| L(s) = 1 | + 3·3-s + 6·9-s + 11-s + 6·13-s + 4·17-s − 6·19-s + 3·23-s + 9·27-s − 4·29-s + 9·31-s + 3·33-s − 7·37-s + 18·39-s − 2·41-s + 6·43-s + 12·47-s − 7·49-s + 12·51-s − 2·53-s − 18·57-s − 9·59-s + 8·61-s − 15·67-s + 9·69-s + 3·71-s + 6·73-s + 6·79-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s + 0.301·11-s + 1.66·13-s + 0.970·17-s − 1.37·19-s + 0.625·23-s + 1.73·27-s − 0.742·29-s + 1.61·31-s + 0.522·33-s − 1.15·37-s + 2.88·39-s − 0.312·41-s + 0.914·43-s + 1.75·47-s − 49-s + 1.68·51-s − 0.274·53-s − 2.38·57-s − 1.17·59-s + 1.02·61-s − 1.83·67-s + 1.08·69-s + 0.356·71-s + 0.702·73-s + 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.957705785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.957705785\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 15 T + p T^{2} \) | 1.67.p |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−7.949505765250478195925162885680, −7.26458536740216410658564566936, −6.47701329068176971306649348794, −5.85511364088302078392141301240, −4.69906586254426622366963327746, −3.94356042955627036968641023012, −3.46607261358809807043008886291, −2.74519612173240814567128325463, −1.84136936728960182732366405957, −1.08235378457247380255367379688,
1.08235378457247380255367379688, 1.84136936728960182732366405957, 2.74519612173240814567128325463, 3.46607261358809807043008886291, 3.94356042955627036968641023012, 4.69906586254426622366963327746, 5.85511364088302078392141301240, 6.47701329068176971306649348794, 7.26458536740216410658564566936, 7.949505765250478195925162885680
