| L(s) = 1 | − 5-s − 3·9-s + 2·11-s − 4·13-s + 17-s + 5·19-s + 9·23-s − 4·25-s − 6·29-s + 8·31-s − 11·37-s − 4·41-s − 43-s + 3·45-s − 2·47-s + 6·53-s − 2·55-s + 9·59-s + 6·61-s + 4·65-s − 7·67-s + 71-s + 4·73-s − 8·79-s + 9·81-s − 4·83-s − 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s + 0.603·11-s − 1.10·13-s + 0.242·17-s + 1.14·19-s + 1.87·23-s − 4/5·25-s − 1.11·29-s + 1.43·31-s − 1.80·37-s − 0.624·41-s − 0.152·43-s + 0.447·45-s − 0.291·47-s + 0.824·53-s − 0.269·55-s + 1.17·59-s + 0.768·61-s + 0.496·65-s − 0.855·67-s + 0.118·71-s + 0.468·73-s − 0.900·79-s + 81-s − 0.439·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 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
−7.43316832123095824552581854288, −7.16944354565385241128712198613, −6.25820759041119387538936776974, −5.28345281243972624418273515776, −5.02866721281013756272124867657, −3.85067301730031915156356164414, −3.20649607198224199715067645321, −2.43750576977275347280275841530, −1.20116384355878213550195591800, 0,
1.20116384355878213550195591800, 2.43750576977275347280275841530, 3.20649607198224199715067645321, 3.85067301730031915156356164414, 5.02866721281013756272124867657, 5.28345281243972624418273515776, 6.25820759041119387538936776974, 7.16944354565385241128712198613, 7.43316832123095824552581854288
