| L(s) = 1 | − 7.67i·5-s + 7.21·7-s − 6.05i·11-s + 2.29·13-s + 21.8i·17-s − 34.8·19-s + 21.5i·23-s − 33.8·25-s + 10.9i·29-s − 37.6·31-s − 55.3i·35-s − 34.8·37-s + 13.3i·41-s − 60.5·43-s + 3.34i·47-s + ⋯ |
| L(s) = 1 | − 1.53i·5-s + 1.03·7-s − 0.550i·11-s + 0.176·13-s + 1.28i·17-s − 1.83·19-s + 0.935i·23-s − 1.35·25-s + 0.376i·29-s − 1.21·31-s − 1.58i·35-s − 0.941·37-s + 0.325i·41-s − 1.40·43-s + 0.0712i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1653646521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1653646521\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 7.67iT - 25T^{2} \) |
| 7 | \( 1 - 7.21T + 49T^{2} \) |
| 11 | \( 1 + 6.05iT - 121T^{2} \) |
| 13 | \( 1 - 2.29T + 169T^{2} \) |
| 17 | \( 1 - 21.8iT - 289T^{2} \) |
| 19 | \( 1 + 34.8T + 361T^{2} \) |
| 23 | \( 1 - 21.5iT - 529T^{2} \) |
| 29 | \( 1 - 10.9iT - 841T^{2} \) |
| 31 | \( 1 + 37.6T + 961T^{2} \) |
| 37 | \( 1 + 34.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 13.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 60.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 3.34iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 35.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 37.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 25.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 25.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 37.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 77.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 31.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 55.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.43iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 52.8T + 9.40e3T^{2} \) |
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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
−8.774541316936398418364093934346, −8.430580263267797810275969857121, −7.932979708980077307369039781865, −6.71938969554553728718814283942, −5.73976304261416433189767063009, −5.14759059947159197818585301954, −4.34117701233936619589955429138, −3.62379354200063764061093911927, −1.89712766815863935676855724508, −1.37860624715850079558914968625,
0.03696628152694583932163027123, 1.88108547611341321762756175076, 2.50828525692627220961548292370, 3.59888085864097632307226130175, 4.52641080856793438460239160440, 5.34676552629887062379209305892, 6.53643053692015153635777369127, 6.88037071137104206876312922223, 7.72540184214794728604020391302, 8.452215914244197126245105014859
