| L(s) = 1 | − 3-s + 9-s + 2·11-s − 4·13-s + 4·17-s − 19-s + 2·23-s − 5·25-s − 27-s + 2·29-s − 2·33-s − 6·37-s + 4·39-s + 6·41-s − 8·43-s − 4·51-s + 2·53-s + 57-s + 4·59-s − 10·61-s − 14·67-s − 2·69-s − 12·71-s − 10·73-s + 5·75-s + 10·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 0.970·17-s − 0.229·19-s + 0.417·23-s − 25-s − 0.192·27-s + 0.371·29-s − 0.348·33-s − 0.986·37-s + 0.640·39-s + 0.937·41-s − 1.21·43-s − 0.560·51-s + 0.274·53-s + 0.132·57-s + 0.520·59-s − 1.28·61-s − 1.71·67-s − 0.240·69-s − 1.42·71-s − 1.17·73-s + 0.577·75-s + 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7709056223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7709056223\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.05007579906227, −12.64320618617304, −12.07245689869825, −11.77797451324020, −11.55642881141761, −10.64193395803451, −10.38447484107296, −9.957119765095429, −9.373638910826884, −9.012245765539740, −8.377319797023123, −7.654768713683317, −7.486324556801241, −6.823027450816283, −6.377007525841563, −5.775323647939712, −5.369702836776396, −4.792864431378548, −4.299845826601888, −3.737340386285760, −3.058586600345013, −2.529618532910838, −1.641797236166663, −1.278443712194209, −0.2659176290454117,
0.2659176290454117, 1.278443712194209, 1.641797236166663, 2.529618532910838, 3.058586600345013, 3.737340386285760, 4.299845826601888, 4.792864431378548, 5.369702836776396, 5.775323647939712, 6.377007525841563, 6.823027450816283, 7.486324556801241, 7.654768713683317, 8.377319797023123, 9.012245765539740, 9.373638910826884, 9.957119765095429, 10.38447484107296, 10.64193395803451, 11.55642881141761, 11.77797451324020, 12.07245689869825, 12.64320618617304, 13.05007579906227
