| L(s) = 1 | − 5-s − 3·9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s + 25-s − 6·29-s + 4·31-s − 37-s − 6·41-s − 4·43-s + 3·45-s + 8·47-s − 7·49-s + 10·53-s − 4·55-s − 4·59-s + 10·61-s − 2·65-s + 8·67-s + 10·73-s + 4·79-s + 9·81-s + 2·85-s + 2·89-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.164·37-s − 0.937·41-s − 0.609·43-s + 0.447·45-s + 1.16·47-s − 49-s + 1.37·53-s − 0.539·55-s − 0.520·59-s + 1.28·61-s − 0.248·65-s + 0.977·67-s + 1.17·73-s + 0.450·79-s + 81-s + 0.216·85-s + 0.211·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.602782834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.602782834\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 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 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.734531122003643087724049825453, −8.129825714537106723073907080795, −7.18886119243338118366578821365, −6.49664616322372714025027522541, −5.73020676976688720839306571574, −4.88856578354223850475874803229, −3.82494672208517844438568110208, −3.30347465535627875246494353467, −2.06558083903480564080310734566, −0.78024997926188957232556441579,
0.78024997926188957232556441579, 2.06558083903480564080310734566, 3.30347465535627875246494353467, 3.82494672208517844438568110208, 4.88856578354223850475874803229, 5.73020676976688720839306571574, 6.49664616322372714025027522541, 7.18886119243338118366578821365, 8.129825714537106723073907080795, 8.734531122003643087724049825453
