| L(s) = 1 | + 3-s + 2·5-s − 4·7-s + 9-s + 11-s + 2·15-s − 4·17-s + 19-s − 4·21-s − 7·23-s − 25-s + 27-s − 5·29-s + 33-s − 8·35-s − 8·37-s − 8·41-s + 43-s + 2·45-s + 7·47-s + 9·49-s − 4·51-s + 8·53-s + 2·55-s + 57-s + 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.516·15-s − 0.970·17-s + 0.229·19-s − 0.872·21-s − 1.45·23-s − 1/5·25-s + 0.192·27-s − 0.928·29-s + 0.174·33-s − 1.35·35-s − 1.31·37-s − 1.24·41-s + 0.152·43-s + 0.298·45-s + 1.02·47-s + 9/7·49-s − 0.560·51-s + 1.09·53-s + 0.269·55-s + 0.132·57-s + 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.392459997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.392459997\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.74405109491625, −13.62109050607976, −12.88869264322536, −12.63139762813548, −11.96375463531172, −11.52978526764729, −10.68928385373098, −10.13160752478192, −9.980996341479986, −9.353061083888241, −8.979923642156258, −8.566029537030832, −7.781761341354950, −7.202842508189932, −6.618535356035683, −6.359857727610323, −5.644629059659574, −5.296322950596339, −4.247162670918420, −3.833150023739342, −3.332739441628208, −2.556661820621464, −2.105337719916939, −1.499264295551046, −0.3384439223575945,
0.3384439223575945, 1.499264295551046, 2.105337719916939, 2.556661820621464, 3.332739441628208, 3.833150023739342, 4.247162670918420, 5.296322950596339, 5.644629059659574, 6.359857727610323, 6.618535356035683, 7.202842508189932, 7.781761341354950, 8.566029537030832, 8.979923642156258, 9.353061083888241, 9.980996341479986, 10.13160752478192, 10.68928385373098, 11.52978526764729, 11.96375463531172, 12.63139762813548, 12.88869264322536, 13.62109050607976, 13.74405109491625
