| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 3·11-s − 2·13-s − 15-s − 17-s − 19-s + 21-s + 2·23-s + 25-s + 27-s + 5·29-s − 6·31-s − 3·33-s − 35-s + 37-s − 2·39-s + 5·41-s − 4·43-s − 45-s − 13·47-s − 6·49-s − 51-s + 11·53-s + 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s − 0.258·15-s − 0.242·17-s − 0.229·19-s + 0.218·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s + 0.928·29-s − 1.07·31-s − 0.522·33-s − 0.169·35-s + 0.164·37-s − 0.320·39-s + 0.780·41-s − 0.609·43-s − 0.149·45-s − 1.89·47-s − 6/7·49-s − 0.140·51-s + 1.51·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.083517553920619891559579827311, −7.44983884473116758311550028817, −6.83157561914238722165659910331, −5.77984598389767781672517790666, −4.88515854832004268144168999284, −4.35113750685884003274653216809, −3.25648294685258233934926988556, −2.59118279103198049381730248293, −1.53973951741459012301822936869, 0,
1.53973951741459012301822936869, 2.59118279103198049381730248293, 3.25648294685258233934926988556, 4.35113750685884003274653216809, 4.88515854832004268144168999284, 5.77984598389767781672517790666, 6.83157561914238722165659910331, 7.44983884473116758311550028817, 8.083517553920619891559579827311
