| L(s) = 1 | + 5-s + 13-s − 2·17-s + 3·19-s + 6·23-s + 25-s − 6·29-s − 7·31-s − 37-s − 4·41-s − 11·43-s + 6·47-s − 4·53-s + 14·59-s + 6·61-s + 65-s + 3·67-s + 7·73-s − 9·79-s − 16·83-s − 2·85-s + 2·89-s + 3·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.277·13-s − 0.485·17-s + 0.688·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s − 1.25·31-s − 0.164·37-s − 0.624·41-s − 1.67·43-s + 0.875·47-s − 0.549·53-s + 1.82·59-s + 0.768·61-s + 0.124·65-s + 0.366·67-s + 0.819·73-s − 1.01·79-s − 1.75·83-s − 0.216·85-s + 0.211·89-s + 0.307·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.203838370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.203838370\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.43780175585885, −12.98570537175805, −12.59607159535074, −11.89070732326315, −11.44157004539641, −11.01368211013368, −10.61302192616719, −9.964760522783601, −9.450838229942936, −9.213204658776053, −8.385592696176897, −8.318836824445549, −7.295386293176914, −7.030855060751498, −6.643956893029723, −5.805377730664493, −5.378614672155324, −5.095325830444615, −4.256191975601684, −3.705559145081336, −3.172226422862643, −2.535483535350225, −1.817464534017646, −1.333858271745248, −0.4408227624165588,
0.4408227624165588, 1.333858271745248, 1.817464534017646, 2.535483535350225, 3.172226422862643, 3.705559145081336, 4.256191975601684, 5.095325830444615, 5.378614672155324, 5.805377730664493, 6.643956893029723, 7.030855060751498, 7.295386293176914, 8.318836824445549, 8.385592696176897, 9.213204658776053, 9.450838229942936, 9.964760522783601, 10.61302192616719, 11.01368211013368, 11.44157004539641, 11.89070732326315, 12.59607159535074, 12.98570537175805, 13.43780175585885
