| L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s + 11-s − 4·13-s + 2·15-s − 2·17-s + 4·19-s − 2·21-s − 4·23-s − 25-s − 27-s + 10·29-s − 33-s − 4·35-s + 37-s + 4·39-s − 4·43-s − 2·45-s − 3·49-s + 2·51-s − 2·53-s − 2·55-s − 4·57-s − 8·59-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.174·33-s − 0.676·35-s + 0.164·37-s + 0.640·39-s − 0.609·43-s − 0.298·45-s − 3/7·49-s + 0.280·51-s − 0.274·53-s − 0.269·55-s − 0.529·57-s − 1.04·59-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.16769012993441, −13.97401792692482, −13.26333470868294, −12.43682334946380, −12.28629115748270, −11.68060193913119, −11.45457293154934, −10.96272404936820, −10.16386605385880, −9.933020755635346, −9.298975283537446, −8.494112328640070, −8.185733818724952, −7.532176998068722, −7.259921068235274, −6.523919434934634, −6.062124379521489, −5.256491192392918, −4.748054495791763, −4.474902429848697, −3.749198986375890, −3.079827245860469, −2.312307396035664, −1.591528078616956, −0.7802815083214842, 0,
0.7802815083214842, 1.591528078616956, 2.312307396035664, 3.079827245860469, 3.749198986375890, 4.474902429848697, 4.748054495791763, 5.256491192392918, 6.062124379521489, 6.523919434934634, 7.259921068235274, 7.532176998068722, 8.185733818724952, 8.494112328640070, 9.298975283537446, 9.933020755635346, 10.16386605385880, 10.96272404936820, 11.45457293154934, 11.68060193913119, 12.28629115748270, 12.43682334946380, 13.26333470868294, 13.97401792692482, 14.16769012993441
