| L(s) = 1 | + 2·3-s − 5-s + 7-s + 9-s − 11-s − 13-s − 2·15-s + 17-s + 2·21-s − 3·23-s − 4·25-s − 4·27-s − 5·29-s + 2·31-s − 2·33-s − 35-s + 6·37-s − 2·39-s + 3·41-s + 5·43-s − 45-s − 6·49-s + 2·51-s − 2·53-s + 55-s − 7·59-s − 5·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.516·15-s + 0.242·17-s + 0.436·21-s − 0.625·23-s − 4/5·25-s − 0.769·27-s − 0.928·29-s + 0.359·31-s − 0.348·33-s − 0.169·35-s + 0.986·37-s − 0.320·39-s + 0.468·41-s + 0.762·43-s − 0.149·45-s − 6/7·49-s + 0.280·51-s − 0.274·53-s + 0.134·55-s − 0.911·59-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.361072956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.361072956\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.39800148044406, −12.89004022791890, −12.41184052152500, −11.90911063346157, −11.32563392216684, −11.06652249002069, −10.36918119814669, −9.796448957483840, −9.411151472642036, −9.002638696808274, −8.355661104485009, −7.894568288429000, −7.695072356388655, −7.289228189480466, −6.368489889957484, −5.957472206804215, −5.359098937120068, −4.630965081768256, −4.247937322323568, −3.507215847267434, −3.278548493261801, −2.352835910340325, −2.169431297775619, −1.340685576180072, −0.4048747596353470,
0.4048747596353470, 1.340685576180072, 2.169431297775619, 2.352835910340325, 3.278548493261801, 3.507215847267434, 4.247937322323568, 4.630965081768256, 5.359098937120068, 5.957472206804215, 6.368489889957484, 7.289228189480466, 7.695072356388655, 7.894568288429000, 8.355661104485009, 9.002638696808274, 9.411151472642036, 9.796448957483840, 10.36918119814669, 11.06652249002069, 11.32563392216684, 11.90911063346157, 12.41184052152500, 12.89004022791890, 13.39800148044406
