L(s) = 1 | + 1.73·3-s + 8.89i·5-s + 2.82i·7-s + 2.99·9-s − 18.2·11-s + 5.79i·13-s + 15.4i·15-s − 21.5·17-s + 18.2·19-s + 4.89i·21-s − 33.3i·23-s − 54.1·25-s + 5.19·27-s + 4.49i·29-s + 2.25i·31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.77i·5-s + 0.404i·7-s + 0.333·9-s − 1.65·11-s + 0.445i·13-s + 1.02i·15-s − 1.27·17-s + 0.960·19-s + 0.233i·21-s − 1.45i·23-s − 2.16·25-s + 0.192·27-s + 0.154i·29-s + 0.0728i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.543597 + 1.31236i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.543597 + 1.31236i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
good | 5 | \( 1 - 8.89iT - 25T^{2} \) |
| 7 | \( 1 - 2.82iT - 49T^{2} \) |
| 11 | \( 1 + 18.2T + 121T^{2} \) |
| 13 | \( 1 - 5.79iT - 169T^{2} \) |
| 17 | \( 1 + 21.5T + 289T^{2} \) |
| 19 | \( 1 - 18.2T + 361T^{2} \) |
| 23 | \( 1 + 33.3iT - 529T^{2} \) |
| 29 | \( 1 - 4.49iT - 841T^{2} \) |
| 31 | \( 1 - 2.25iT - 961T^{2} \) |
| 37 | \( 1 - 43.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 1.59T + 1.68e3T^{2} \) |
| 43 | \( 1 - 63.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 72.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 70.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 34.6T + 3.48e3T^{2} \) |
| 61 | \( 1 - 63.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 3.24T + 4.48e3T^{2} \) |
| 71 | \( 1 - 68.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 35.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 42.2T + 6.88e3T^{2} \) |
| 89 | \( 1 - 5.19T + 7.92e3T^{2} \) |
| 97 | \( 1 + 26.8T + 9.40e3T^{2} \) |
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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
−11.14542889455126646091982571868, −10.62912705965786827211784867521, −9.792117506047038547521619084379, −8.640933342173182980023484772423, −7.61409454282058622358729147779, −6.90201880219013170882823205398, −5.87228877023309869152501390125, −4.40148409664798519766239810780, −2.85337990190841641218506003606, −2.48245251814859696055176850910,
0.55453423134577743180133730555, 2.13647827428484131105236094576, 3.75019784939746879223032331572, 4.93335492315791671880524551106, 5.56453226194417313722642746541, 7.40543289888976544612656408124, 8.030163122758195829753593191908, 8.925952750557589324188453998116, 9.650131052228941980580741284767, 10.68063862225455790121470763464