| L(s) = 1 | + 1.73i·3-s + 2.23·5-s + 2.64i·7-s − 2.99·9-s + 0.994i·11-s + 11.7·13-s + 3.87i·15-s − 14.5·17-s − 31.6i·19-s − 4.58·21-s − 17.8i·23-s + 5.00·25-s − 5.19i·27-s + 49.5·29-s − 46.7i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447·5-s + 0.377i·7-s − 0.333·9-s + 0.0903i·11-s + 0.902·13-s + 0.258i·15-s − 0.857·17-s − 1.66i·19-s − 0.218·21-s − 0.775i·23-s + 0.200·25-s − 0.192i·27-s + 1.70·29-s − 1.50i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.176769733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.176769733\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 - 0.994iT - 121T^{2} \) |
| 13 | \( 1 - 11.7T + 169T^{2} \) |
| 17 | \( 1 + 14.5T + 289T^{2} \) |
| 19 | \( 1 + 31.6iT - 361T^{2} \) |
| 23 | \( 1 + 17.8iT - 529T^{2} \) |
| 29 | \( 1 - 49.5T + 841T^{2} \) |
| 31 | \( 1 + 46.7iT - 961T^{2} \) |
| 37 | \( 1 - 47.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.75T + 1.68e3T^{2} \) |
| 43 | \( 1 - 65.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 2.11iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.48T + 2.80e3T^{2} \) |
| 59 | \( 1 - 47.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 76.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 33.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 48.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 42.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.74iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 150.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 155.T + 9.40e3T^{2} \) |
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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
−9.103981765759353586114675817955, −8.653931109555551706642322181731, −7.68234197360250809159307326550, −6.40687793634923659437801568489, −6.15214580494752822421940301226, −4.81906724958255973301606789742, −4.40754982308734450941096408702, −3.03573853104743068555408977958, −2.28356633075212274613785214703, −0.70753493372014628113270012030,
1.00063962203862690756868399509, 1.88941752865460562949444818930, 3.12844391972850776566336646004, 4.06520106365489769833456246836, 5.19693945451314644590591983542, 6.11650162553028433548364511216, 6.64140833614032706454176725822, 7.59415899507815083943264892216, 8.400073410887911425832158904921, 8.984466438263813103997885896531
