L(s) = 1 | + (−2.41 − 2.41i)5-s − 1.53i·7-s + (3.37 + 3.37i)11-s + (0.414 − 0.414i)13-s − 2.82·17-s + (0.317 − 0.317i)19-s − 5.86i·23-s + 6.65i·25-s + (3.24 − 3.24i)29-s + 7.39·31-s + (−3.69 + 3.69i)35-s + (−3.58 − 3.58i)37-s − 4i·41-s + (−1.84 − 1.84i)43-s + 7.39·47-s + ⋯ |
L(s) = 1 | + (−1.07 − 1.07i)5-s − 0.578i·7-s + (1.01 + 1.01i)11-s + (0.114 − 0.114i)13-s − 0.685·17-s + (0.0727 − 0.0727i)19-s − 1.22i·23-s + 1.33i·25-s + (0.602 − 0.602i)29-s + 1.32·31-s + (−0.624 + 0.624i)35-s + (−0.589 − 0.589i)37-s − 0.624i·41-s + (−0.281 − 0.281i)43-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.162900645\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.162900645\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.41 + 2.41i)T + 5iT^{2} \) |
| 7 | \( 1 + 1.53iT - 7T^{2} \) |
| 11 | \( 1 + (-3.37 - 3.37i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.414 + 0.414i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + (-0.317 + 0.317i)T - 19iT^{2} \) |
| 23 | \( 1 + 5.86iT - 23T^{2} \) |
| 29 | \( 1 + (-3.24 + 3.24i)T - 29iT^{2} \) |
| 31 | \( 1 - 7.39T + 31T^{2} \) |
| 37 | \( 1 + (3.58 + 3.58i)T + 37iT^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + (1.84 + 1.84i)T + 43iT^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + (-5.24 - 5.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.84 - 1.84i)T + 59iT^{2} \) |
| 61 | \( 1 + (9.24 - 9.24i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.07 - 7.07i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.9iT - 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 6.12T + 79T^{2} \) |
| 83 | \( 1 + (-2.48 + 2.48i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.828iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.057089312386634577832817772927, −7.34005422444555565566636421666, −6.78271518965496887067031199803, −5.87877332386198114727329611294, −4.61113150178944075821419606803, −4.44613269311099368338738908283, −3.79947150252392346967911430929, −2.50987456814370423180048380665, −1.28688801243571720454641933233, −0.38280853569707423071652775847,
1.15868430742329960546886703553, 2.53925616891296166537170518390, 3.32565087695571024815320955614, 3.85108961508792794142348350679, 4.79784346011652978234276523661, 5.86044182627609456022079909138, 6.52428669895417413634555793946, 7.03782807752930849214312981330, 7.902801479486522916507383249227, 8.548696307787177407049321673746