| L(s) = 1 | + 3-s − 2·5-s − 2·7-s − 2·9-s − 4·11-s + 4·13-s − 2·15-s − 7·17-s − 3·19-s − 2·21-s − 25-s − 5·27-s − 4·29-s − 6·31-s − 4·33-s + 4·35-s + 2·37-s + 4·39-s + 6·41-s − 5·43-s + 4·45-s + 10·47-s − 3·49-s − 7·51-s + 8·55-s − 3·57-s + 5·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.755·7-s − 2/3·9-s − 1.20·11-s + 1.10·13-s − 0.516·15-s − 1.69·17-s − 0.688·19-s − 0.436·21-s − 1/5·25-s − 0.962·27-s − 0.742·29-s − 1.07·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s + 0.640·39-s + 0.937·41-s − 0.762·43-s + 0.596·45-s + 1.45·47-s − 3/7·49-s − 0.980·51-s + 1.07·55-s − 0.397·57-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5583094182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5583094182\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.82360946212207186754799580348, −7.29333166571947583680191588719, −6.35502301594063207434041650553, −5.87018085067170705593877591911, −4.92837701158713264345699921336, −3.97238880261002908995114838571, −3.58323926950548706842187913180, −2.67894680171947318095506928823, −2.04164282340789982393768170948, −0.32727354925912527472258326024,
0.32727354925912527472258326024, 2.04164282340789982393768170948, 2.67894680171947318095506928823, 3.58323926950548706842187913180, 3.97238880261002908995114838571, 4.92837701158713264345699921336, 5.87018085067170705593877591911, 6.35502301594063207434041650553, 7.29333166571947583680191588719, 7.82360946212207186754799580348
