| L(s) = 1 | − 5-s − 7-s + 11-s − 2·13-s + 2·17-s + 4·19-s − 4·23-s − 4·25-s + 6·29-s − 3·31-s + 35-s − 4·37-s + 6·43-s + 2·47-s − 6·49-s − 3·53-s − 55-s − 4·59-s − 8·61-s + 2·65-s + 2·67-s + 6·71-s − 7·73-s − 77-s − 8·79-s + 9·83-s − 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 4/5·25-s + 1.11·29-s − 0.538·31-s + 0.169·35-s − 0.657·37-s + 0.914·43-s + 0.291·47-s − 6/7·49-s − 0.412·53-s − 0.134·55-s − 0.520·59-s − 1.02·61-s + 0.248·65-s + 0.244·67-s + 0.712·71-s − 0.819·73-s − 0.113·77-s − 0.900·79-s + 0.987·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−8.050100998818257336993219851544, −7.58650361754200810022033992258, −6.77881545003433188894440611768, −5.97629865190862322169908364868, −5.19590723901610650808869978368, −4.26935121830958918846934579750, −3.50701121063537275942899711628, −2.64886062866746601925031687396, −1.40257266022074623092089392269, 0,
1.40257266022074623092089392269, 2.64886062866746601925031687396, 3.50701121063537275942899711628, 4.26935121830958918846934579750, 5.19590723901610650808869978368, 5.97629865190862322169908364868, 6.77881545003433188894440611768, 7.58650361754200810022033992258, 8.050100998818257336993219851544
