| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s − 6·13-s + 15-s − 2·17-s − 4·19-s − 23-s + 25-s − 27-s + 6·29-s + 4·33-s + 6·37-s + 6·39-s − 10·41-s + 4·43-s − 45-s + 8·47-s + 2·51-s + 6·53-s + 4·55-s + 4·57-s + 4·59-s + 2·61-s + 6·65-s − 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.696·33-s + 0.986·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s + 0.280·51-s + 0.824·53-s + 0.539·55-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.744·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008083193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008083193\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 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 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.54581601168682, −12.44064094230123, −11.85643098728433, −11.49647315090109, −10.83554698704193, −10.56165041467308, −10.00636780525771, −9.818739444519027, −9.026394354904563, −8.524309140513766, −8.090537807063128, −7.556505497346277, −7.165578670172772, −6.721790126284977, −6.144000569830323, −5.564244296597077, −5.085009549616079, −4.579060465624567, −4.355663010557618, −3.545672083469949, −2.861506405482857, −2.304723780683876, −2.016532226148021, −0.8025884640246662, −0.3688060917666437,
0.3688060917666437, 0.8025884640246662, 2.016532226148021, 2.304723780683876, 2.861506405482857, 3.545672083469949, 4.355663010557618, 4.579060465624567, 5.085009549616079, 5.564244296597077, 6.144000569830323, 6.721790126284977, 7.165578670172772, 7.556505497346277, 8.090537807063128, 8.524309140513766, 9.026394354904563, 9.818739444519027, 10.00636780525771, 10.56165041467308, 10.83554698704193, 11.49647315090109, 11.85643098728433, 12.44064094230123, 12.54581601168682
