| L(s) = 1 | − 5-s + 4·7-s − 3·9-s − 4·11-s − 13-s + 2·17-s + 4·19-s + 25-s − 6·29-s + 4·31-s − 4·35-s − 6·37-s + 10·41-s + 8·43-s + 3·45-s − 12·47-s + 9·49-s + 10·53-s + 4·55-s + 12·59-s − 6·61-s − 12·63-s + 65-s + 4·67-s + 4·71-s + 2·73-s − 16·77-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s − 1.20·11-s − 0.277·13-s + 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.676·35-s − 0.986·37-s + 1.56·41-s + 1.21·43-s + 0.447·45-s − 1.75·47-s + 9/7·49-s + 1.37·53-s + 0.539·55-s + 1.56·59-s − 0.768·61-s − 1.51·63-s + 0.124·65-s + 0.488·67-s + 0.474·71-s + 0.234·73-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683579269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683579269\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.309244290450919804794333553753, −7.67329570964766411027870248746, −7.35680121007472984855602818238, −5.99129441215593003127101172838, −5.27287808314464585859698784451, −4.91090016218149243406995408016, −3.82516275228838992667910123065, −2.86476400565642674836506023363, −2.05091823105766722106475095759, −0.73077687596790736687239317964,
0.73077687596790736687239317964, 2.05091823105766722106475095759, 2.86476400565642674836506023363, 3.82516275228838992667910123065, 4.91090016218149243406995408016, 5.27287808314464585859698784451, 5.99129441215593003127101172838, 7.35680121007472984855602818238, 7.67329570964766411027870248746, 8.309244290450919804794333553753
