| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s + 6·11-s + 13-s − 16-s + 4·17-s + 4·19-s − 20-s − 6·22-s + 25-s − 26-s − 4·29-s − 5·32-s − 4·34-s + 8·37-s − 4·38-s + 3·40-s + 6·41-s + 2·43-s − 6·44-s − 2·47-s − 50-s − 52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 1.80·11-s + 0.277·13-s − 1/4·16-s + 0.970·17-s + 0.917·19-s − 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.196·26-s − 0.742·29-s − 0.883·32-s − 0.685·34-s + 1.31·37-s − 0.648·38-s + 0.474·40-s + 0.937·41-s + 0.304·43-s − 0.904·44-s − 0.291·47-s − 0.141·50-s − 0.138·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.171207067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.171207067\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−14.94592610730018, −14.62780648944623, −14.15986403185442, −13.70090784778651, −13.08855102591104, −12.61508619308808, −11.74369892498967, −11.56781784128011, −10.77346171416092, −10.10492365208316, −9.693748658878026, −9.168469597856171, −8.908565040777614, −8.143864522857227, −7.486697168100683, −7.108811610500319, −6.170415723704214, −5.803162805653339, −5.037010325249706, −4.262418168267092, −3.798677961456105, −3.064816844950615, −1.964036434796249, −1.214088471362279, −0.7816968203664426,
0.7816968203664426, 1.214088471362279, 1.964036434796249, 3.064816844950615, 3.798677961456105, 4.262418168267092, 5.037010325249706, 5.803162805653339, 6.170415723704214, 7.108811610500319, 7.486697168100683, 8.143864522857227, 8.908565040777614, 9.168469597856171, 9.693748658878026, 10.10492365208316, 10.77346171416092, 11.56781784128011, 11.74369892498967, 12.61508619308808, 13.08855102591104, 13.70090784778651, 14.15986403185442, 14.62780648944623, 14.94592610730018
