| L(s) = 1 | + 2·3-s + 9-s − 3·11-s + 2·13-s + 4·17-s − 3·23-s − 4·27-s − 29-s − 2·31-s − 6·33-s + 7·37-s + 4·39-s − 2·41-s + 43-s + 12·47-s + 8·51-s + 6·53-s − 6·59-s + 6·61-s + 7·67-s − 6·69-s + 3·71-s − 2·73-s + 5·79-s − 11·81-s − 6·83-s − 2·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.904·11-s + 0.554·13-s + 0.970·17-s − 0.625·23-s − 0.769·27-s − 0.185·29-s − 0.359·31-s − 1.04·33-s + 1.15·37-s + 0.640·39-s − 0.312·41-s + 0.152·43-s + 1.75·47-s + 1.12·51-s + 0.824·53-s − 0.781·59-s + 0.768·61-s + 0.855·67-s − 0.722·69-s + 0.356·71-s − 0.234·73-s + 0.562·79-s − 1.22·81-s − 0.658·83-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.461516121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.461516121\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.98313231216118, −13.64056751004933, −13.10526588466802, −12.70115115179880, −12.06213394931678, −11.58663062267413, −10.89512016573107, −10.48693976754276, −9.922400092408142, −9.388641682439128, −8.988767160671671, −8.309451360943412, −7.982920550348539, −7.599501245436746, −6.997005696379197, −6.220983900310445, −5.598497875199260, −5.306842838210071, −4.321519576791098, −3.844274986466629, −3.309350204077422, −2.631443328246187, −2.262585273449984, −1.418772108041912, −0.5605513517673386,
0.5605513517673386, 1.418772108041912, 2.262585273449984, 2.631443328246187, 3.309350204077422, 3.844274986466629, 4.321519576791098, 5.306842838210071, 5.598497875199260, 6.220983900310445, 6.997005696379197, 7.599501245436746, 7.982920550348539, 8.309451360943412, 8.988767160671671, 9.388641682439128, 9.922400092408142, 10.48693976754276, 10.89512016573107, 11.58663062267413, 12.06213394931678, 12.70115115179880, 13.10526588466802, 13.64056751004933, 13.98313231216118
