| L(s) = 1 | + 5·7-s + 5·13-s − 7·19-s − 5·25-s + 11·31-s − 37-s + 5·43-s + 18·49-s + 14·61-s − 16·67-s − 10·73-s + 17·79-s + 25·91-s − 19·97-s + 20·103-s − 19·109-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 1.38·13-s − 1.60·19-s − 25-s + 1.97·31-s − 0.164·37-s + 0.762·43-s + 18/7·49-s + 1.79·61-s − 1.95·67-s − 1.17·73-s + 1.91·79-s + 2.62·91-s − 1.92·97-s + 1.97·103-s − 1.81·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022296538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022296538\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 19 T + p T^{2} \) | 1.97.t |
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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
−10.23722071309935247727298348837, −8.885100087967757116394169801081, −8.333992987459830023944164057176, −7.76727946355254846029070703935, −6.52139454380345394631337752802, −5.69511621612207163474256411692, −4.61166736570082881718643515307, −3.96932792582439025171338750491, −2.32578529085747485075021286451, −1.27516089803264212584639399656,
1.27516089803264212584639399656, 2.32578529085747485075021286451, 3.96932792582439025171338750491, 4.61166736570082881718643515307, 5.69511621612207163474256411692, 6.52139454380345394631337752802, 7.76727946355254846029070703935, 8.333992987459830023944164057176, 8.885100087967757116394169801081, 10.23722071309935247727298348837
