| L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s − 6·11-s − 13-s + 16-s − 3·17-s − 2·19-s − 3·20-s − 6·22-s + 4·25-s − 26-s − 6·29-s + 4·31-s + 32-s − 3·34-s − 7·37-s − 2·38-s − 3·40-s − 43-s − 6·44-s + 3·47-s + 4·50-s − 52-s + 18·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s − 1.80·11-s − 0.277·13-s + 1/4·16-s − 0.727·17-s − 0.458·19-s − 0.670·20-s − 1.27·22-s + 4/5·25-s − 0.196·26-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.514·34-s − 1.15·37-s − 0.324·38-s − 0.474·40-s − 0.152·43-s − 0.904·44-s + 0.437·47-s + 0.565·50-s − 0.138·52-s + 2.42·55-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
| \(L(1)\) |
\(\approx\) |
\(1.012572441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012572441\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.15426388221805, −15.69568757854266, −15.32984193310870, −14.99579272241240, −14.14384385699382, −13.44310401436843, −13.09319309422100, −12.28494425734336, −12.12785726025217, −11.11618964892605, −10.93633242047812, −10.30575510035556, −9.462672275250506, −8.536902758946745, −8.038796962133949, −7.526513265431088, −6.977497322424188, −6.169530127661402, −5.286114768065388, −4.840240904771855, −4.123186683165709, −3.456867024218628, −2.704259344093753, −1.970244998707438, −0.3864530768850686,
0.3864530768850686, 1.970244998707438, 2.704259344093753, 3.456867024218628, 4.123186683165709, 4.840240904771855, 5.286114768065388, 6.169530127661402, 6.977497322424188, 7.526513265431088, 8.038796962133949, 8.536902758946745, 9.462672275250506, 10.30575510035556, 10.93633242047812, 11.11618964892605, 12.12785726025217, 12.28494425734336, 13.09319309422100, 13.44310401436843, 14.14384385699382, 14.99579272241240, 15.32984193310870, 15.69568757854266, 16.15426388221805
