| L(s) = 1 | − 2-s + 4-s + 4·5-s − 8-s − 4·10-s + 4·11-s − 13-s + 16-s − 2·17-s + 4·19-s + 4·20-s − 4·22-s − 2·23-s + 11·25-s + 26-s − 6·29-s − 4·31-s − 32-s + 2·34-s + 4·37-s − 4·38-s − 4·40-s + 6·41-s − 4·43-s + 4·44-s + 2·46-s − 11·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.353·8-s − 1.26·10-s + 1.20·11-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.894·20-s − 0.852·22-s − 0.417·23-s + 11/5·25-s + 0.196·26-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.342·34-s + 0.657·37-s − 0.648·38-s − 0.632·40-s + 0.937·41-s − 0.609·43-s + 0.603·44-s + 0.294·46-s − 1.55·50-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
| \(L(1)\) |
\(\approx\) |
\(2.511128048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.511128048\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−16.59249352437820, −16.20343866133380, −15.23197222043570, −14.67523471130990, −14.15803884928300, −13.72661946075261, −12.96171510394202, −12.56501276066892, −11.59532290323907, −11.26049519567206, −10.44670345040291, −9.853521409006295, −9.394055999514484, −9.123916129300815, −8.365986728836603, −7.424175384039495, −6.881638826826102, −6.215331296653371, −5.735424692226267, −5.072238933736294, −4.058530445603365, −3.138501949592007, −2.195865294904756, −1.726958895111398, −0.8412634021820966,
0.8412634021820966, 1.726958895111398, 2.195865294904756, 3.138501949592007, 4.058530445603365, 5.072238933736294, 5.735424692226267, 6.215331296653371, 6.881638826826102, 7.424175384039495, 8.365986728836603, 9.123916129300815, 9.394055999514484, 9.853521409006295, 10.44670345040291, 11.26049519567206, 11.59532290323907, 12.56501276066892, 12.96171510394202, 13.72661946075261, 14.15803884928300, 14.67523471130990, 15.23197222043570, 16.20343866133380, 16.59249352437820
