| L(s) = 1 | − 4·7-s + 5·13-s − 7·19-s − 5·25-s − 7·31-s − 10·37-s − 13·43-s + 9·49-s − 13·61-s + 11·67-s + 17·73-s + 17·79-s − 20·91-s − 19·97-s − 7·103-s + 17·109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.38·13-s − 1.60·19-s − 25-s − 1.25·31-s − 1.64·37-s − 1.98·43-s + 9/7·49-s − 1.66·61-s + 1.34·67-s + 1.98·73-s + 1.91·79-s − 2.09·91-s − 1.92·97-s − 0.689·103-s + 1.62·109-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 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 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 13 T + p T^{2} \) | 1.43.n |
| 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 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 17 T + p T^{2} \) | 1.73.ar |
| 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
−9.560406478930196694623363606584, −8.836919150676161187346044032312, −8.048301102429563301256457101837, −6.72127363228842394258218909510, −6.37963360432194233363479609739, −5.37914816586411840428812893207, −3.91733393921845546677555767878, −3.37170556741266467753859084934, −1.91268222393441456759898930404, 0,
1.91268222393441456759898930404, 3.37170556741266467753859084934, 3.91733393921845546677555767878, 5.37914816586411840428812893207, 6.37963360432194233363479609739, 6.72127363228842394258218909510, 8.048301102429563301256457101837, 8.836919150676161187346044032312, 9.560406478930196694623363606584
