| L(s) = 1 | − 5-s − 11-s − 2·13-s − 6·17-s + 5·19-s + 3·23-s − 4·25-s − 2·29-s − 5·31-s + 3·37-s + 3·41-s + 2·43-s + 10·47-s − 8·53-s + 55-s + 10·59-s − 4·61-s + 2·65-s − 10·67-s − 13·71-s + 14·73-s + 2·79-s − 6·83-s + 6·85-s − 17·89-s − 5·95-s − 4·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 1.45·17-s + 1.14·19-s + 0.625·23-s − 4/5·25-s − 0.371·29-s − 0.898·31-s + 0.493·37-s + 0.468·41-s + 0.304·43-s + 1.45·47-s − 1.09·53-s + 0.134·55-s + 1.30·59-s − 0.512·61-s + 0.248·65-s − 1.22·67-s − 1.54·71-s + 1.63·73-s + 0.225·79-s − 0.658·83-s + 0.650·85-s − 1.80·89-s − 0.512·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.153716141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.153716141\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 17 T + p T^{2} \) | 1.89.r |
| 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
−15.58589738211322, −15.12223603102802, −14.58501419743357, −13.89570193961795, −13.42231234709085, −12.89481121321372, −12.28233419310354, −11.76466361102190, −11.04301038134209, −10.91753738807970, −9.967998283216074, −9.424849181511540, −8.974542215021362, −8.273563238418095, −7.490262122283073, −7.312845728955364, −6.533890130581444, −5.717966809465267, −5.235248748597492, −4.403417107063559, −3.972104032773087, −3.036342954148758, −2.446195400662350, −1.558438883926281, −0.4338891794812387,
0.4338891794812387, 1.558438883926281, 2.446195400662350, 3.036342954148758, 3.972104032773087, 4.403417107063559, 5.235248748597492, 5.717966809465267, 6.533890130581444, 7.312845728955364, 7.490262122283073, 8.273563238418095, 8.974542215021362, 9.424849181511540, 9.967998283216074, 10.91753738807970, 11.04301038134209, 11.76466361102190, 12.28233419310354, 12.89481121321372, 13.42231234709085, 13.89570193961795, 14.58501419743357, 15.12223603102802, 15.58589738211322
