| L(s) = 1 | + 3-s − 7-s + 9-s + 5·11-s + 13-s + 3·17-s + 19-s − 21-s − 23-s + 27-s − 29-s + 2·31-s + 5·33-s − 4·37-s + 39-s − 5·41-s − 10·43-s − 6·47-s + 49-s + 3·51-s − 6·53-s + 57-s − 13·59-s − 3·61-s − 63-s + 15·67-s − 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s + 0.727·17-s + 0.229·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.185·29-s + 0.359·31-s + 0.870·33-s − 0.657·37-s + 0.160·39-s − 0.780·41-s − 1.52·43-s − 0.875·47-s + 1/7·49-s + 0.420·51-s − 0.824·53-s + 0.132·57-s − 1.69·59-s − 0.384·61-s − 0.125·63-s + 1.83·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 13 T + p T^{2} \) | 1.59.n |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.73342384417774, −12.98387231745891, −12.66205736449891, −12.04539775731020, −11.62621999698623, −11.36723760208353, −10.46325549592449, −10.10662292277912, −9.709870871815578, −9.106803601429708, −8.812741625043623, −8.287363455832070, −7.702909325838825, −7.221520415891038, −6.651607214068031, −6.225131357958950, −5.809095170345379, −4.848635077524331, −4.639698586561833, −3.742413732058131, −3.371606358347432, −3.112850025322140, −2.040365046451070, −1.593856878495549, −1.002577413094559, 0,
1.002577413094559, 1.593856878495549, 2.040365046451070, 3.112850025322140, 3.371606358347432, 3.742413732058131, 4.639698586561833, 4.848635077524331, 5.809095170345379, 6.225131357958950, 6.651607214068031, 7.221520415891038, 7.702909325838825, 8.287363455832070, 8.812741625043623, 9.106803601429708, 9.709870871815578, 10.10662292277912, 10.46325549592449, 11.36723760208353, 11.62621999698623, 12.04539775731020, 12.66205736449891, 12.98387231745891, 13.73342384417774
