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 | \( 1 - 22.4iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.27170911640132820099956290985, −14.09907473386160379182323141455, −13.75953545220165563995112091797, −12.21153878685675940118220140065, −11.20332124666384339933741293856, −9.895381696635400358434481100478, −7.58003047582494965399361367001, −6.52930374092958023186628093231, −4.54391709569482988669188469873, −3.42855380757235375829437411522,
3.17327027439020235664910501686, 5.01968940999722986554726678164, 5.86672440114740604001970269545, 8.140350270184260463931036343619, 9.341901056857532974839818189989, 11.57831972666020740592539212873, 12.28627678179624555870757487878, 13.09973144649670298051484729971, 14.45756399092038643472564129422, 15.49101780229463246507384250784