.23 − 6.76i)14-s + 4.00·16-s + 5.49·17-s + 4.24i·18-s + 33.2i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.500·4-s − 0.408i·6-s + (−0.683 − 0.730i)7-s − 0.353i·8-s + 0.333·9-s + 0.941·11-s + 0.288·12-s + 0.0465·13-s + (0.516 − 0.483i)14-s + 0.250·16-s + 0.323·17-s + 0.235i·18-s + 1.74i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3255026177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3255026177\) |
| \(L(2)\) |
|
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 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (4.78 + 5.11i)T \) |
| good | 11 | \( 1 - 10.3T + 121T^{2} \) |
| 13 | \( 1 - 0.605T + 169T^{2} \) |
| 17 | \( 1 - 5.49T + 289T^{2} \) |
| 19 | \( 1 - 33.2iT - 361T^{2} \) |
| 23 | \( 1 + 14.4iT - 529T^{2} \) |
| 29 | \( 1 + 42.0T + 841T^{2} \) |
| 31 | \( 1 - 0.540iT - 961T^{2} \) |
| 37 | \( 1 - 13.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 13.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 53.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 19.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 29.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 74.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 12.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 42.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 66.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 27.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 126.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 30.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 164.T + 9.40e3T^{2} \) |
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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
−9.574238819479701984340253486002, −8.553171530485657272107168838983, −7.59958571875838372613979152726, −6.84224717467995822231459818784, −6.15439023367089675040727269645, −5.37783789669881361783163497778, −4.11851271971978288767308298726, −3.55295134005497779168416009252, −1.51954119512648510138437510718, −0.11888136274499469156891275383,
1.31200799422361091191576243753, 2.64623644787665429384429731526, 3.64643687519223574889811517621, 4.71434781598126519768577582979, 5.65722203264497305370986132734, 6.47211206064873586215485001043, 7.37755328447136394608936024352, 8.640804190620938619818784564443, 9.426485585034454487689029341433, 9.783211716408963148722437141191
