| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s − 4·11-s + 2·13-s + 2·15-s − 6·17-s + 4·19-s + 4·21-s − 25-s − 27-s − 2·29-s + 4·31-s + 4·33-s + 8·35-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s − 2·45-s + 8·47-s + 9·49-s + 6·51-s − 10·53-s + 8·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s + 1.35·35-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s + 0.840·51-s − 1.37·53-s + 1.07·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 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.k |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−12.03546217958314609201730143237, −11.10886283085595504556202509202, −10.20586918892848667331432436971, −9.133901360741508328144538474134, −7.83182394757322498939219938639, −6.80946897903754876349072202165, −5.77092473843782707280190480124, −4.31158789218116792910899233548, −3.00571163091752467289605693825, 0,
3.00571163091752467289605693825, 4.31158789218116792910899233548, 5.77092473843782707280190480124, 6.80946897903754876349072202165, 7.83182394757322498939219938639, 9.133901360741508328144538474134, 10.20586918892848667331432436971, 11.10886283085595504556202509202, 12.03546217958314609201730143237
