| L(s) = 1 | − 9.79·5-s + 10i·7-s − 19.5i·11-s + 70.9·25-s − 29.3·29-s − 38i·31-s − 97.9i·35-s − 51·49-s + 48.9·53-s + 191. i·55-s − 117. i·59-s + 50·73-s + 195.·77-s + 58i·79-s + 97.9i·83-s + ⋯ |
| L(s) = 1 | − 1.95·5-s + 1.42i·7-s − 1.78i·11-s + 2.83·25-s − 1.01·29-s − 1.22i·31-s − 2.79i·35-s − 1.04·49-s + 0.924·53-s + 3.49i·55-s − 1.99i·59-s + 0.684·73-s + 2.54·77-s + 0.734i·79-s + 1.18i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6437106826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6437106826\) |
| \(L(2)\) |
|
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 | \( 1 + 9.79T + 25T^{2} \) |
| 7 | \( 1 - 10iT - 49T^{2} \) |
| 11 | \( 1 + 19.5iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 29.3T + 841T^{2} \) |
| 31 | \( 1 + 38iT - 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 117. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 50T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 97.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 + 190T + 9.40e3T^{2} \) |
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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
−8.755643810876821656357699615587, −8.265172331840330271340176517639, −7.75850732177185775401915625606, −6.71716144630320663255630419520, −5.82150250724435124139643067920, −5.11553320465664845746111863737, −3.94899011098152153020771940081, −3.38723878150185355565469140358, −2.47551979418242481886955964312, −0.72059049115024612794924143916,
0.25034872120441938495022209497, 1.44567942849851958235677034953, 3.01790591740906781322941013768, 4.09134338724035898720703239832, 4.22435810731441905349335624962, 5.15578815112845455471770408134, 6.81605989549089768692532183055, 7.22228360659911710397242940490, 7.62407096467797421757000730434, 8.434241145504740766237558367615
