| L(s) = 1 | + (−1.44 − 2.62i)3-s + (−2.67 − 4.22i)5-s + 1.30i·7-s + (−4.83 + 7.59i)9-s + 9.69·11-s + 15.3·13-s + (−7.23 + 13.1i)15-s + 9.62·17-s + 9.66i·19-s + (3.41 − 1.87i)21-s + 16.1·23-s + (−10.6 + 22.6i)25-s + (26.9 + 1.74i)27-s + 47.2·29-s − 40.6·31-s + ⋯ |
| L(s) = 1 | + (−0.481 − 0.876i)3-s + (−0.535 − 0.844i)5-s + 0.185i·7-s + (−0.536 + 0.843i)9-s + 0.881·11-s + 1.18·13-s + (−0.482 + 0.875i)15-s + 0.566·17-s + 0.508i·19-s + (0.162 − 0.0893i)21-s + 0.703·23-s + (−0.426 + 0.904i)25-s + (0.997 + 0.0644i)27-s + 1.63·29-s − 1.31·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.580289836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580289836\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.44 + 2.62i)T \) |
| 5 | \( 1 + (2.67 + 4.22i)T \) |
| good | 7 | \( 1 - 1.30iT - 49T^{2} \) |
| 11 | \( 1 - 9.69T + 121T^{2} \) |
| 13 | \( 1 - 15.3T + 169T^{2} \) |
| 17 | \( 1 - 9.62T + 289T^{2} \) |
| 19 | \( 1 - 9.66iT - 361T^{2} \) |
| 23 | \( 1 - 16.1T + 529T^{2} \) |
| 29 | \( 1 - 47.2T + 841T^{2} \) |
| 31 | \( 1 + 40.6T + 961T^{2} \) |
| 37 | \( 1 - 43.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 4.20iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 48.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 9.87iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 70.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 92.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 25.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 41.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 17.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 92.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 54.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 102. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 97.2iT - 9.40e3T^{2} \) |
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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
−9.485084187606103408329444746241, −8.614194737506666285038054323009, −8.077940507088444478600567643406, −7.10111507858478188088210457639, −6.21251874595076548491269114519, −5.44891860893960208285547526007, −4.39436989587703382636289274012, −3.29106888525105149924393731577, −1.61727446814253003482257129305, −0.790485604221303604112693221610,
0.937502510016542363141999168937, 2.95449017086996941476170488150, 3.76351281866970715815049654790, 4.51720657258502374228124262167, 5.78095624874665630965160333570, 6.50745998965199706767769745345, 7.32514281910435270043501506987, 8.536805071205170435906258906339, 9.183790689716850251730283065096, 10.24755600867427759620389872091
