L(s) = 1 | − 2.82i·3-s − 5.00·9-s − 2.82i·11-s + 6·17-s + 8.48i·19-s + 5·25-s + 5.65i·27-s − 8.00·33-s − 6·41-s + 8.48i·43-s − 7·49-s − 16.9i·51-s + 24·57-s − 14.1i·59-s + 8.48i·67-s + ⋯ |
L(s) = 1 | − 1.63i·3-s − 1.66·9-s − 0.852i·11-s + 1.45·17-s + 1.94i·19-s + 25-s + 1.08i·27-s − 1.39·33-s − 0.937·41-s + 1.29i·43-s − 49-s − 2.37i·51-s + 3.17·57-s − 1.84i·59-s + 1.03i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.748881 - 0.748881i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.748881 - 0.748881i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 2.82iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 8.48iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 8.48iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−12.92089563107896110519093354136, −12.29818754311778415065996876260, −11.36515988012176547866224364384, −10.00565361958677993380557336950, −8.380648284949726943323366842492, −7.76869240934951829307603994212, −6.51950332514357795625674257850, −5.58550133389113302414045527469, −3.23595485619640913873366119897, −1.38179389167272920808910458982,
3.08634036933586957896715177110, 4.48789758656622716181904592878, 5.35336554189979674456025177029, 7.08971988569328979813016041294, 8.664242019433138992834480113785, 9.588916288933705161532427497691, 10.35406548918393499584234425732, 11.29975411628232970704714828288, 12.44943601301639991206556802577, 13.82686932364188423095463472958