| L(s) = 1 | − 2·4-s + 7-s − 3·11-s + 4·13-s + 4·16-s − 3·17-s + 19-s − 2·28-s − 6·29-s − 4·31-s − 2·37-s + 6·41-s + 43-s + 6·44-s − 3·47-s − 6·49-s − 8·52-s + 12·53-s + 6·59-s − 61-s − 8·64-s + 4·67-s + 6·68-s − 6·71-s + 7·73-s − 2·76-s − 3·77-s + ⋯ |
| L(s) = 1 | − 4-s + 0.377·7-s − 0.904·11-s + 1.10·13-s + 16-s − 0.727·17-s + 0.229·19-s − 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s + 0.152·43-s + 0.904·44-s − 0.437·47-s − 6/7·49-s − 1.10·52-s + 1.64·53-s + 0.781·59-s − 0.128·61-s − 64-s + 0.488·67-s + 0.727·68-s − 0.712·71-s + 0.819·73-s − 0.229·76-s − 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 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.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−8.119095666179137364472777237002, −7.49798376596251597554397220643, −6.51369974795999795736995570396, −5.54613367347237377002063134384, −5.17199046098683380620115379184, −4.14903220539879145832106272998, −3.62000165820744285556163438193, −2.45040466975635997887579513137, −1.28001607419382822143833582519, 0,
1.28001607419382822143833582519, 2.45040466975635997887579513137, 3.62000165820744285556163438193, 4.14903220539879145832106272998, 5.17199046098683380620115379184, 5.54613367347237377002063134384, 6.51369974795999795736995570396, 7.49798376596251597554397220643, 8.119095666179137364472777237002
