| L(s) = 1 | − 2·5-s − 4·7-s − 11-s − 4·13-s − 6·17-s − 2·19-s − 6·23-s − 25-s − 2·29-s − 8·31-s + 8·35-s − 6·37-s − 2·41-s + 6·43-s + 2·47-s + 9·49-s − 6·53-s + 2·55-s + 8·59-s + 12·61-s + 8·65-s − 4·67-s + 14·71-s − 10·73-s + 4·77-s − 8·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 0.301·11-s − 1.10·13-s − 1.45·17-s − 0.458·19-s − 1.25·23-s − 1/5·25-s − 0.371·29-s − 1.43·31-s + 1.35·35-s − 0.986·37-s − 0.312·41-s + 0.914·43-s + 0.291·47-s + 9/7·49-s − 0.824·53-s + 0.269·55-s + 1.04·59-s + 1.53·61-s + 0.992·65-s − 0.488·67-s + 1.66·71-s − 1.17·73-s + 0.455·77-s − 0.900·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 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
−7.14553397308808796144989472946, −6.85232483972482117507115236442, −5.96428832059467357853195786803, −5.20600307749914949530578180690, −4.10651821997780985470584323025, −3.82055440693277025750311035124, −2.76293612676071792007297692671, −2.05217675695395051603615365581, 0, 0,
2.05217675695395051603615365581, 2.76293612676071792007297692671, 3.82055440693277025750311035124, 4.10651821997780985470584323025, 5.20600307749914949530578180690, 5.96428832059467357853195786803, 6.85232483972482117507115236442, 7.14553397308808796144989472946
