| L(s) = 1 | + 2·5-s + 4·7-s + 4·11-s + 6·13-s + 2·17-s + 4·19-s − 6·23-s − 25-s − 2·29-s + 6·31-s + 8·35-s − 8·37-s + 10·41-s + 8·43-s + 6·47-s + 9·49-s − 2·53-s + 8·55-s + 10·61-s + 12·65-s − 12·67-s − 2·71-s + 2·73-s + 16·77-s + 2·79-s − 4·83-s + 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s − 1.25·23-s − 1/5·25-s − 0.371·29-s + 1.07·31-s + 1.35·35-s − 1.31·37-s + 1.56·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s − 0.274·53-s + 1.07·55-s + 1.28·61-s + 1.48·65-s − 1.46·67-s − 0.237·71-s + 0.234·73-s + 1.82·77-s + 0.225·79-s − 0.439·83-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.204405031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.204405031\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 563 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.46264954890983, −12.03745756234850, −11.77227318331003, −11.18621043510555, −10.89729933950594, −10.40612984449469, −9.817324200750576, −9.418038659952577, −8.936465174217634, −8.489575241087667, −8.087801517475183, −7.566843405349834, −7.113075269212361, −6.321671601351317, −6.060515375532176, −5.583444996331116, −5.244694066589435, −4.389629649755221, −4.050037998272030, −3.660063047052989, −2.833498208205387, −2.185754987996192, −1.549130950813123, −1.319184027963317, −0.7525000604606973,
0.7525000604606973, 1.319184027963317, 1.549130950813123, 2.185754987996192, 2.833498208205387, 3.660063047052989, 4.050037998272030, 4.389629649755221, 5.244694066589435, 5.583444996331116, 6.060515375532176, 6.321671601351317, 7.113075269212361, 7.566843405349834, 8.087801517475183, 8.489575241087667, 8.936465174217634, 9.418038659952577, 9.817324200750576, 10.40612984449469, 10.89729933950594, 11.18621043510555, 11.77227318331003, 12.03745756234850, 12.46264954890983
