| L(s) = 1 | − 2·4-s + 4·5-s − 11-s − 5·13-s + 4·16-s + 4·17-s + 3·19-s − 8·20-s − 8·23-s + 11·25-s + 4·29-s − 31-s + 7·37-s − 4·41-s + 43-s + 2·44-s − 8·47-s + 10·52-s + 12·53-s − 4·55-s + 2·61-s − 8·64-s − 20·65-s + 3·67-s − 8·68-s + 4·71-s + 11·73-s + ⋯ |
| L(s) = 1 | − 4-s + 1.78·5-s − 0.301·11-s − 1.38·13-s + 16-s + 0.970·17-s + 0.688·19-s − 1.78·20-s − 1.66·23-s + 11/5·25-s + 0.742·29-s − 0.179·31-s + 1.15·37-s − 0.624·41-s + 0.152·43-s + 0.301·44-s − 1.16·47-s + 1.38·52-s + 1.64·53-s − 0.539·55-s + 0.256·61-s − 64-s − 2.48·65-s + 0.366·67-s − 0.970·68-s + 0.474·71-s + 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.021595669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.021595669\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.222669137408852996614785953991, −7.74677482369755447454752802918, −6.67755237620063103486628067058, −5.92434616135715426908803112020, −5.24685262606952990138384332699, −4.91771166260872014695720184214, −3.76825417834086991094417396602, −2.72469116806906951969494172293, −1.95894456188638180107141105007, −0.796888666086561984140950249880,
0.796888666086561984140950249880, 1.95894456188638180107141105007, 2.72469116806906951969494172293, 3.76825417834086991094417396602, 4.91771166260872014695720184214, 5.24685262606952990138384332699, 5.92434616135715426908803112020, 6.67755237620063103486628067058, 7.74677482369755447454752802918, 8.222669137408852996614785953991
