L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s − 15-s − 2·17-s − 4·19-s + 8·23-s + 25-s − 27-s − 6·29-s − 4·33-s + 10·37-s − 6·41-s − 4·43-s + 45-s − 7·49-s + 2·51-s + 6·53-s + 4·55-s + 4·57-s − 12·59-s − 10·61-s + 12·67-s − 8·69-s + 12·71-s + 10·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.258·15-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.696·33-s + 1.64·37-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 49-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s − 1.56·59-s − 1.28·61-s + 1.46·67-s − 0.963·69-s + 1.42·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.15775140663806, −13.77381936397007, −12.98447537265224, −12.88412638546304, −12.32763528877056, −11.57956946174260, −11.17015009609513, −10.99988763569419, −10.22213851195401, −9.682006304108123, −9.206014874739976, −8.863921855542933, −8.174428951739439, −7.523888355257506, −6.860529783633451, −6.461424907723106, −6.174328198630980, −5.325712465281563, −4.926486143022130, −4.276499054948290, −3.744331747213941, −3.024909791990722, −2.236196313338033, −1.570968174312003, −0.9689476328862510, 0,
0.9689476328862510, 1.570968174312003, 2.236196313338033, 3.024909791990722, 3.744331747213941, 4.276499054948290, 4.926486143022130, 5.325712465281563, 6.174328198630980, 6.461424907723106, 6.860529783633451, 7.523888355257506, 8.174428951739439, 8.863921855542933, 9.206014874739976, 9.682006304108123, 10.22213851195401, 10.99988763569419, 11.17015009609513, 11.57956946174260, 12.32763528877056, 12.88412638546304, 12.98447537265224, 13.77381936397007, 14.15775140663806