| L(s) = 1 | − 2·4-s − 5-s + 7-s + 5·11-s − 5·13-s + 4·16-s − 17-s − 5·19-s + 2·20-s + 23-s − 4·25-s − 2·28-s + 6·29-s − 6·31-s − 35-s + 4·37-s − 7·41-s − 7·43-s − 10·44-s − 6·47-s + 49-s + 10·52-s − 6·53-s − 5·55-s − 14·59-s − 8·64-s + 5·65-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.377·7-s + 1.50·11-s − 1.38·13-s + 16-s − 0.242·17-s − 1.14·19-s + 0.447·20-s + 0.208·23-s − 4/5·25-s − 0.377·28-s + 1.11·29-s − 1.07·31-s − 0.169·35-s + 0.657·37-s − 1.09·41-s − 1.06·43-s − 1.50·44-s − 0.875·47-s + 1/7·49-s + 1.38·52-s − 0.824·53-s − 0.674·55-s − 1.82·59-s − 64-s + 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−9.425392701566482270296626907091, −8.692150001217748984746063838156, −7.969127769610394273522197763302, −7.01545680207915144778802109603, −6.07343061968960142265880942129, −4.79428223153719685759494266821, −4.36810445385342074666389399007, −3.32489094027495759351806808784, −1.69787774382846645869535268108, 0,
1.69787774382846645869535268108, 3.32489094027495759351806808784, 4.36810445385342074666389399007, 4.79428223153719685759494266821, 6.07343061968960142265880942129, 7.01545680207915144778802109603, 7.969127769610394273522197763302, 8.692150001217748984746063838156, 9.425392701566482270296626907091
