| L(s) = 1 | − 3-s + 7-s + 9-s − 3·11-s − 2·13-s + 2·19-s − 21-s + 7·23-s − 27-s − 3·29-s − 6·31-s + 3·33-s − 3·37-s + 2·39-s + 5·43-s + 2·47-s + 49-s + 2·53-s − 2·57-s − 10·59-s − 8·61-s + 63-s − 9·67-s − 7·69-s + 9·71-s − 8·73-s − 3·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.458·19-s − 0.218·21-s + 1.45·23-s − 0.192·27-s − 0.557·29-s − 1.07·31-s + 0.522·33-s − 0.493·37-s + 0.320·39-s + 0.762·43-s + 0.291·47-s + 1/7·49-s + 0.274·53-s − 0.264·57-s − 1.30·59-s − 1.02·61-s + 0.125·63-s − 1.09·67-s − 0.842·69-s + 1.06·71-s − 0.936·73-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.73437464907961643851036901764, −7.46287741183482478046701039351, −6.61161774930155236019901290654, −5.61022432071550987249179652844, −5.17016472851633920308383166012, −4.44917986958329971027596031720, −3.35743140263865319414545423618, −2.44190990683377745139512896175, −1.32194709001282814131560780686, 0,
1.32194709001282814131560780686, 2.44190990683377745139512896175, 3.35743140263865319414545423618, 4.44917986958329971027596031720, 5.17016472851633920308383166012, 5.61022432071550987249179652844, 6.61161774930155236019901290654, 7.46287741183482478046701039351, 7.73437464907961643851036901764
