| L(s) = 1 | − 3·5-s − 7-s − 3·9-s + 2·11-s − 13-s − 4·17-s + 19-s + 7·23-s + 4·25-s + 7·29-s + 5·31-s + 3·35-s + 4·37-s − 6·41-s − 9·43-s + 9·45-s + 7·47-s + 49-s + 11·53-s − 6·55-s − 2·61-s + 3·63-s + 3·65-s + 10·67-s + 7·73-s − 2·77-s − 79-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s − 9-s + 0.603·11-s − 0.277·13-s − 0.970·17-s + 0.229·19-s + 1.45·23-s + 4/5·25-s + 1.29·29-s + 0.898·31-s + 0.507·35-s + 0.657·37-s − 0.937·41-s − 1.37·43-s + 1.34·45-s + 1.02·47-s + 1/7·49-s + 1.51·53-s − 0.809·55-s − 0.256·61-s + 0.377·63-s + 0.372·65-s + 1.22·67-s + 0.819·73-s − 0.227·77-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9710247088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9710247088\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−9.331145111588057450204254811998, −8.613139122500934973351512042328, −8.073835726380055285731119779302, −6.98310044294046265095518743449, −6.50563998187962013416614283559, −5.23123780073428658383368888507, −4.38489274585589665409638399826, −3.44945532176692215171653318021, −2.62011047372528918091083422316, −0.68655072835500181647361608872,
0.68655072835500181647361608872, 2.62011047372528918091083422316, 3.44945532176692215171653318021, 4.38489274585589665409638399826, 5.23123780073428658383368888507, 6.50563998187962013416614283559, 6.98310044294046265095518743449, 8.073835726380055285731119779302, 8.613139122500934973351512042328, 9.331145111588057450204254811998
