| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 11-s + 2·13-s − 15-s − 2·21-s + 7·23-s − 4·25-s − 27-s + 9·31-s + 33-s + 2·35-s − 10·37-s − 2·39-s − 4·43-s + 45-s − 3·49-s − 53-s − 55-s + 59-s + 12·61-s + 2·63-s + 2·65-s − 12·67-s − 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.436·21-s + 1.45·23-s − 4/5·25-s − 0.192·27-s + 1.61·31-s + 0.174·33-s + 0.338·35-s − 1.64·37-s − 0.320·39-s − 0.609·43-s + 0.149·45-s − 3/7·49-s − 0.137·53-s − 0.134·55-s + 0.130·59-s + 1.53·61-s + 0.251·63-s + 0.248·65-s − 1.46·67-s − 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.22718945996259, −12.81306660659923, −12.03622831641269, −11.88904130489455, −11.29483756934570, −10.89856917827620, −10.47880328056910, −9.974494979887589, −9.594134019745501, −8.868910676985639, −8.492616217067817, −8.081257167736756, −7.430915019335437, −6.944506664753653, −6.491028086470941, −5.953103588908723, −5.410922170194254, −5.021098205554178, −4.582434465484198, −3.951706421543620, −3.278563555806575, −2.726598507095876, −1.967608387039675, −1.449070989347856, −0.8993491254208363, 0,
0.8993491254208363, 1.449070989347856, 1.967608387039675, 2.726598507095876, 3.278563555806575, 3.951706421543620, 4.582434465484198, 5.021098205554178, 5.410922170194254, 5.953103588908723, 6.491028086470941, 6.944506664753653, 7.430915019335437, 8.081257167736756, 8.492616217067817, 8.868910676985639, 9.594134019745501, 9.974494979887589, 10.47880328056910, 10.89856917827620, 11.29483756934570, 11.88904130489455, 12.03622831641269, 12.81306660659923, 13.22718945996259
