| L(s) = 1 | + 2·3-s + 7-s + 9-s + 11-s − 4·13-s + 6·19-s + 2·21-s − 3·23-s − 4·27-s + 3·29-s + 2·33-s − 9·37-s − 8·39-s + 2·41-s + 9·43-s + 6·47-s + 49-s − 6·53-s + 12·57-s + 8·59-s + 10·61-s + 63-s − 67-s − 6·69-s + 7·71-s − 2·73-s + 77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 1.37·19-s + 0.436·21-s − 0.625·23-s − 0.769·27-s + 0.557·29-s + 0.348·33-s − 1.47·37-s − 1.28·39-s + 0.312·41-s + 1.37·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s + 1.58·57-s + 1.04·59-s + 1.28·61-s + 0.125·63-s − 0.122·67-s − 0.722·69-s + 0.830·71-s − 0.234·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.306403587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.306403587\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 9 T + p T^{2} \) | 1.37.j |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−16.41203648741156, −15.62769936625472, −15.44208764662034, −14.47794988762351, −14.17404004439568, −14.01318358463154, −13.12991251217455, −12.48530960079685, −11.85362234587871, −11.45649289834394, −10.47929080822966, −9.971654351126163, −9.300531571600378, −8.908196680689290, −8.172102503300677, −7.576259171880658, −7.226224248187426, −6.275923153598266, −5.406440143329527, −4.854952829554821, −3.918701631310406, −3.345939800257365, −2.501921396479668, −1.969594535539960, −0.7849297436093460,
0.7849297436093460, 1.969594535539960, 2.501921396479668, 3.345939800257365, 3.918701631310406, 4.854952829554821, 5.406440143329527, 6.275923153598266, 7.226224248187426, 7.576259171880658, 8.172102503300677, 8.908196680689290, 9.300531571600378, 9.971654351126163, 10.47929080822966, 11.45649289834394, 11.85362234587871, 12.48530960079685, 13.12991251217455, 14.01318358463154, 14.17404004439568, 14.47794988762351, 15.44208764662034, 15.62769936625472, 16.41203648741156
