| L(s) = 1 | − 3-s + 5-s + 3·7-s + 9-s + 11-s − 6·13-s − 15-s + 17-s + 19-s − 3·21-s − 6·23-s + 25-s − 27-s − 5·29-s + 2·31-s − 33-s + 3·35-s − 11·37-s + 6·39-s − 9·41-s + 45-s − 5·47-s + 2·49-s − 51-s − 3·53-s + 55-s − 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.258·15-s + 0.242·17-s + 0.229·19-s − 0.654·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 0.359·31-s − 0.174·33-s + 0.507·35-s − 1.80·37-s + 0.960·39-s − 1.40·41-s + 0.149·45-s − 0.729·47-s + 2/7·49-s − 0.140·51-s − 0.412·53-s + 0.134·55-s − 0.132·57-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 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.982393337556492219002104333941, −7.31706918834567317308530137799, −6.64453879555839533878075930448, −5.66319503664164599650087198443, −5.09754573678134694976380027318, −4.54051358927375399882161974333, −3.46329854129581577195971176587, −2.17492015147028769328847304195, −1.55599823806811570026361624687, 0,
1.55599823806811570026361624687, 2.17492015147028769328847304195, 3.46329854129581577195971176587, 4.54051358927375399882161974333, 5.09754573678134694976380027318, 5.66319503664164599650087198443, 6.64453879555839533878075930448, 7.31706918834567317308530137799, 7.982393337556492219002104333941
