| L(s) = 1 | − 2·7-s + 3·11-s + 13-s − 6·17-s + 2·19-s + 3·23-s + 6·29-s − 4·31-s + 7·37-s − 2·43-s − 3·47-s − 3·49-s + 6·53-s + 15·59-s + 5·61-s − 2·67-s + 9·71-s − 2·73-s − 6·77-s − 10·79-s + 12·83-s + 18·89-s − 2·91-s − 17·97-s − 6·101-s + 4·103-s + 3·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.904·11-s + 0.277·13-s − 1.45·17-s + 0.458·19-s + 0.625·23-s + 1.11·29-s − 0.718·31-s + 1.15·37-s − 0.304·43-s − 0.437·47-s − 3/7·49-s + 0.824·53-s + 1.95·59-s + 0.640·61-s − 0.244·67-s + 1.06·71-s − 0.234·73-s − 0.683·77-s − 1.12·79-s + 1.31·83-s + 1.90·89-s − 0.209·91-s − 1.72·97-s − 0.597·101-s + 0.394·103-s + 0.290·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.660515670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.660515670\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−8.902577249157073414380558168260, −8.231330619005234080616622430198, −7.05525826539387688225534951207, −6.65878470813573638073037977066, −5.90323147283558325005985769616, −4.85305843094606377641117698092, −4.02134541433755646714847258667, −3.19195339335968397918746588984, −2.15381345080132225520788663485, −0.807734145259017760584365766093,
0.807734145259017760584365766093, 2.15381345080132225520788663485, 3.19195339335968397918746588984, 4.02134541433755646714847258667, 4.85305843094606377641117698092, 5.90323147283558325005985769616, 6.65878470813573638073037977066, 7.05525826539387688225534951207, 8.231330619005234080616622430198, 8.902577249157073414380558168260
