| L(s) = 1 | − 3-s − 7-s − 2·9-s + 11-s + 4·13-s + 5·17-s − 19-s + 21-s + 8·23-s − 5·25-s + 5·27-s + 3·29-s + 31-s − 33-s + 3·37-s − 4·39-s + 6·41-s + 2·47-s − 6·49-s − 5·51-s + 57-s − 3·59-s − 3·61-s + 2·63-s − 14·67-s − 8·69-s + 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 1.10·13-s + 1.21·17-s − 0.229·19-s + 0.218·21-s + 1.66·23-s − 25-s + 0.962·27-s + 0.557·29-s + 0.179·31-s − 0.174·33-s + 0.493·37-s − 0.640·39-s + 0.937·41-s + 0.291·47-s − 6/7·49-s − 0.700·51-s + 0.132·57-s − 0.390·59-s − 0.384·61-s + 0.251·63-s − 1.71·67-s − 0.963·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| 83 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.97186963728059, −13.51616770942826, −12.90460541667296, −12.58379659277643, −11.92629715157079, −11.49254183136180, −11.21048751226003, −10.45572490536204, −10.31428850737177, −9.445003616776111, −9.035576294615776, −8.664533687182076, −7.809546526083625, −7.668769008438544, −6.726104992109563, −6.278570847233905, −5.982311848944969, −5.358378667845802, −4.858035673933638, −4.144647150186273, −3.470647619973426, −3.059944085638470, −2.403222629766752, −1.311493052769368, −0.9577666642213191, 0,
0.9577666642213191, 1.311493052769368, 2.403222629766752, 3.059944085638470, 3.470647619973426, 4.144647150186273, 4.858035673933638, 5.358378667845802, 5.982311848944969, 6.278570847233905, 6.726104992109563, 7.668769008438544, 7.809546526083625, 8.664533687182076, 9.035576294615776, 9.445003616776111, 10.31428850737177, 10.45572490536204, 11.21048751226003, 11.49254183136180, 11.92629715157079, 12.58379659277643, 12.90460541667296, 13.51616770942826, 13.97186963728059
