| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s + 4·11-s − 13-s + 16-s − 2·17-s + 4·19-s − 2·20-s + 4·22-s + 4·23-s − 25-s − 26-s + 2·29-s + 32-s − 2·34-s − 2·37-s + 4·38-s − 2·40-s + 2·41-s + 4·43-s + 4·44-s + 4·46-s − 12·47-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s + 1.20·11-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.447·20-s + 0.852·22-s + 0.834·23-s − 1/5·25-s − 0.196·26-s + 0.371·29-s + 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.648·38-s − 0.316·40-s + 0.312·41-s + 0.609·43-s + 0.603·44-s + 0.589·46-s − 1.75·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.045547784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.045547784\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 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.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−16.23501398980454, −15.87368340281751, −15.15954328097834, −14.84582301723071, −14.05194945427202, −13.82634209366865, −12.84177533573739, −12.51221011402026, −11.76336961463657, −11.41963733947729, −11.00215408312169, −10.03207374511305, −9.440862994706839, −8.796526948706556, −8.004273935613916, −7.488476012214550, −6.706501279057586, −6.401076586760148, −5.320894532838322, −4.819906696400552, −3.999475383644771, −3.567420791014992, −2.775830327509546, −1.735445755461261, −0.7358066219325504,
0.7358066219325504, 1.735445755461261, 2.775830327509546, 3.567420791014992, 3.999475383644771, 4.819906696400552, 5.320894532838322, 6.401076586760148, 6.706501279057586, 7.488476012214550, 8.004273935613916, 8.796526948706556, 9.440862994706839, 10.03207374511305, 11.00215408312169, 11.41963733947729, 11.76336961463657, 12.51221011402026, 12.84177533573739, 13.82634209366865, 14.05194945427202, 14.84582301723071, 15.15954328097834, 15.87368340281751, 16.23501398980454
