| L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s − 2·10-s + 11-s − 13-s − 16-s − 6·17-s + 4·19-s + 2·20-s + 22-s + 8·23-s − 25-s − 26-s + 10·29-s + 5·32-s − 6·34-s + 6·37-s + 4·38-s + 6·40-s + 10·41-s + 4·43-s − 44-s + 8·46-s + 8·47-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s − 0.632·10-s + 0.301·11-s − 0.277·13-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.447·20-s + 0.213·22-s + 1.66·23-s − 1/5·25-s − 0.196·26-s + 1.85·29-s + 0.883·32-s − 1.02·34-s + 0.986·37-s + 0.648·38-s + 0.948·40-s + 1.56·41-s + 0.609·43-s − 0.150·44-s + 1.17·46-s + 1.16·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63063 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63063 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.255618918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.255618918\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 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.ak |
| 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 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.09151159117557, −13.75649329283915, −13.36884691531430, −12.70373793921179, −12.27438820088887, −11.89759307798313, −11.34934394668559, −10.82249666304682, −10.33301547705350, −9.357752960160030, −9.144162002547139, −8.774210397598662, −7.924316089960056, −7.554803633118058, −6.917281013310072, −6.265776248918129, −5.830011976945118, −4.943406061691046, −4.580325945466165, −4.230831622243434, −3.512135639698846, −2.878098742567390, −2.427335175381795, −1.078145068200788, −0.5426099377477248,
0.5426099377477248, 1.078145068200788, 2.427335175381795, 2.878098742567390, 3.512135639698846, 4.230831622243434, 4.580325945466165, 4.943406061691046, 5.830011976945118, 6.265776248918129, 6.917281013310072, 7.554803633118058, 7.924316089960056, 8.774210397598662, 9.144162002547139, 9.357752960160030, 10.33301547705350, 10.82249666304682, 11.34934394668559, 11.89759307798313, 12.27438820088887, 12.70373793921179, 13.36884691531430, 13.75649329283915, 14.09151159117557
