| L(s) = 1 | − 3-s + 2·7-s + 9-s − 11-s − 6·17-s − 4·19-s − 2·21-s − 6·23-s − 5·25-s − 27-s + 6·29-s + 8·31-s + 33-s + 10·37-s − 6·41-s − 8·43-s − 6·47-s − 3·49-s + 6·51-s + 4·57-s + 8·61-s + 2·63-s − 4·67-s + 6·69-s + 6·71-s − 2·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.45·17-s − 0.917·19-s − 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.174·33-s + 1.64·37-s − 0.937·41-s − 1.21·43-s − 0.875·47-s − 3/7·49-s + 0.840·51-s + 0.529·57-s + 1.02·61-s + 0.251·63-s − 0.488·67-s + 0.722·69-s + 0.712·71-s − 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8672849491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8672849491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.82840833109233, −13.29939350358715, −13.00266170171282, −12.32406526008920, −11.68789195120939, −11.51425833480319, −11.05741497154287, −10.34657830237525, −9.994755565506961, −9.606187057262055, −8.591409111434842, −8.332973707935836, −8.023380180955298, −7.182551889726091, −6.656540056044655, −6.158879095392211, −5.799278997477264, −4.837440399313594, −4.595644866522986, −4.186824066766062, −3.331701732658877, −2.438734300027935, −2.027931484578098, −1.302663797523145, −0.3093400225099523,
0.3093400225099523, 1.302663797523145, 2.027931484578098, 2.438734300027935, 3.331701732658877, 4.186824066766062, 4.595644866522986, 4.837440399313594, 5.799278997477264, 6.158879095392211, 6.656540056044655, 7.182551889726091, 8.023380180955298, 8.332973707935836, 8.591409111434842, 9.606187057262055, 9.994755565506961, 10.34657830237525, 11.05741497154287, 11.51425833480319, 11.68789195120939, 12.32406526008920, 13.00266170171282, 13.29939350358715, 13.82840833109233
