| L(s) = 1 | + 2·5-s + 2·7-s + 3·11-s − 13-s + 3·17-s − 2·19-s + 3·23-s − 25-s + 8·29-s + 31-s + 4·35-s − 8·37-s − 41-s + 43-s − 3·49-s + 53-s + 6·55-s − 4·59-s − 2·65-s + 7·67-s − 6·71-s + 4·73-s + 6·77-s + 8·79-s − 83-s + 6·85-s + 14·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 0.904·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.625·23-s − 1/5·25-s + 1.48·29-s + 0.179·31-s + 0.676·35-s − 1.31·37-s − 0.156·41-s + 0.152·43-s − 3/7·49-s + 0.137·53-s + 0.809·55-s − 0.520·59-s − 0.248·65-s + 0.855·67-s − 0.712·71-s + 0.468·73-s + 0.683·77-s + 0.900·79-s − 0.109·83-s + 0.650·85-s + 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.350385461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.350385461\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 43 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−9.454965000395228915733534541285, −8.701652992186140299268037829714, −7.941847924140247265760484564724, −6.91554887728147456894540367119, −6.22176148737971503762693697714, −5.29287508548958766419413587760, −4.56048735025800706814497570912, −3.39033787312621070244809504024, −2.16937736085584771329388457103, −1.21293335275974913766887560152,
1.21293335275974913766887560152, 2.16937736085584771329388457103, 3.39033787312621070244809504024, 4.56048735025800706814497570912, 5.29287508548958766419413587760, 6.22176148737971503762693697714, 6.91554887728147456894540367119, 7.941847924140247265760484564724, 8.701652992186140299268037829714, 9.454965000395228915733534541285
