| L(s) = 1 | − 3-s − 5-s − 3·7-s − 2·9-s + 11-s + 6·13-s + 15-s − 7·17-s + 5·19-s + 3·21-s + 6·23-s + 25-s + 5·27-s − 5·29-s + 3·31-s − 33-s + 3·35-s − 3·37-s − 6·39-s + 2·41-s + 4·43-s + 2·45-s + 2·47-s + 2·49-s + 7·51-s + 53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.13·7-s − 2/3·9-s + 0.301·11-s + 1.66·13-s + 0.258·15-s − 1.69·17-s + 1.14·19-s + 0.654·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s − 0.928·29-s + 0.538·31-s − 0.174·33-s + 0.507·35-s − 0.493·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 0.291·47-s + 2/7·49-s + 0.980·51-s + 0.137·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−8.365818347054701609982302093181, −7.26291614725309666178558100598, −6.58619282920726274072965958167, −6.08123432043213436496165948563, −5.30093363546630192553859846323, −4.27131168297609269556807675426, −3.45536279391829989915086256097, −2.76395446960541053188187699258, −1.17453234798128096107230740542, 0,
1.17453234798128096107230740542, 2.76395446960541053188187699258, 3.45536279391829989915086256097, 4.27131168297609269556807675426, 5.30093363546630192553859846323, 6.08123432043213436496165948563, 6.58619282920726274072965958167, 7.26291614725309666178558100598, 8.365818347054701609982302093181
