| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s − 6·11-s + 13-s + 14-s + 16-s − 4·19-s + 20-s − 6·22-s + 25-s + 26-s + 28-s − 4·31-s + 32-s + 35-s − 4·37-s − 4·38-s + 40-s + 6·41-s − 10·43-s − 6·44-s − 6·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 0.718·31-s + 0.176·32-s + 0.169·35-s − 0.657·37-s − 0.648·38-s + 0.158·40-s + 0.937·41-s − 1.52·43-s − 0.904·44-s − 0.875·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.42849162600199506565088365996, −6.68434403272278136187454356173, −5.92785639999487263152577498234, −5.30111742700512971099579474970, −4.82740032861511687235799177066, −3.95846143949050257400102763232, −3.03018154500921595917340112315, −2.34974082290313804694710734993, −1.56510557791233952704378575125, 0,
1.56510557791233952704378575125, 2.34974082290313804694710734993, 3.03018154500921595917340112315, 3.95846143949050257400102763232, 4.82740032861511687235799177066, 5.30111742700512971099579474970, 5.92785639999487263152577498234, 6.68434403272278136187454356173, 7.42849162600199506565088365996
