| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 4·13-s − 2·17-s − 6·19-s + 21-s + 4·23-s + 27-s − 4·29-s − 33-s − 6·37-s − 4·39-s + 8·43-s + 4·47-s + 49-s − 2·51-s + 6·53-s − 6·57-s + 4·59-s + 6·61-s + 63-s + 4·67-s + 4·69-s − 77-s − 6·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.485·17-s − 1.37·19-s + 0.218·21-s + 0.834·23-s + 0.192·27-s − 0.742·29-s − 0.174·33-s − 0.986·37-s − 0.640·39-s + 1.21·43-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.794·57-s + 0.520·59-s + 0.768·61-s + 0.125·63-s + 0.488·67-s + 0.481·69-s − 0.113·77-s − 0.675·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.096818489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.096818489\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−15.40433504719545, −14.91626312933079, −14.39403935784813, −14.04410772772613, −13.15077969544819, −12.90348684206091, −12.36148942876730, −11.63621738198747, −11.07851167481854, −10.45175087194917, −10.06717468059672, −9.220993005015920, −8.852681375747513, −8.296338641013856, −7.594243805222177, −7.117388250411131, −6.592347023219778, −5.639594607901416, −5.125318715039172, −4.358066152776124, −3.929322346539789, −2.909983328992262, −2.346601394894104, −1.755802403486283, −0.5398995633505966,
0.5398995633505966, 1.755802403486283, 2.346601394894104, 2.909983328992262, 3.929322346539789, 4.358066152776124, 5.125318715039172, 5.639594607901416, 6.592347023219778, 7.117388250411131, 7.594243805222177, 8.296338641013856, 8.852681375747513, 9.220993005015920, 10.06717468059672, 10.45175087194917, 11.07851167481854, 11.63621738198747, 12.36148942876730, 12.90348684206091, 13.15077969544819, 14.04410772772613, 14.39403935784813, 14.91626312933079, 15.40433504719545
