| L(s) = 1 | − 7-s − 7·13-s − 7·19-s + 11·31-s − 10·37-s − 13·43-s − 6·49-s − 61-s + 11·67-s − 10·73-s − 4·79-s + 7·91-s − 19·97-s + 20·103-s + 17·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.94·13-s − 1.60·19-s + 1.97·31-s − 1.64·37-s − 1.98·43-s − 6/7·49-s − 0.128·61-s + 1.34·67-s − 1.17·73-s − 0.450·79-s + 0.733·91-s − 1.92·97-s + 1.97·103-s + 1.62·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 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 | \( 1 + 10 T + p T^{2} \) | 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 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 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
−10.01185297753129750762994211064, −8.810318287138666656620561570522, −8.068267125649044096169699508794, −7.00418644170583896469357710759, −6.41391381288026324076920371033, −5.13958532772580839484877653078, −4.42116330171299264580891094162, −3.08601112889137016020769949109, −2.05095977025298622950154668732, 0,
2.05095977025298622950154668732, 3.08601112889137016020769949109, 4.42116330171299264580891094162, 5.13958532772580839484877653078, 6.41391381288026324076920371033, 7.00418644170583896469357710759, 8.068267125649044096169699508794, 8.810318287138666656620561570522, 10.01185297753129750762994211064
