| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 13-s − 15-s + 4·17-s + 2·19-s − 2·21-s + 6·23-s + 25-s − 27-s + 4·31-s + 2·35-s − 2·37-s + 39-s − 6·41-s + 4·43-s + 45-s + 4·47-s − 3·49-s − 4·51-s − 10·53-s − 2·57-s + 8·59-s + 6·61-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.970·17-s + 0.458·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s + 0.718·31-s + 0.338·35-s − 0.328·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 0.583·47-s − 3/7·49-s − 0.560·51-s − 1.37·53-s − 0.264·57-s + 1.04·59-s + 0.768·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.167894624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.167894624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.991785692066881989630658961876, −7.29217966111429915708361251067, −6.64423311018434721084215502611, −5.81027128270552578570912853476, −5.13991218018497787389765808906, −4.73679541195531233051277867723, −3.63042363848013260991655415374, −2.73882448578956584605360995055, −1.66284645711473060254756457206, −0.845916286672162984679868780104,
0.845916286672162984679868780104, 1.66284645711473060254756457206, 2.73882448578956584605360995055, 3.63042363848013260991655415374, 4.73679541195531233051277867723, 5.13991218018497787389765808906, 5.81027128270552578570912853476, 6.64423311018434721084215502611, 7.29217966111429915708361251067, 7.991785692066881989630658961876
