| L(s) = 1 | − 2·5-s + 4·11-s − 2·13-s + 6·17-s + 23-s − 25-s + 2·29-s + 4·31-s + 6·37-s − 6·41-s − 12·43-s + 12·47-s − 6·53-s − 8·55-s + 4·59-s + 10·61-s + 4·65-s − 4·67-s − 16·71-s − 2·73-s − 8·79-s + 16·83-s − 12·85-s + 6·89-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.554·13-s + 1.45·17-s + 0.208·23-s − 1/5·25-s + 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s − 1.82·43-s + 1.75·47-s − 0.824·53-s − 1.07·55-s + 0.520·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s − 1.89·71-s − 0.234·73-s − 0.900·79-s + 1.75·83-s − 1.30·85-s + 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.53782643675204, −13.01808069210946, −12.29162352848933, −12.01345756741671, −11.75542904646199, −11.34237695373164, −10.61246972396748, −10.08665987558826, −9.767814423238589, −9.205434986836740, −8.593869134683008, −8.220452064479257, −7.627666828434914, −7.315381015383027, −6.660141955109967, −6.272419354658317, −5.558509579663567, −5.087651535501685, −4.365763741911146, −4.055963817591134, −3.367447629871431, −3.025347611625558, −2.187844389332399, −1.371486733563656, −0.8824973290569547, 0,
0.8824973290569547, 1.371486733563656, 2.187844389332399, 3.025347611625558, 3.367447629871431, 4.055963817591134, 4.365763741911146, 5.087651535501685, 5.558509579663567, 6.272419354658317, 6.660141955109967, 7.315381015383027, 7.627666828434914, 8.220452064479257, 8.593869134683008, 9.205434986836740, 9.767814423238589, 10.08665987558826, 10.61246972396748, 11.34237695373164, 11.75542904646199, 12.01345756741671, 12.29162352848933, 13.01808069210946, 13.53782643675204
