| L(s) = 1 | + 2·3-s + 9-s − 6·13-s − 6·17-s − 6·23-s − 4·27-s − 4·31-s + 2·37-s − 12·39-s + 12·41-s − 12·43-s + 6·47-s − 7·49-s − 12·51-s − 6·53-s − 14·67-s − 12·69-s − 6·73-s + 12·79-s − 11·81-s − 12·83-s + 6·89-s − 8·93-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.66·13-s − 1.45·17-s − 1.25·23-s − 0.769·27-s − 0.718·31-s + 0.328·37-s − 1.92·39-s + 1.87·41-s − 1.82·43-s + 0.875·47-s − 49-s − 1.68·51-s − 0.824·53-s − 1.71·67-s − 1.44·69-s − 0.702·73-s + 1.35·79-s − 1.22·81-s − 1.31·83-s + 0.635·89-s − 0.829·93-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.328613366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.328613366\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.49872135838623, −14.21335762072932, −13.56949659519476, −13.11716613625848, −12.64004484417628, −12.01470920735223, −11.52716767715212, −10.92933593503705, −10.28129931941351, −9.681213011645336, −9.362555639205994, −8.797660721477533, −8.298158877897307, −7.576783845825524, −7.451945115287870, −6.609726150543681, −6.035247898925175, −5.315257203465292, −4.500584649519306, −4.263188149597764, −3.334172719817250, −2.811983503141930, −2.117865065097744, −1.828895596470555, −0.3426772032242629,
0.3426772032242629, 1.828895596470555, 2.117865065097744, 2.811983503141930, 3.334172719817250, 4.263188149597764, 4.500584649519306, 5.315257203465292, 6.035247898925175, 6.609726150543681, 7.451945115287870, 7.576783845825524, 8.298158877897307, 8.797660721477533, 9.362555639205994, 9.681213011645336, 10.28129931941351, 10.92933593503705, 11.52716767715212, 12.01470920735223, 12.64004484417628, 13.11716613625848, 13.56949659519476, 14.21335762072932, 14.49872135838623
