| L(s) = 1 | − 3-s + 2·5-s + 3·7-s + 9-s + 13-s − 2·15-s − 2·17-s − 7·19-s − 3·21-s − 23-s − 25-s − 27-s + 4·29-s + 6·35-s − 2·37-s − 39-s + 4·43-s + 2·45-s − 12·47-s + 2·49-s + 2·51-s + 2·53-s + 7·57-s − 5·59-s − 5·61-s + 3·63-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.13·7-s + 1/3·9-s + 0.277·13-s − 0.516·15-s − 0.485·17-s − 1.60·19-s − 0.654·21-s − 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.742·29-s + 1.01·35-s − 0.328·37-s − 0.160·39-s + 0.609·43-s + 0.298·45-s − 1.75·47-s + 2/7·49-s + 0.280·51-s + 0.274·53-s + 0.927·57-s − 0.650·59-s − 0.640·61-s + 0.377·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32244 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32244 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 2687 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.21596531283638, −14.82487298807830, −14.19158277293891, −13.76274806608148, −13.22831123006742, −12.66169847189678, −12.16272400990056, −11.45583315092213, −11.07495249670442, −10.54834788224874, −10.11636250718987, −9.429693177041068, −8.797820638088538, −8.288069930174517, −7.763052665271474, −6.958411513332782, −6.254034950556637, −6.090252793241145, −5.211471423267503, −4.719080617294663, −4.252255222269746, −3.353433938305424, −2.249133979429907, −1.916712320455758, −1.143029802763313, 0,
1.143029802763313, 1.916712320455758, 2.249133979429907, 3.353433938305424, 4.252255222269746, 4.719080617294663, 5.211471423267503, 6.090252793241145, 6.254034950556637, 6.958411513332782, 7.763052665271474, 8.288069930174517, 8.797820638088538, 9.429693177041068, 10.11636250718987, 10.54834788224874, 11.07495249670442, 11.45583315092213, 12.16272400990056, 12.66169847189678, 13.22831123006742, 13.76274806608148, 14.19158277293891, 14.82487298807830, 15.21596531283638
