| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 6·11-s − 2·13-s + 14-s + 16-s − 3·17-s + 2·19-s + 6·22-s − 2·26-s + 28-s + 8·31-s + 32-s − 3·34-s − 2·37-s + 2·38-s + 3·41-s + 10·43-s + 6·44-s + 3·47-s − 6·49-s − 2·52-s − 12·53-s + 56-s + 6·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.80·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 1.27·22-s − 0.392·26-s + 0.188·28-s + 1.43·31-s + 0.176·32-s − 0.514·34-s − 0.328·37-s + 0.324·38-s + 0.468·41-s + 1.52·43-s + 0.904·44-s + 0.437·47-s − 6/7·49-s − 0.277·52-s − 1.64·53-s + 0.133·56-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.622919523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.622919523\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−8.346212246314974574226701398773, −7.61773432256632124956979349889, −6.69485747773736211209483326488, −6.37893985382487209328328175036, −5.37545541415504014341306579918, −4.53274674419110166644396580674, −4.03536650303338836080745036959, −3.05371668490022537971453314603, −2.05687187057551879747467060410, −1.04585618057385714309721390590,
1.04585618057385714309721390590, 2.05687187057551879747467060410, 3.05371668490022537971453314603, 4.03536650303338836080745036959, 4.53274674419110166644396580674, 5.37545541415504014341306579918, 6.37893985382487209328328175036, 6.69485747773736211209483326488, 7.61773432256632124956979349889, 8.346212246314974574226701398773
