| L(s) = 1 | − 2·3-s − 2·4-s + 5-s − 4·7-s + 9-s + 4·12-s + 4·13-s − 2·15-s + 4·16-s − 17-s − 5·19-s − 2·20-s + 8·21-s + 23-s + 25-s + 4·27-s + 8·28-s − 7·29-s + 4·31-s − 4·35-s − 2·36-s + 6·37-s − 8·39-s + 45-s − 7·47-s − 8·48-s + 9·49-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.15·12-s + 1.10·13-s − 0.516·15-s + 16-s − 0.242·17-s − 1.14·19-s − 0.447·20-s + 1.74·21-s + 0.208·23-s + 1/5·25-s + 0.769·27-s + 1.51·28-s − 1.29·29-s + 0.718·31-s − 0.676·35-s − 1/3·36-s + 0.986·37-s − 1.28·39-s + 0.149·45-s − 1.02·47-s − 1.15·48-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.99138000309836, −14.52125051455633, −13.77427150175278, −13.26127622332492, −12.98881338281872, −12.63666316175966, −11.98227812507170, −11.25916182854838, −10.85169269271480, −10.29918293830595, −9.717380518417185, −9.393548801811795, −8.667342596025192, −8.345482919996935, −7.395628350257439, −6.585697274461859, −6.237439437737247, −5.900781189617171, −5.282216828803224, −4.581828041895711, −3.989289473389525, −3.369308398688899, −2.667917784680667, −1.543472109421876, −0.6546841716290127, 0,
0.6546841716290127, 1.543472109421876, 2.667917784680667, 3.369308398688899, 3.989289473389525, 4.581828041895711, 5.282216828803224, 5.900781189617171, 6.237439437737247, 6.585697274461859, 7.395628350257439, 8.345482919996935, 8.667342596025192, 9.393548801811795, 9.717380518417185, 10.29918293830595, 10.85169269271480, 11.25916182854838, 11.98227812507170, 12.63666316175966, 12.98881338281872, 13.26127622332492, 13.77427150175278, 14.52125051455633, 14.99138000309836
