L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.18 + 1.26i)3-s + (−0.499 − 0.866i)4-s + 0.792i·5-s + (−0.5 − 1.65i)6-s + (2.5 + 0.866i)7-s + 0.999·8-s + (−0.186 − 2.99i)9-s + (−0.686 − 0.396i)10-s + (−2.18 + 3.78i)11-s + (1.68 + 0.396i)12-s + (−3.5 + 0.866i)13-s + (−2 + 1.73i)14-s + (−1 − 0.939i)15-s + (−0.5 + 0.866i)16-s + (2.18 + 3.78i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.684 + 0.728i)3-s + (−0.249 − 0.433i)4-s + 0.354i·5-s + (−0.204 − 0.677i)6-s + (0.944 + 0.327i)7-s + 0.353·8-s + (−0.0620 − 0.998i)9-s + (−0.216 − 0.125i)10-s + (−0.659 + 1.14i)11-s + (0.486 + 0.114i)12-s + (−0.970 + 0.240i)13-s + (−0.534 + 0.462i)14-s + (−0.258 − 0.242i)15-s + (−0.125 + 0.216i)16-s + (0.530 + 0.918i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0682331 - 0.633000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0682331 - 0.633000i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.18 - 1.26i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| 13 | \( 1 + (3.5 - 0.866i)T \) |
good | 5 | \( 1 - 0.792iT - 5T^{2} \) |
| 11 | \( 1 + (2.18 - 3.78i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.18 - 3.78i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.18 - 2.05i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.68 + 2.12i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.18 + 1.26i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + (10.1 + 5.84i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (8.18 + 4.72i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 0.939iT - 47T^{2} \) |
| 53 | \( 1 + 2.22iT - 53T^{2} \) |
| 59 | \( 1 + (5.31 - 3.06i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.1 - 5.84i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.05 - 13.9i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 + 1.58iT - 83T^{2} \) |
| 89 | \( 1 + (9.30 + 5.37i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.372 - 0.644i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.95377306254011928489423664216, −10.33955603857537753581796758095, −9.657142016348176602972845026222, −8.617949492185138340879946278381, −7.61141488750175650607224273534, −6.82551194206921806856566443833, −5.53646853375080175619108399257, −5.03586786604847722971286312233, −3.92063457148674118812806012423, −1.99450657905141344499325334863,
0.44578731320238724266552267906, 1.79035854669682259963359720508, 3.18597415995688318318606313287, 4.98211054160208841873850350271, 5.32258941404889102390313224497, 6.92655921548222456488448209823, 7.77209431337103679357293691406, 8.381517559335769011946216015863, 9.565294721918741011296529708882, 10.67624715171640553261291751918