L(s) = 1 | + (−0.258 + 0.965i)2-s + (0.965 − 0.258i)3-s + (−0.866 − 0.499i)4-s + i·6-s + (1.87 + 1.86i)7-s + (0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s + (−2.74 + 4.75i)11-s + (−0.965 − 0.258i)12-s + (2.41 + 2.41i)13-s + (−2.28 + 1.33i)14-s + (0.500 + 0.866i)16-s + (0.548 + 2.04i)17-s + (0.258 + 0.965i)18-s + (−3.49 − 6.05i)19-s + ⋯ |
L(s) = 1 | + (−0.183 + 0.683i)2-s + (0.557 − 0.149i)3-s + (−0.433 − 0.249i)4-s + 0.408i·6-s + (0.709 + 0.704i)7-s + (0.249 − 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.827 + 1.43i)11-s + (−0.278 − 0.0747i)12-s + (0.670 + 0.670i)13-s + (−0.611 + 0.355i)14-s + (0.125 + 0.216i)16-s + (0.132 + 0.496i)17-s + (0.0610 + 0.227i)18-s + (−0.802 − 1.38i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.636014554\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.636014554\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.87 - 1.86i)T \) |
good | 11 | \( 1 + (2.74 - 4.75i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.41 - 2.41i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.548 - 2.04i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (3.49 + 6.05i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.69 + 0.454i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 0.684iT - 29T^{2} \) |
| 31 | \( 1 + (-4.82 - 2.78i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.53 - 9.46i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 2.50iT - 41T^{2} \) |
| 43 | \( 1 + (-1.95 + 1.95i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.40 - 0.912i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.43 - 9.08i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (5.08 - 8.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.01 + 0.585i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.61 + 2.57i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + (4.70 - 1.26i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.21 + 4.16i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.05 - 4.05i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.59 + 6.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.1 + 13.1i)T - 97iT^{2} \) |
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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
−10.03134686691640413841125592519, −8.973935209462239090427895678418, −8.586543165023876459121708556801, −7.69991131791121597237581736625, −6.95434831862798213311321706367, −6.06348397093690543252253557474, −4.87769480278847466415168558249, −4.33535885942409932552279918947, −2.67647926740971132344528561780, −1.67198831943769035359969067039,
0.75862732957774951431961051024, 2.15864699441625006331241913426, 3.35155261758893187765760799189, 4.00689502855195183925520303973, 5.20908396296449465466107555234, 6.12778735796122858546217633783, 7.64085560446783619558228500319, 8.138810817720527575805198801672, 8.688013876108313477799938981313, 9.849719029293657707695258363331