| L(s) = 1 | − 3-s + 4·5-s − 7-s + 9-s + 2·11-s − 2·13-s − 4·15-s + 2·17-s − 2·19-s + 21-s − 23-s + 11·25-s − 27-s + 6·29-s − 2·33-s − 4·35-s − 4·37-s + 2·39-s − 10·41-s − 10·43-s + 4·45-s + 8·47-s + 49-s − 2·51-s + 8·55-s + 2·57-s + 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s − 1.03·15-s + 0.485·17-s − 0.458·19-s + 0.218·21-s − 0.208·23-s + 11/5·25-s − 0.192·27-s + 1.11·29-s − 0.348·33-s − 0.676·35-s − 0.657·37-s + 0.320·39-s − 1.56·41-s − 1.52·43-s + 0.596·45-s + 1.16·47-s + 1/7·49-s − 0.280·51-s + 1.07·55-s + 0.264·57-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.639357747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.639357747\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.11250493123680, −14.34181452912047, −14.09562501844634, −13.48959633278780, −13.03833761289225, −12.47583170998624, −11.95462855631337, −11.48366549775537, −10.49139287342206, −10.20600132614857, −9.967035495583828, −9.192511984807795, −8.807595058997936, −8.080919078520068, −7.044894641283240, −6.758224816417108, −6.170384579651069, −5.711976949682093, −5.064262865987873, −4.619896714793823, −3.603386048688958, −2.902666256761587, −2.066594322990921, −1.568183472242722, −0.6440254825650223,
0.6440254825650223, 1.568183472242722, 2.066594322990921, 2.902666256761587, 3.603386048688958, 4.619896714793823, 5.064262865987873, 5.711976949682093, 6.170384579651069, 6.758224816417108, 7.044894641283240, 8.080919078520068, 8.807595058997936, 9.192511984807795, 9.967035495583828, 10.20600132614857, 10.49139287342206, 11.48366549775537, 11.95462855631337, 12.47583170998624, 13.03833761289225, 13.48959633278780, 14.09562501844634, 14.34181452912047, 15.11250493123680
