| L(s) = 1 | + 3-s − 2·4-s − 7-s + 9-s + 11-s − 2·12-s + 13-s + 4·16-s − 4·19-s − 21-s − 3·23-s + 27-s + 2·28-s + 8·29-s + 4·31-s + 33-s − 2·36-s + 3·37-s + 39-s + 9·41-s + 8·43-s − 2·44-s − 10·47-s + 4·48-s − 6·49-s − 2·52-s + 53-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 0.277·13-s + 16-s − 0.917·19-s − 0.218·21-s − 0.625·23-s + 0.192·27-s + 0.377·28-s + 1.48·29-s + 0.718·31-s + 0.174·33-s − 1/3·36-s + 0.493·37-s + 0.160·39-s + 1.40·41-s + 1.21·43-s − 0.301·44-s − 1.45·47-s + 0.577·48-s − 6/7·49-s − 0.277·52-s + 0.137·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.675824214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.675824214\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 15 T + p T^{2} \) | 1.97.p |
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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
−12.85973636743495, −12.51546399165274, −11.94398445246410, −11.31823757950264, −10.89901437933989, −10.21825890287441, −9.870471115648369, −9.589412797156783, −9.006046213274926, −8.540442090773113, −8.163939736340997, −7.909736816290349, −7.097587224598394, −6.595442060141729, −6.145533624382587, −5.680975743088996, −4.920247443199147, −4.513431608699839, −4.017872640159872, −3.676243027730835, −2.927809408367629, −2.515799323987322, −1.768124703203761, −0.9772905591174997, −0.5135021113735553,
0.5135021113735553, 0.9772905591174997, 1.768124703203761, 2.515799323987322, 2.927809408367629, 3.676243027730835, 4.017872640159872, 4.513431608699839, 4.920247443199147, 5.680975743088996, 6.145533624382587, 6.595442060141729, 7.097587224598394, 7.909736816290349, 8.163939736340997, 8.540442090773113, 9.006046213274926, 9.589412797156783, 9.870471115648369, 10.21825890287441, 10.89901437933989, 11.31823757950264, 11.94398445246410, 12.51546399165274, 12.85973636743495
