| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 2·11-s + 13-s + 15-s − 4·17-s + 19-s + 21-s − 3·23-s − 4·25-s − 27-s + 3·29-s + 5·31-s − 2·33-s + 35-s − 8·37-s − 39-s − 10·41-s + 43-s − 45-s − 13·47-s + 49-s + 4·51-s − 53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s − 0.970·17-s + 0.229·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s − 0.192·27-s + 0.557·29-s + 0.898·31-s − 0.348·33-s + 0.169·35-s − 1.31·37-s − 0.160·39-s − 1.56·41-s + 0.152·43-s − 0.149·45-s − 1.89·47-s + 1/7·49-s + 0.560·51-s − 0.137·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−9.532366981200356213997890003779, −8.619423495770954651617910608696, −7.79849941895422628515061307566, −6.68780104430679236304835437256, −6.28663769946687387854667854925, −5.07596806523130964387263477013, −4.19321729294232862568998017202, −3.24727161345737161805736947131, −1.68295655341299300219024027250, 0,
1.68295655341299300219024027250, 3.24727161345737161805736947131, 4.19321729294232862568998017202, 5.07596806523130964387263477013, 6.28663769946687387854667854925, 6.68780104430679236304835437256, 7.79849941895422628515061307566, 8.619423495770954651617910608696, 9.532366981200356213997890003779
