| L(s) = 1 | + 5-s − 3·11-s + 13-s − 3·17-s + 19-s + 2·23-s − 4·25-s + 2·29-s + 31-s + 2·37-s − 4·43-s + 7·47-s − 7·49-s − 3·55-s + 6·59-s + 9·61-s + 65-s + 3·67-s − 71-s − 8·73-s − 79-s − 5·83-s − 3·85-s + 6·89-s + 95-s − 7·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.229·19-s + 0.417·23-s − 4/5·25-s + 0.371·29-s + 0.179·31-s + 0.328·37-s − 0.609·43-s + 1.02·47-s − 49-s − 0.404·55-s + 0.781·59-s + 1.15·61-s + 0.124·65-s + 0.366·67-s − 0.118·71-s − 0.936·73-s − 0.112·79-s − 0.548·83-s − 0.325·85-s + 0.635·89-s + 0.102·95-s − 0.710·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 31 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−15.98539506467212, −15.62690684515709, −15.04471536008820, −14.41431503897725, −13.80093671396152, −13.30064400123886, −12.96720460455346, −12.27893571275368, −11.51429395728937, −11.16339371782538, −10.39302174794895, −10.00005816298966, −9.381187124642371, −8.682872208030064, −8.197572711154195, −7.509436684258750, −6.874436355144633, −6.221450693550554, −5.590081645856324, −5.016145507533342, −4.308350035209268, −3.519112074049666, −2.678621299971479, −2.114054547331342, −1.132629448043231, 0,
1.132629448043231, 2.114054547331342, 2.678621299971479, 3.519112074049666, 4.308350035209268, 5.016145507533342, 5.590081645856324, 6.221450693550554, 6.874436355144633, 7.509436684258750, 8.197572711154195, 8.682872208030064, 9.381187124642371, 10.00005816298966, 10.39302174794895, 11.16339371782538, 11.51429395728937, 12.27893571275368, 12.96720460455346, 13.30064400123886, 13.80093671396152, 14.41431503897725, 15.04471536008820, 15.62690684515709, 15.98539506467212
