L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.28 + 2.21i)3-s + (−0.499 − 0.866i)4-s + (−1.28 − 2.21i)6-s − 0.438·7-s + 0.999·8-s + (−1.78 − 3.08i)9-s + 11-s + 2.56·12-s + (−1 − 1.73i)13-s + (0.219 − 0.379i)14-s + (−0.5 + 0.866i)16-s + (2.56 − 4.43i)17-s + 3.56·18-s + (−2.5 − 3.57i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.739 + 1.28i)3-s + (−0.249 − 0.433i)4-s + (−0.522 − 0.905i)6-s − 0.165·7-s + 0.353·8-s + (−0.593 − 1.02i)9-s + 0.301·11-s + 0.739·12-s + (−0.277 − 0.480i)13-s + (0.0585 − 0.101i)14-s + (−0.125 + 0.216i)16-s + (0.621 − 1.07i)17-s + 0.839·18-s + (−0.573 − 0.819i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.720631 + 0.108940i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.720631 + 0.108940i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.5 + 3.57i)T \) |
good | 3 | \( 1 + (1.28 - 2.21i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.56 + 4.43i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (2.34 + 4.05i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 4.68T + 37T^{2} \) |
| 41 | \( 1 + (-3.06 + 5.30i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.56 + 2.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.43 - 2.49i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.78 + 6.54i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.28 - 12.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.56 + 4.43i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.71 - 8.17i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (8.12 - 14.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.842 - 1.45i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.56 + 9.63i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + (-1.34 - 2.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.842 + 1.45i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.979078343451387869442993613943, −9.449758665055224898959183682739, −8.548760555123878621234908043410, −7.54308161069149032378403271335, −6.50339896577056259295196510060, −5.73845687576166860524115295903, −4.82585311783983490856143275476, −4.22050970552036277765789174689, −2.79327854421939094898649892545, −0.49198785928873035586696560899,
1.16614741241938070408549416878, 2.03685003497668402038170409333, 3.47549000116529370054881429046, 4.67860278308229281341404030444, 6.09619932431761870005228223442, 6.40431852113881690202360225857, 7.71353917563814611499713700706, 8.034391012194054811197451773689, 9.291999136530214864891043819226, 10.08591534982631190882523816716