L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (2.18 + 0.473i)5-s + 0.999i·6-s + (0.827 + 0.478i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−0.682 − 2.12i)10-s + 0.252·11-s + (0.866 − 0.499i)12-s + (−0.858 + 1.48i)13-s − 0.956i·14-s + (−1.65 − 1.50i)15-s + (−0.5 − 0.866i)16-s + (−2.88 − 4.99i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.977 + 0.211i)5-s + 0.408i·6-s + (0.312 + 0.180i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.215 − 0.673i)10-s + 0.0762·11-s + (0.249 − 0.144i)12-s + (−0.238 + 0.412i)13-s − 0.255i·14-s + (−0.427 − 0.387i)15-s + (−0.125 − 0.216i)16-s + (−0.698 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.381189405\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381189405\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-2.18 - 0.473i)T \) |
| 37 | \( 1 + (-3.20 - 5.16i)T \) |
good | 7 | \( 1 + (-0.827 - 0.478i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 0.252T + 11T^{2} \) |
| 13 | \( 1 + (0.858 - 1.48i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.88 + 4.99i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.89 - 2.24i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 - 6.74iT - 29T^{2} \) |
| 31 | \( 1 + 0.279iT - 31T^{2} \) |
| 41 | \( 1 + (2.96 - 5.13i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 1.27T + 43T^{2} \) |
| 47 | \( 1 - 6.17iT - 47T^{2} \) |
| 53 | \( 1 + (-7.10 + 4.10i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-13.0 + 7.52i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.43 - 1.98i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.518 - 0.299i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.17 + 10.6i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 10.7iT - 73T^{2} \) |
| 79 | \( 1 + (2.20 + 1.27i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.5 + 7.23i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.84 + 3.95i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−9.755688141709094576079377836582, −9.244341363858801269266428308678, −8.301623000871946210298043238640, −7.14323309237755843310320605636, −6.60972648351765833569310335467, −5.33586786639330150357256616837, −4.81315060430499051420036000453, −3.22196527941200832874124037412, −2.21303638511191670093688992821, −1.13445261338672183410223482396,
0.923726576746801729858935999304, 2.32360892794864643242114436079, 3.98470402903641291531416719269, 5.04319776647245003970970791287, 5.64243800611358480449544380825, 6.49292018353419446632917601674, 7.28759786595031153039256754836, 8.358767168794576167702197878859, 9.108925780582193971949043093967, 9.838436360432215572413650375815