| L(s) = 1 | − 3-s − 7-s + 9-s − 2·11-s − 2·13-s + 8·17-s − 2·19-s + 21-s − 27-s + 6·29-s − 6·31-s + 2·33-s − 8·37-s + 2·39-s + 6·41-s + 8·43-s − 4·47-s + 49-s − 8·51-s + 2·53-s + 2·57-s − 8·59-s − 10·61-s − 63-s − 12·67-s + 14·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.94·17-s − 0.458·19-s + 0.218·21-s − 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.348·33-s − 1.31·37-s + 0.320·39-s + 0.937·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s − 1.12·51-s + 0.274·53-s + 0.264·57-s − 1.04·59-s − 1.28·61-s − 0.125·63-s − 1.46·67-s + 1.66·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 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.ag |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−15.32433059167934, −14.68722439840101, −14.23447711161550, −13.75283888806561, −12.92793600690093, −12.61388943598187, −12.09525342164326, −11.76816047074256, −10.83165487480116, −10.43514294160287, −10.15840276812647, −9.331994930985513, −8.994591800303279, −8.013207559218460, −7.624597120465984, −7.173476744560128, −6.296342466035333, −5.933406699550861, −5.210576901594139, −4.837069471419365, −3.964519775955479, −3.286385661777856, −2.679957141118064, −1.764105399346455, −0.9006338875616792, 0,
0.9006338875616792, 1.764105399346455, 2.679957141118064, 3.286385661777856, 3.964519775955479, 4.837069471419365, 5.210576901594139, 5.933406699550861, 6.296342466035333, 7.173476744560128, 7.624597120465984, 8.013207559218460, 8.994591800303279, 9.331994930985513, 10.15840276812647, 10.43514294160287, 10.83165487480116, 11.76816047074256, 12.09525342164326, 12.61388943598187, 12.92793600690093, 13.75283888806561, 14.23447711161550, 14.68722439840101, 15.32433059167934
