| L(s) = 1 | + 2·7-s − 11-s − 6·13-s − 2·19-s − 8·23-s − 5·25-s − 2·29-s + 4·31-s − 2·37-s + 10·41-s − 6·43-s − 4·47-s − 3·49-s + 4·53-s + 4·59-s + 2·61-s − 8·67-s − 12·71-s + 2·73-s − 2·77-s − 14·79-s − 4·83-s − 12·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s − 1.66·13-s − 0.458·19-s − 1.66·23-s − 25-s − 0.371·29-s + 0.718·31-s − 0.328·37-s + 1.56·41-s − 0.914·43-s − 0.583·47-s − 3/7·49-s + 0.549·53-s + 0.520·59-s + 0.256·61-s − 0.977·67-s − 1.42·71-s + 0.234·73-s − 0.227·77-s − 1.57·79-s − 0.439·83-s − 1.25·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.30035073768886, −13.02530088004361, −12.21131084958655, −12.13096114918851, −11.57363165169232, −11.21000692016095, −10.45058759095247, −10.19550132766251, −9.652317218742348, −9.396730947304498, −8.464407743236974, −8.267614751039379, −7.729212426323570, −7.332423453487955, −6.865723460122552, −6.061030393882544, −5.784841357793176, −5.138900147320657, −4.656929087772220, −4.212966452667671, −3.728997147959773, −2.787600804551188, −2.434762397206341, −1.854351060320906, −1.289070033546151, 0, 0,
1.289070033546151, 1.854351060320906, 2.434762397206341, 2.787600804551188, 3.728997147959773, 4.212966452667671, 4.656929087772220, 5.138900147320657, 5.784841357793176, 6.061030393882544, 6.865723460122552, 7.332423453487955, 7.729212426323570, 8.267614751039379, 8.464407743236974, 9.396730947304498, 9.652317218742348, 10.19550132766251, 10.45058759095247, 11.21000692016095, 11.57363165169232, 12.13096114918851, 12.21131084958655, 13.02530088004361, 13.30035073768886
