| L(s) = 1 | − 3-s − 5-s + 5·7-s − 2·9-s − 2·13-s + 15-s − 3·17-s − 7·19-s − 5·21-s + 6·23-s + 25-s + 5·27-s + 3·29-s + 7·31-s − 5·35-s − 7·37-s + 2·39-s − 6·41-s + 8·43-s + 2·45-s − 6·47-s + 18·49-s + 3·51-s − 3·53-s + 7·57-s + 6·59-s + 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.88·7-s − 2/3·9-s − 0.554·13-s + 0.258·15-s − 0.727·17-s − 1.60·19-s − 1.09·21-s + 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s + 1.25·31-s − 0.845·35-s − 1.15·37-s + 0.320·39-s − 0.937·41-s + 1.21·43-s + 0.298·45-s − 0.875·47-s + 18/7·49-s + 0.420·51-s − 0.412·53-s + 0.927·57-s + 0.781·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 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
−7.31255377689667285544960604677, −6.69782273601934473967951306674, −5.94383595872833382547546504030, −4.97301817963993372284724772183, −4.81481731445867365118394815554, −4.11145300839382180016745493035, −2.88083826991005901339736267452, −2.14120006002455429236720334822, −1.16153294336403454289660329971, 0,
1.16153294336403454289660329971, 2.14120006002455429236720334822, 2.88083826991005901339736267452, 4.11145300839382180016745493035, 4.81481731445867365118394815554, 4.97301817963993372284724772183, 5.94383595872833382547546504030, 6.69782273601934473967951306674, 7.31255377689667285544960604677
