| L(s) = 1 | + 3-s − 5-s − 2·7-s − 2·9-s + 5·13-s − 15-s + 17-s − 3·19-s − 2·21-s + 2·23-s + 25-s − 5·27-s − 3·29-s + 7·31-s + 2·35-s − 8·37-s + 5·39-s − 2·41-s − 10·43-s + 2·45-s − 3·47-s − 3·49-s + 51-s − 53-s − 3·57-s + 59-s + 11·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 1.38·13-s − 0.258·15-s + 0.242·17-s − 0.688·19-s − 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.962·27-s − 0.557·29-s + 1.25·31-s + 0.338·35-s − 1.31·37-s + 0.800·39-s − 0.312·41-s − 1.52·43-s + 0.298·45-s − 0.437·47-s − 3/7·49-s + 0.140·51-s − 0.137·53-s − 0.397·57-s + 0.130·59-s + 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−8.643545488035701999947666461119, −7.87995894220606401167740582559, −6.87656053290699676723240345966, −6.24469883627112638659209119691, −5.43041561451280541566524009805, −4.28076411629263570061152179260, −3.41223872086910546489705288912, −2.93838509703017962383403876391, −1.57442252944188894255343632843, 0,
1.57442252944188894255343632843, 2.93838509703017962383403876391, 3.41223872086910546489705288912, 4.28076411629263570061152179260, 5.43041561451280541566524009805, 6.24469883627112638659209119691, 6.87656053290699676723240345966, 7.87995894220606401167740582559, 8.643545488035701999947666461119
