| L(s) = 1 | + 5-s + 7-s − 11-s + 4·13-s − 7·17-s − 5·19-s + 9·23-s + 25-s + 9·29-s − 2·31-s + 35-s − 2·37-s + 12·41-s + 9·43-s − 2·47-s + 49-s + 5·53-s − 55-s − 59-s + 7·61-s + 4·65-s + 2·67-s − 4·71-s + 14·73-s − 77-s − 12·79-s + 5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.301·11-s + 1.10·13-s − 1.69·17-s − 1.14·19-s + 1.87·23-s + 1/5·25-s + 1.67·29-s − 0.359·31-s + 0.169·35-s − 0.328·37-s + 1.87·41-s + 1.37·43-s − 0.291·47-s + 1/7·49-s + 0.686·53-s − 0.134·55-s − 0.130·59-s + 0.896·61-s + 0.496·65-s + 0.244·67-s − 0.474·71-s + 1.63·73-s − 0.113·77-s − 1.35·79-s + 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.796093134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.796093134\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.10278984400708, −12.70871333543543, −12.05958041492373, −11.34199654917277, −11.06995191121434, −10.64834302129241, −10.45194083417571, −9.564654581588221, −9.108833942885472, −8.669259782162783, −8.496496455195359, −7.804335013208885, −7.070860820606018, −6.811399226044048, −6.177676970815978, −5.898241955046681, −5.126627987927550, −4.653543193532074, −4.264869878912043, −3.656564035967594, −2.841830258839541, −2.429092700957162, −1.912918498554688, −1.051522369417850, −0.6152631406874672,
0.6152631406874672, 1.051522369417850, 1.912918498554688, 2.429092700957162, 2.841830258839541, 3.656564035967594, 4.264869878912043, 4.653543193532074, 5.126627987927550, 5.898241955046681, 6.177676970815978, 6.811399226044048, 7.070860820606018, 7.804335013208885, 8.496496455195359, 8.669259782162783, 9.108833942885472, 9.564654581588221, 10.45194083417571, 10.64834302129241, 11.06995191121434, 11.34199654917277, 12.05958041492373, 12.70871333543543, 13.10278984400708
