| L(s) = 1 | − 5-s + 2·7-s + 5·11-s + 4·13-s + 4·17-s + 19-s − 23-s − 4·25-s − 2·29-s + 4·31-s − 2·35-s + 9·37-s + 8·41-s + 4·43-s + 8·47-s − 3·49-s − 9·53-s − 5·55-s − 59-s + 5·61-s − 4·65-s − 13·67-s + 2·71-s + 6·73-s + 10·77-s − 83-s − 4·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 1.50·11-s + 1.10·13-s + 0.970·17-s + 0.229·19-s − 0.208·23-s − 4/5·25-s − 0.371·29-s + 0.718·31-s − 0.338·35-s + 1.47·37-s + 1.24·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s − 1.23·53-s − 0.674·55-s − 0.130·59-s + 0.640·61-s − 0.496·65-s − 1.58·67-s + 0.237·71-s + 0.702·73-s + 1.13·77-s − 0.109·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.632909282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.632909282\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−14.48965815926952, −14.14779456570008, −13.72268049628201, −13.00432123488111, −12.42720018524138, −11.87990590450402, −11.48419916939375, −11.14061118229234, −10.54029990208001, −9.775064886475791, −9.313515508593786, −8.854375909094530, −8.088630019427246, −7.824188006728263, −7.265357763879556, −6.395893170294133, −6.036932092492346, −5.507671051495642, −4.534038569324584, −4.167129677943824, −3.644084740186532, −2.960991802708806, −1.986661682692860, −1.259468396809284, −0.7837189292609485,
0.7837189292609485, 1.259468396809284, 1.986661682692860, 2.960991802708806, 3.644084740186532, 4.167129677943824, 4.534038569324584, 5.507671051495642, 6.036932092492346, 6.395893170294133, 7.265357763879556, 7.824188006728263, 8.088630019427246, 8.854375909094530, 9.313515508593786, 9.775064886475791, 10.54029990208001, 11.14061118229234, 11.48419916939375, 11.87990590450402, 12.42720018524138, 13.00432123488111, 13.72268049628201, 14.14779456570008, 14.48965815926952
