L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.337 + 1.69i)3-s − 1.00i·4-s + (2.19 − 2.19i)5-s + (−0.962 − 1.44i)6-s + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)8-s + (−2.77 − 1.14i)9-s + 3.09i·10-s + (1.27 + 1.27i)11-s + (1.69 + 0.337i)12-s + (−3.23 + 1.58i)13-s − 1.00i·14-s + (2.98 + 4.46i)15-s − 1.00·16-s + 3.68·17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.194 + 0.980i)3-s − 0.500i·4-s + (0.980 − 0.980i)5-s + (−0.392 − 0.587i)6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + (−0.923 − 0.382i)9-s + 0.980i·10-s + (0.384 + 0.384i)11-s + (0.490 + 0.0974i)12-s + (−0.897 + 0.440i)13-s − 0.267i·14-s + (0.770 + 1.15i)15-s − 0.250·16-s + 0.892·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0338 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0338 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.850076 + 0.821753i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.850076 + 0.821753i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.337 - 1.69i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (3.23 - 1.58i)T \) |
good | 5 | \( 1 + (-2.19 + 2.19i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1.27 - 1.27i)T + 11iT^{2} \) |
| 17 | \( 1 - 3.68T + 17T^{2} \) |
| 19 | \( 1 + (-5.26 - 5.26i)T + 19iT^{2} \) |
| 23 | \( 1 - 8.15T + 23T^{2} \) |
| 29 | \( 1 - 9.50iT - 29T^{2} \) |
| 31 | \( 1 + (0.125 + 0.125i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.328 - 0.328i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.21 - 2.21i)T - 41iT^{2} \) |
| 43 | \( 1 - 1.43iT - 43T^{2} \) |
| 47 | \( 1 + (0.805 + 0.805i)T + 47iT^{2} \) |
| 53 | \( 1 + 5.59iT - 53T^{2} \) |
| 59 | \( 1 + (-1.49 - 1.49i)T + 59iT^{2} \) |
| 61 | \( 1 + 8.14T + 61T^{2} \) |
| 67 | \( 1 + (10.6 + 10.6i)T + 67iT^{2} \) |
| 71 | \( 1 + (0.752 - 0.752i)T - 71iT^{2} \) |
| 73 | \( 1 + (-3.59 + 3.59i)T - 73iT^{2} \) |
| 79 | \( 1 - 4.38T + 79T^{2} \) |
| 83 | \( 1 + (-9.35 + 9.35i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.19 - 3.19i)T + 89iT^{2} \) |
| 97 | \( 1 + (-1.79 - 1.79i)T + 97iT^{2} \) |
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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
−10.63381288856660900525336888291, −9.776136506460649447385515815498, −9.372837417102640915340964561659, −8.756582089209074405713617885384, −7.48088837608944936845350959856, −6.29596311338459615350585462504, −5.25405456507213699908457932198, −4.94217562170613891296059503260, −3.25970535848757692973950401620, −1.42840454817281527534580902899,
0.946536480488979162954107322543, 2.50048578588406985200145409669, 3.13314125143481705650662543856, 5.19237432279635881490334040548, 6.21854439921630229321193843368, 7.11387244123781051244734013320, 7.65643092614784873538321726700, 9.021740661239789027744391760445, 9.780910114104967531174998174308, 10.60335413655622057753937805616