| L(s) = 1 | + 2·3-s + 9-s − 11-s + 3·13-s + 4·17-s + 19-s − 3·23-s − 4·27-s − 5·29-s − 3·31-s − 2·33-s − 12·37-s + 6·39-s + 8·41-s − 5·43-s + 8·47-s − 7·49-s + 8·51-s − 10·53-s + 2·57-s − 8·59-s − 10·61-s + 14·67-s − 6·69-s − 5·71-s + 4·73-s − 8·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.970·17-s + 0.229·19-s − 0.625·23-s − 0.769·27-s − 0.928·29-s − 0.538·31-s − 0.348·33-s − 1.97·37-s + 0.960·39-s + 1.24·41-s − 0.762·43-s + 1.16·47-s − 49-s + 1.12·51-s − 1.37·53-s + 0.264·57-s − 1.04·59-s − 1.28·61-s + 1.71·67-s − 0.722·69-s − 0.593·71-s + 0.468·73-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.83777584867871, −15.63553918796968, −14.93211898446785, −14.34070073142401, −13.95509725551447, −13.60940186196324, −12.75708643745009, −12.48367171124581, −11.63519575324999, −11.04529168310686, −10.50146936958633, −9.714990221344994, −9.357190226513690, −8.646265201105130, −8.227946538332811, −7.608687237054009, −7.178922352125165, −6.153464002451589, −5.688330394346905, −4.942104971104290, −3.976864111289612, −3.450157638712218, −2.966437141975170, −2.004784107612482, −1.400585189027015, 0,
1.400585189027015, 2.004784107612482, 2.966437141975170, 3.450157638712218, 3.976864111289612, 4.942104971104290, 5.688330394346905, 6.153464002451589, 7.178922352125165, 7.608687237054009, 8.227946538332811, 8.646265201105130, 9.357190226513690, 9.714990221344994, 10.50146936958633, 11.04529168310686, 11.63519575324999, 12.48367171124581, 12.75708643745009, 13.60940186196324, 13.95509725551447, 14.34070073142401, 14.93211898446785, 15.63553918796968, 15.83777584867871
