| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s − 15-s − 2·17-s + 4·19-s − 4·21-s + 8·23-s + 25-s − 27-s − 6·29-s + 4·35-s + 6·37-s − 10·41-s + 4·43-s + 45-s + 8·47-s + 9·49-s + 2·51-s + 10·53-s − 4·57-s + 6·61-s + 4·63-s − 4·67-s − 8·69-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.872·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.676·35-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s − 0.529·57-s + 0.768·61-s + 0.503·63-s − 0.488·67-s − 0.963·69-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.102612563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.102612563\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 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 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.87114247186786, −14.27128105186120, −13.66981776452645, −13.28099269186838, −12.72998279115699, −11.97924871255123, −11.60849138523407, −11.06186423143917, −10.79510823463208, −10.13535543299975, −9.411608265776341, −9.005504933824223, −8.364111660854386, −7.769945027320821, −7.124238007476809, −6.830321776687621, −5.819429699641103, −5.435991778922042, −4.974692211503937, −4.401967151519942, −3.693414476446511, −2.757558142592397, −2.058160321039696, −1.341244879248390, −0.7281305196615273,
0.7281305196615273, 1.341244879248390, 2.058160321039696, 2.757558142592397, 3.693414476446511, 4.401967151519942, 4.974692211503937, 5.435991778922042, 5.819429699641103, 6.830321776687621, 7.124238007476809, 7.769945027320821, 8.364111660854386, 9.005504933824223, 9.411608265776341, 10.13535543299975, 10.79510823463208, 11.06186423143917, 11.60849138523407, 11.97924871255123, 12.72998279115699, 13.28099269186838, 13.66981776452645, 14.27128105186120, 14.87114247186786
