| 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 + (−14.9 + 0.944i)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 + (−0.998 + 0.0629i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3721184697\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3721184697\) |
| \(L(2)\) |
|
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
−10.36121517993872389557301455918, −9.719784317724812917351655912959, −8.888132686468886069558336067683, −7.898008865725633410423929976121, −6.75703585296356589673768050757, −5.88489224075791076828711969059, −5.39303862572048615961761868011, −4.03640988216295684215368536666, −3.24741917902129553703422162705, −1.95622438625625316822402021948,
0.12198170398276750846285078024, 1.40621427088226559003378120278, 2.38578824649227916185822268663, 3.98155135754578250076945825741, 5.20726239595107223907604341590, 5.76565846522574365918561902402, 6.65819119958662847733946113441, 7.66033241617850672179613432988, 8.397539512786408070665395679833, 9.163232588592487172928582956829
