| L(s) = 1 | − 2·3-s − 7-s + 9-s + 11-s + 4·13-s + 6·19-s + 2·21-s + 3·23-s + 4·27-s + 3·29-s − 2·33-s + 9·37-s − 8·39-s + 2·41-s − 9·43-s − 6·47-s + 49-s + 6·53-s − 12·57-s + 8·59-s + 10·61-s − 63-s + 67-s − 6·69-s + 7·71-s + 2·73-s − 77-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1.37·19-s + 0.436·21-s + 0.625·23-s + 0.769·27-s + 0.557·29-s − 0.348·33-s + 1.47·37-s − 1.28·39-s + 0.312·41-s − 1.37·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.58·57-s + 1.04·59-s + 1.28·61-s − 0.125·63-s + 0.122·67-s − 0.722·69-s + 0.830·71-s + 0.234·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.478668636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.478668636\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−16.53760880658075, −16.04630662631086, −15.57198420146109, −14.77296736919652, −14.20775772005557, −13.41693483604715, −13.12184215472429, −12.29096283406776, −11.80612206250579, −11.17207649344246, −11.02425630669693, −9.946612093231680, −9.744635257598048, −8.759920874634058, −8.289161035935557, −7.381054176627832, −6.638950538079668, −6.307287892763881, −5.481199525736786, −5.122277601937348, −4.163446163403056, −3.429338120823396, −2.650724120402114, −1.301806638509399, −0.6855720107106096,
0.6855720107106096, 1.301806638509399, 2.650724120402114, 3.429338120823396, 4.163446163403056, 5.122277601937348, 5.481199525736786, 6.307287892763881, 6.638950538079668, 7.381054176627832, 8.289161035935557, 8.759920874634058, 9.744635257598048, 9.946612093231680, 11.02425630669693, 11.17207649344246, 11.80612206250579, 12.29096283406776, 13.12184215472429, 13.41693483604715, 14.20775772005557, 14.77296736919652, 15.57198420146109, 16.04630662631086, 16.53760880658075
