L(s) = 1 | − 0.732·3-s − 2.73·5-s − 7-s − 2.46·9-s − 1.46·11-s − 2.73·13-s + 2·15-s + 7.46·17-s + 6.19·19-s + 0.732·21-s + 8.92·23-s + 2.46·25-s + 4·27-s − 3.46·29-s + 2.53·31-s + 1.07·33-s + 2.73·35-s − 4.53·37-s + 2·39-s − 3.46·41-s + 2.53·43-s + 6.73·45-s − 1.46·47-s + 49-s − 5.46·51-s − 6·53-s + 4·55-s + ⋯ |
L(s) = 1 | − 0.422·3-s − 1.22·5-s − 0.377·7-s − 0.821·9-s − 0.441·11-s − 0.757·13-s + 0.516·15-s + 1.81·17-s + 1.42·19-s + 0.159·21-s + 1.86·23-s + 0.492·25-s + 0.769·27-s − 0.643·29-s + 0.455·31-s + 0.186·33-s + 0.461·35-s − 0.745·37-s + 0.320·39-s − 0.541·41-s + 0.386·43-s + 1.00·45-s − 0.213·47-s + 0.142·49-s − 0.765·51-s − 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8335094276\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8335094276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 - 6.19T + 19T^{2} \) |
| 23 | \( 1 - 8.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2.53T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 7.26T + 83T^{2} \) |
| 89 | \( 1 - 4.92T + 89T^{2} \) |
| 97 | \( 1 - 0.535T + 97T^{2} \) |
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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
−10.14541775262149454025394527724, −9.340826296968830460302565618334, −8.257341331299566460173286881641, −7.57924197624754994115573037650, −6.88068789654443728763285234285, −5.48294420086263651486097595276, −5.05981136329614530614496399976, −3.54918125436149410005033932467, −2.94423163878843719037928893307, −0.73134025224868805701632684494,
0.73134025224868805701632684494, 2.94423163878843719037928893307, 3.54918125436149410005033932467, 5.05981136329614530614496399976, 5.48294420086263651486097595276, 6.88068789654443728763285234285, 7.57924197624754994115573037650, 8.257341331299566460173286881641, 9.340826296968830460302565618334, 10.14541775262149454025394527724