| L(s) = 1 | − 2i·7-s − 13.8·13-s − 27.7i·19-s − 25·25-s + 46i·31-s + 69.2·37-s + 83.1i·43-s + 45·49-s − 96.9·61-s − 55.4i·67-s − 46·73-s + 142i·79-s + 27.7i·91-s + 2·97-s + 194i·103-s + ⋯ |
| L(s) = 1 | − 0.285i·7-s − 1.06·13-s − 1.45i·19-s − 25-s + 1.48i·31-s + 1.87·37-s + 1.93i·43-s + 0.918·49-s − 1.59·61-s − 0.827i·67-s − 0.630·73-s + 1.79i·79-s + 0.304i·91-s + 0.0206·97-s + 1.88i·103-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.058949275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058949275\) |
| \(L(2)\) |
|
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 | \( 1 + 25T^{2} \) |
| 7 | \( 1 + 2iT - 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 13.8T + 169T^{2} \) |
| 17 | \( 1 + 289T^{2} \) |
| 19 | \( 1 + 27.7iT - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 + 841T^{2} \) |
| 31 | \( 1 - 46iT - 961T^{2} \) |
| 37 | \( 1 - 69.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 1.68e3T^{2} \) |
| 43 | \( 1 - 83.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 96.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 55.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 46T + 5.32e3T^{2} \) |
| 79 | \( 1 - 142iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−9.199949878825242069011807091443, −8.140461060635000237395919531927, −7.47837940255453052231973262845, −6.78046821677098106199552207326, −5.92868798608084289000154036913, −4.89221071113872100082874327917, −4.37128960434190520622242616431, −3.13138767326344879571441330650, −2.33901227208363476252769218598, −0.993176949362384907866768054133,
0.28621413222160152279031065412, 1.80454615586364721233489680523, 2.64270932463618340768017923252, 3.81463205791533725876342778387, 4.54352273170685158708011796442, 5.68998749069041600488985731644, 6.05007662987097755302700461408, 7.32475472948798387275439228136, 7.74024141772456527662060958424, 8.611278119119510307171859422581
