| L(s) = 1 | − 2·5-s − 4·7-s + 2·13-s − 6·17-s + 4·23-s − 25-s − 29-s + 6·31-s + 8·35-s + 4·37-s − 6·41-s + 4·43-s + 9·49-s + 2·53-s − 2·59-s − 4·65-s + 12·67-s + 14·73-s + 10·79-s + 6·83-s + 12·85-s + 6·89-s − 8·91-s + 2·97-s + 10·101-s + 16·103-s − 6·107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 0.554·13-s − 1.45·17-s + 0.834·23-s − 1/5·25-s − 0.185·29-s + 1.07·31-s + 1.35·35-s + 0.657·37-s − 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.274·53-s − 0.260·59-s − 0.496·65-s + 1.46·67-s + 1.63·73-s + 1.12·79-s + 0.658·83-s + 1.30·85-s + 0.635·89-s − 0.838·91-s + 0.203·97-s + 0.995·101-s + 1.57·103-s − 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9089969718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9089969718\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−9.106225363181316348178210347859, −8.399677141320715672839793538349, −7.52043856670607732679232521565, −6.61702591112359586751690983464, −6.28137091025114493851163801786, −5.00674246954936362071730692498, −4.02625653879928883131179215294, −3.39626721380068378961545581487, −2.40120939874958364919050027678, −0.60369033234400906253951676774,
0.60369033234400906253951676774, 2.40120939874958364919050027678, 3.39626721380068378961545581487, 4.02625653879928883131179215294, 5.00674246954936362071730692498, 6.28137091025114493851163801786, 6.61702591112359586751690983464, 7.52043856670607732679232521565, 8.399677141320715672839793538349, 9.106225363181316348178210347859
