| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s − 4·11-s − 2·13-s + 2·15-s + 4·21-s + 23-s − 25-s − 27-s + 2·29-s + 4·33-s + 8·35-s + 10·37-s + 2·39-s + 6·41-s − 8·43-s − 2·45-s + 8·47-s + 9·49-s − 6·53-s + 8·55-s + 4·59-s − 14·61-s − 4·63-s + 4·65-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 + 0.872·21-s + 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s + 1.35·35-s + 1.64·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.824·53-s + 1.07·55-s + 0.520·59-s − 1.79·61-s − 0.503·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 319056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 319056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8942207006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8942207006\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| 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.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.56980063115339, −12.11893543298752, −11.95914767908092, −11.12325693442349, −10.90675289984953, −10.37368248238650, −9.911234731319695, −9.522113980957887, −9.109028064873131, −8.407200767371365, −7.802172849399670, −7.595108682728078, −7.090993801123034, −6.514315570405885, −6.063168850297600, −5.695337561470304, −5.002148478842602, −4.539299950397750, −4.088189856001834, −3.401420446395968, −2.968634213370541, −2.542751071010093, −1.766164626505076, −0.6345661882889972, −0.4240185457956980,
0.4240185457956980, 0.6345661882889972, 1.766164626505076, 2.542751071010093, 2.968634213370541, 3.401420446395968, 4.088189856001834, 4.539299950397750, 5.002148478842602, 5.695337561470304, 6.063168850297600, 6.514315570405885, 7.090993801123034, 7.595108682728078, 7.802172849399670, 8.407200767371365, 9.109028064873131, 9.522113980957887, 9.911234731319695, 10.37368248238650, 10.90675289984953, 11.12325693442349, 11.95914767908092, 12.11893543298752, 12.56980063115339
