| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 2·11-s − 14-s + 16-s + 17-s − 2·19-s + 2·20-s + 2·22-s + 8·23-s − 25-s − 28-s + 8·31-s + 32-s + 34-s − 2·35-s − 4·37-s − 2·38-s + 2·40-s + 6·41-s + 4·43-s + 2·44-s + 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.603·11-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.458·19-s + 0.447·20-s + 0.426·22-s + 1.66·23-s − 1/5·25-s − 0.188·28-s + 1.43·31-s + 0.176·32-s + 0.171·34-s − 0.338·35-s − 0.657·37-s − 0.324·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + 0.301·44-s + 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.409269234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.409269234\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−9.193438041525969459140423565967, −8.359570697179676719728215901866, −7.25240739827983854207639165382, −6.55628051685253958930913791783, −5.94819319792275289673190098158, −5.12300888609309699278993633706, −4.25931768077630798940248043382, −3.23306469703830825571019602514, −2.37230827909562924229408640307, −1.19502502056681126609413306691,
1.19502502056681126609413306691, 2.37230827909562924229408640307, 3.23306469703830825571019602514, 4.25931768077630798940248043382, 5.12300888609309699278993633706, 5.94819319792275289673190098158, 6.55628051685253958930913791783, 7.25240739827983854207639165382, 8.359570697179676719728215901866, 9.193438041525969459140423565967
