| L(s) = 1 | + 3·5-s − 7-s + 3·11-s − 13-s − 7·19-s − 23-s + 4·25-s + 2·29-s − 4·31-s − 3·35-s + 4·37-s − 7·41-s − 5·43-s + 4·47-s + 49-s + 4·53-s + 9·55-s − 12·59-s + 12·61-s − 3·65-s + 8·67-s + 4·71-s − 16·73-s − 3·77-s − 4·79-s + 16·83-s + 91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 0.904·11-s − 0.277·13-s − 1.60·19-s − 0.208·23-s + 4/5·25-s + 0.371·29-s − 0.718·31-s − 0.507·35-s + 0.657·37-s − 1.09·41-s − 0.762·43-s + 0.583·47-s + 1/7·49-s + 0.549·53-s + 1.21·55-s − 1.56·59-s + 1.53·61-s − 0.372·65-s + 0.977·67-s + 0.474·71-s − 1.87·73-s − 0.341·77-s − 0.450·79-s + 1.75·83-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.661742309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.661742309\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−14.20285646511399, −13.53701776169394, −13.04717574127122, −12.85508759080233, −12.02692686802321, −11.76970526837103, −10.92858109900743, −10.49911677277416, −10.03497321552204, −9.566842526088372, −9.077767887466242, −8.650220699534151, −8.067436612249274, −7.249824386851912, −6.650683262325393, −6.371154118944900, −5.842302252182183, −5.269307596339400, −4.592391920878602, −4.000548498825961, −3.359562043988918, −2.544452721359389, −2.012565477759382, −1.506940891677032, −0.5194067772780255,
0.5194067772780255, 1.506940891677032, 2.012565477759382, 2.544452721359389, 3.359562043988918, 4.000548498825961, 4.592391920878602, 5.269307596339400, 5.842302252182183, 6.371154118944900, 6.650683262325393, 7.249824386851912, 8.067436612249274, 8.650220699534151, 9.077767887466242, 9.566842526088372, 10.03497321552204, 10.49911677277416, 10.92858109900743, 11.76970526837103, 12.02692686802321, 12.85508759080233, 13.04717574127122, 13.53701776169394, 14.20285646511399
