| L(s) = 1 | + 2-s − 4-s + 4·5-s − 3·8-s + 4·10-s + 11-s − 16-s − 17-s − 4·19-s − 4·20-s + 22-s + 6·23-s + 11·25-s + 2·29-s + 4·31-s + 5·32-s − 34-s − 8·37-s − 4·38-s − 12·40-s − 8·43-s − 44-s + 6·46-s + 8·47-s − 7·49-s + 11·50-s − 10·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.06·8-s + 1.26·10-s + 0.301·11-s − 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.894·20-s + 0.213·22-s + 1.25·23-s + 11/5·25-s + 0.371·29-s + 0.718·31-s + 0.883·32-s − 0.171·34-s − 1.31·37-s − 0.648·38-s − 1.89·40-s − 1.21·43-s − 0.150·44-s + 0.884·46-s + 1.16·47-s − 49-s + 1.55·50-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 284427 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 284427 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 3 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.00819835306575, −12.66841908277363, −12.33477938301991, −11.69300513904695, −11.00010975759145, −10.76022386116947, −10.04546500001916, −9.709390469439352, −9.424467970169874, −8.783714288793659, −8.504011327293396, −8.071767273783063, −6.845540776994032, −6.783247259924725, −6.400564014626401, −5.675048749265004, −5.332937246862219, −5.012274630986095, −4.398953968666671, −3.905035907198913, −3.117481084726872, −2.779013727981287, −2.147463666060599, −1.531336044656143, −0.9399667134962462, 0,
0.9399667134962462, 1.531336044656143, 2.147463666060599, 2.779013727981287, 3.117481084726872, 3.905035907198913, 4.398953968666671, 5.012274630986095, 5.332937246862219, 5.675048749265004, 6.400564014626401, 6.783247259924725, 6.845540776994032, 8.071767273783063, 8.504011327293396, 8.783714288793659, 9.424467970169874, 9.709390469439352, 10.04546500001916, 10.76022386116947, 11.00010975759145, 11.69300513904695, 12.33477938301991, 12.66841908277363, 13.00819835306575
