| L(s) = 1 | − 1.41·2-s + (−0.236 + 2.99i)3-s + 2.00·4-s + (0.335 − 4.22i)6-s + 2.64i·7-s − 2.82·8-s + (−8.88 − 1.41i)9-s − 9.89i·11-s + (−0.473 + 5.98i)12-s + 6.33i·13-s − 3.74i·14-s + 4.00·16-s − 5.17·17-s + (12.5 + 2.00i)18-s − 10.2·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.0789 + 0.996i)3-s + 0.500·4-s + (0.0558 − 0.704i)6-s + 0.377i·7-s − 0.353·8-s + (−0.987 − 0.157i)9-s − 0.899i·11-s + (−0.0394 + 0.498i)12-s + 0.486i·13-s − 0.267i·14-s + 0.250·16-s − 0.304·17-s + (0.698 + 0.111i)18-s − 0.541·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8138670976\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8138670976\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (0.236 - 2.99i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 + 9.89iT - 121T^{2} \) |
| 13 | \( 1 - 6.33iT - 169T^{2} \) |
| 17 | \( 1 + 5.17T + 289T^{2} \) |
| 19 | \( 1 + 10.2T + 361T^{2} \) |
| 23 | \( 1 - 15.2T + 529T^{2} \) |
| 29 | \( 1 + 15.1iT - 841T^{2} \) |
| 31 | \( 1 + 13.0T + 961T^{2} \) |
| 37 | \( 1 + 35.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 63.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 19.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 10.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 72.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 0.266iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 17.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 5.55iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 139. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 32.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 2.91T + 6.24e3T^{2} \) |
| 83 | \( 1 + 88.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 167. iT - 9.40e3T^{2} \) |
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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
−9.515821344191564906255819554986, −8.890548367962909742287697299185, −8.394986166068115104008156311458, −7.21536202064749402731646698316, −6.15877909967427165283965000180, −5.46123335588405118200257063050, −4.29922209695960994155287933964, −3.29859588151124790846072490257, −2.18712687806149552380515053385, −0.37860275129560677556164791119,
1.00207206662898950953725705312, 2.06825327172805119495025924195, 3.14910361674074161443081741620, 4.64946298201521389906577844251, 5.76939218549307033749630565870, 6.78479601368508890611231113470, 7.22909047343007973348797257662, 8.123999768188698654277147351887, 8.788301677504265445490131048610, 9.784908246511967398054663597233
