| L(s) = 1 | + 2·7-s + 11-s − 8·19-s − 8·23-s − 10·29-s + 8·31-s + 10·37-s + 2·41-s + 6·43-s − 8·47-s − 3·49-s + 14·53-s + 4·59-s + 10·61-s − 4·67-s + 8·73-s + 2·77-s − 4·79-s + 10·83-s − 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s − 1.83·19-s − 1.66·23-s − 1.85·29-s + 1.43·31-s + 1.64·37-s + 0.312·41-s + 0.914·43-s − 1.16·47-s − 3/7·49-s + 1.92·53-s + 0.520·59-s + 1.28·61-s − 0.488·67-s + 0.936·73-s + 0.227·77-s − 0.450·79-s + 1.09·83-s − 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 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
−16.01779293295251, −15.18778815757003, −14.87845040722049, −14.42229919687073, −13.84351109208128, −13.08101693495940, −12.83239165731789, −11.97735603237845, −11.53935588770816, −11.07114759662003, −10.39534376619484, −9.880150717904109, −9.231279729677794, −8.537494990915650, −8.045476337264412, −7.629886438998717, −6.703171222760361, −6.193206427869566, −5.628580184621927, −4.797256323116616, −4.093135703792394, −3.826819864387522, −2.459708852759450, −2.124390536110365, −1.140899384722113, 0,
1.140899384722113, 2.124390536110365, 2.459708852759450, 3.826819864387522, 4.093135703792394, 4.797256323116616, 5.628580184621927, 6.193206427869566, 6.703171222760361, 7.629886438998717, 8.045476337264412, 8.537494990915650, 9.231279729677794, 9.880150717904109, 10.39534376619484, 11.07114759662003, 11.53935588770816, 11.97735603237845, 12.83239165731789, 13.08101693495940, 13.84351109208128, 14.42229919687073, 14.87845040722049, 15.18778815757003, 16.01779293295251
