| L(s) = 1 | + 5-s − 4·7-s − 3·9-s + 4·13-s + 4·17-s + 8·19-s + 4·23-s + 25-s − 8·29-s − 4·31-s − 4·35-s − 6·37-s + 8·41-s + 4·43-s − 3·45-s + 12·47-s + 9·49-s + 10·53-s + 8·61-s + 12·63-s + 4·65-s − 8·67-s + 12·71-s − 12·73-s + 8·79-s + 9·81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 9-s + 1.10·13-s + 0.970·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s − 1.48·29-s − 0.718·31-s − 0.676·35-s − 0.986·37-s + 1.24·41-s + 0.609·43-s − 0.447·45-s + 1.75·47-s + 9/7·49-s + 1.37·53-s + 1.02·61-s + 1.51·63-s + 0.496·65-s − 0.977·67-s + 1.42·71-s − 1.40·73-s + 0.900·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.183404030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.183404030\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.74216777585839, −14.25375049424727, −13.61087789286781, −13.43287712696765, −12.81013489407557, −12.19768548748939, −11.80621696936566, −10.98903735990171, −10.73381680845478, −9.962589423207637, −9.411832643966701, −9.100865307980296, −8.653801261517688, −7.632801767693591, −7.344671609057963, −6.613845012318733, −5.939560088826103, −5.538374956519488, −5.290681708212092, −3.862485299454781, −3.583403803175522, −2.979104925211208, −2.392604360845817, −1.237124117981091, −0.5979330824998000,
0.5979330824998000, 1.237124117981091, 2.392604360845817, 2.979104925211208, 3.583403803175522, 3.862485299454781, 5.290681708212092, 5.538374956519488, 5.939560088826103, 6.613845012318733, 7.344671609057963, 7.632801767693591, 8.653801261517688, 9.100865307980296, 9.411832643966701, 9.962589423207637, 10.73381680845478, 10.98903735990171, 11.80621696936566, 12.19768548748939, 12.81013489407557, 13.43287712696765, 13.61087789286781, 14.25375049424727, 14.74216777585839
