| L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s + 5·11-s + 4·13-s − 2·15-s − 5·17-s − 2·19-s − 4·21-s − 4·23-s − 25-s − 27-s − 29-s + 5·31-s − 5·33-s + 8·35-s + 7·37-s − 4·39-s − 41-s + 7·43-s + 2·45-s − 7·47-s + 9·49-s + 5·51-s + 14·53-s + 10·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.50·11-s + 1.10·13-s − 0.516·15-s − 1.21·17-s − 0.458·19-s − 0.872·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.185·29-s + 0.898·31-s − 0.870·33-s + 1.35·35-s + 1.15·37-s − 0.640·39-s − 0.156·41-s + 1.06·43-s + 0.298·45-s − 1.02·47-s + 9/7·49-s + 0.700·51-s + 1.92·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.957909048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.957909048\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.966986265557926570197432680178, −6.97251470849582837880103346059, −6.19642745389320963271087195581, −6.03595879291802229699044108462, −5.03096503540440472438701112935, −4.31360004267648799529321687464, −3.86602836040786568101247257090, −2.31215148686502236781007605524, −1.69942039841040252834657793640, −0.967525573606667506502291802924,
0.967525573606667506502291802924, 1.69942039841040252834657793640, 2.31215148686502236781007605524, 3.86602836040786568101247257090, 4.31360004267648799529321687464, 5.03096503540440472438701112935, 6.03595879291802229699044108462, 6.19642745389320963271087195581, 6.97251470849582837880103346059, 7.966986265557926570197432680178
