| L(s) = 1 | + 2.64·2-s + 3.00·4-s + (−2.64 − 4.24i)5-s + 11.2i·7-s − 2.64·8-s + (−7.00 − 11.2i)10-s − 4.24i·11-s − 11.2i·13-s + 29.6i·14-s − 18.9·16-s + 10.5·17-s + 20·19-s + (−7.93 − 12.7i)20-s − 11.2i·22-s − 5.29·23-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.750·4-s + (−0.529 − 0.848i)5-s + 1.60i·7-s − 0.330·8-s + (−0.700 − 1.12i)10-s − 0.385i·11-s − 0.863i·13-s + 2.12i·14-s − 1.18·16-s + 0.622·17-s + 1.05·19-s + (−0.396 − 0.636i)20-s − 0.510i·22-s − 0.230·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0578i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 + 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.77099 - 0.0512677i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77099 - 0.0512677i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.64 + 4.24i)T \) |
| good | 2 | \( 1 - 2.64T + 4T^{2} \) |
| 7 | \( 1 - 11.2iT - 49T^{2} \) |
| 11 | \( 1 + 4.24iT - 121T^{2} \) |
| 13 | \( 1 + 11.2iT - 169T^{2} \) |
| 17 | \( 1 - 10.5T + 289T^{2} \) |
| 19 | \( 1 - 20T + 361T^{2} \) |
| 23 | \( 1 + 5.29T + 529T^{2} \) |
| 29 | \( 1 - 8.48iT - 841T^{2} \) |
| 31 | \( 1 - 26T + 961T^{2} \) |
| 37 | \( 1 + 33.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 55.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 21.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 84.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 46.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 22T + 3.72e3T^{2} \) |
| 67 | \( 1 - 89.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 67.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 14T + 6.24e3T^{2} \) |
| 83 | \( 1 - 74.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 89.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 22.4iT - 9.40e3T^{2} \) |
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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
−15.49101780229463246507384250784, −14.45756399092038643472564129422, −13.09973144649670298051484729971, −12.28627678179624555870757487878, −11.57831972666020740592539212873, −9.341901056857532974839818189989, −8.140350270184260463931036343619, −5.86672440114740604001970269545, −5.01968940999722986554726678164, −3.17327027439020235664910501686,
3.42855380757235375829437411522, 4.54391709569482988669188469873, 6.52930374092958023186628093231, 7.58003047582494965399361367001, 9.895381696635400358434481100478, 11.20332124666384339933741293856, 12.21153878685675940118220140065, 13.75953545220165563995112091797, 14.09907473386160379182323141455, 15.27170911640132820099956290985
