| L(s) = 1 | − 3·3-s + 3·5-s − 3·7-s + 6·9-s − 9·15-s + 3·17-s − 2·19-s + 9·21-s + 4·23-s + 4·25-s − 9·27-s + 6·29-s − 9·35-s − 5·37-s − 8·41-s − 5·43-s + 18·45-s + 7·47-s + 2·49-s − 9·51-s − 2·53-s + 6·57-s + 8·59-s − 4·61-s − 18·63-s + 4·67-s − 12·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s − 1.13·7-s + 2·9-s − 2.32·15-s + 0.727·17-s − 0.458·19-s + 1.96·21-s + 0.834·23-s + 4/5·25-s − 1.73·27-s + 1.11·29-s − 1.52·35-s − 0.821·37-s − 1.24·41-s − 0.762·43-s + 2.68·45-s + 1.02·47-s + 2/7·49-s − 1.26·51-s − 0.274·53-s + 0.794·57-s + 1.04·59-s − 0.512·61-s − 2.26·63-s + 0.488·67-s − 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8751473889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8751473889\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 113 | \( 1 + T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−12.67144771674575, −12.21072284615836, −11.81975764781116, −11.33105197389702, −10.68056205839241, −10.36193230138180, −10.04716522176079, −9.742936365767380, −9.146312010879544, −8.682438595080909, −8.010914951401675, −7.183099350602477, −6.791500503901173, −6.542680204619149, −6.063836075166690, −5.650577442855069, −5.192031863190181, −4.889445067770098, −4.153685498030074, −3.502275246553881, −2.913309537876444, −2.284658377154520, −1.491118629371979, −1.103245672057307, −0.3050225687940153,
0.3050225687940153, 1.103245672057307, 1.491118629371979, 2.284658377154520, 2.913309537876444, 3.502275246553881, 4.153685498030074, 4.889445067770098, 5.192031863190181, 5.650577442855069, 6.063836075166690, 6.542680204619149, 6.791500503901173, 7.183099350602477, 8.010914951401675, 8.682438595080909, 9.146312010879544, 9.742936365767380, 10.04716522176079, 10.36193230138180, 10.68056205839241, 11.33105197389702, 11.81975764781116, 12.21072284615836, 12.67144771674575
