| L(s) = 1 | − 5-s − 7-s − 11-s + 13-s − 17-s + 4·23-s + 25-s − 29-s − 10·31-s + 35-s + 8·37-s − 2·41-s − 4·43-s + 11·47-s + 49-s + 4·53-s + 55-s − 8·59-s − 10·61-s − 65-s − 4·67-s + 12·71-s + 6·73-s + 77-s − 7·79-s + 8·83-s + 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.301·11-s + 0.277·13-s − 0.242·17-s + 0.834·23-s + 1/5·25-s − 0.185·29-s − 1.79·31-s + 0.169·35-s + 1.31·37-s − 0.312·41-s − 0.609·43-s + 1.60·47-s + 1/7·49-s + 0.549·53-s + 0.134·55-s − 1.04·59-s − 1.28·61-s − 0.124·65-s − 0.488·67-s + 1.42·71-s + 0.702·73-s + 0.113·77-s − 0.787·79-s + 0.878·83-s + 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - T + p T^{2} \) | 1.97.ab |
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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
−15.87601093027008, −15.35341589686052, −14.94979658573733, −14.37821158118494, −13.59674740455603, −13.24206894226392, −12.61567381880471, −12.19577235461641, −11.44942424391362, −10.90854439240280, −10.58123772960752, −9.743940153130950, −9.103095491648532, −8.814353137448339, −7.870823837254521, −7.515169325136221, −6.840786164114800, −6.200965977578187, −5.524788186696835, −4.891028035565079, −4.120357731432609, −3.522783591011590, −2.832744728564297, −2.016337503413102, −0.9924089833752980, 0,
0.9924089833752980, 2.016337503413102, 2.832744728564297, 3.522783591011590, 4.120357731432609, 4.891028035565079, 5.524788186696835, 6.200965977578187, 6.840786164114800, 7.515169325136221, 7.870823837254521, 8.814353137448339, 9.103095491648532, 9.743940153130950, 10.58123772960752, 10.90854439240280, 11.44942424391362, 12.19577235461641, 12.61567381880471, 13.24206894226392, 13.59674740455603, 14.37821158118494, 14.94979658573733, 15.35341589686052, 15.87601093027008
