| L(s) = 1 | − 2·7-s + 2·11-s − 6·13-s + 17-s − 8·19-s + 2·23-s + 6·29-s + 2·31-s + 6·37-s − 2·41-s + 4·43-s − 4·47-s − 3·49-s + 10·53-s + 10·61-s − 8·67-s + 14·71-s − 10·73-s − 4·77-s + 14·79-s − 4·83-s − 6·89-s + 12·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.603·11-s − 1.66·13-s + 0.242·17-s − 1.83·19-s + 0.417·23-s + 1.11·29-s + 0.359·31-s + 0.986·37-s − 0.312·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s + 1.37·53-s + 1.28·61-s − 0.977·67-s + 1.66·71-s − 1.17·73-s − 0.455·77-s + 1.57·79-s − 0.439·83-s − 0.635·89-s + 1.25·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.475538681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.475538681\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.78378949226538, −12.41553089709123, −12.11661838229563, −11.48068500745774, −11.09815543141988, −10.36651135583100, −10.06709757806150, −9.746383737255077, −9.137780333054657, −8.720031612533249, −8.226135317594948, −7.656158835212401, −7.118647408221349, −6.630078465115385, −6.357025178892993, −5.782308545387255, −5.036434391819920, −4.665616654630861, −4.133523579098990, −3.607114643418423, −2.849290308753328, −2.474877883254237, −1.951621495048054, −1.006828258316027, −0.3716698287515472,
0.3716698287515472, 1.006828258316027, 1.951621495048054, 2.474877883254237, 2.849290308753328, 3.607114643418423, 4.133523579098990, 4.665616654630861, 5.036434391819920, 5.782308545387255, 6.357025178892993, 6.630078465115385, 7.118647408221349, 7.656158835212401, 8.226135317594948, 8.720031612533249, 9.137780333054657, 9.746383737255077, 10.06709757806150, 10.36651135583100, 11.09815543141988, 11.48068500745774, 12.11661838229563, 12.41553089709123, 12.78378949226538
