| L(s) = 1 | − 5-s + 7-s − 4·11-s − 2·13-s − 2·17-s + 4·19-s + 25-s + 10·29-s − 35-s + 6·37-s + 6·41-s + 4·43-s − 8·47-s + 49-s − 6·53-s + 4·55-s − 4·59-s − 10·61-s + 2·65-s − 4·67-s − 16·71-s − 14·73-s − 4·77-s − 8·79-s − 4·83-s + 2·85-s − 10·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 1.85·29-s − 0.169·35-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.248·65-s − 0.488·67-s − 1.89·71-s − 1.63·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-s + 0.216·85-s − 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.66466471864948486403991492151, −7.50772977644731328416858015728, −6.42468588937931507640521883985, −5.66434029757852435472137439510, −4.73501558740432269489317707940, −4.43915013676501344026194730311, −3.06400540965228070122045870183, −2.61685710241734262069895622117, −1.29741549150192396760217377166, 0,
1.29741549150192396760217377166, 2.61685710241734262069895622117, 3.06400540965228070122045870183, 4.43915013676501344026194730311, 4.73501558740432269489317707940, 5.66434029757852435472137439510, 6.42468588937931507640521883985, 7.50772977644731328416858015728, 7.66466471864948486403991492151
