| L(s) = 1 | + 2·7-s − 11-s − 2·17-s + 3·19-s + 4·23-s + 3·29-s + 31-s + 10·37-s + 41-s + 10·43-s − 10·47-s − 3·49-s − 6·53-s + 13·59-s + 10·61-s + 8·67-s + 9·71-s + 14·73-s − 2·77-s − 10·83-s + 13·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.301·11-s − 0.485·17-s + 0.688·19-s + 0.834·23-s + 0.557·29-s + 0.179·31-s + 1.64·37-s + 0.156·41-s + 1.52·43-s − 1.45·47-s − 3/7·49-s − 0.824·53-s + 1.69·59-s + 1.28·61-s + 0.977·67-s + 1.06·71-s + 1.63·73-s − 0.227·77-s − 1.09·83-s + 1.37·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.638603476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.638603476\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 13 T + p T^{2} \) | 1.59.an |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.43124276583120, −12.88849090235893, −12.75422491490692, −11.95879541760687, −11.37164577907931, −11.21690219783396, −10.77119083460325, −9.928765824680747, −9.729533524607559, −9.115532882270369, −8.498196781777018, −8.103936888883946, −7.666671800349966, −7.070130876117131, −6.570204545708772, −5.988923182362385, −5.385341839518226, −4.796075047870163, −4.580002596272112, −3.720756615525069, −3.212444439968686, −2.409311993279658, −2.077773676729172, −1.046217952531064, −0.6825753473177828,
0.6825753473177828, 1.046217952531064, 2.077773676729172, 2.409311993279658, 3.212444439968686, 3.720756615525069, 4.580002596272112, 4.796075047870163, 5.385341839518226, 5.988923182362385, 6.570204545708772, 7.070130876117131, 7.666671800349966, 8.103936888883946, 8.498196781777018, 9.115532882270369, 9.729533524607559, 9.928765824680747, 10.77119083460325, 11.21690219783396, 11.37164577907931, 11.95879541760687, 12.75422491490692, 12.88849090235893, 13.43124276583120
