| L(s) = 1 | − 2·4-s − 2·7-s − 2·13-s + 4·16-s − 7·17-s − 6·19-s + 4·28-s − 7·29-s + 3·37-s − 5·43-s − 3·49-s + 4·52-s − 7·53-s + 14·59-s + 61-s − 8·64-s − 2·67-s + 14·68-s + 7·71-s + 3·73-s + 12·76-s − 10·79-s − 14·83-s − 7·89-s + 4·91-s + 12·97-s + 101-s + ⋯ |
| L(s) = 1 | − 4-s − 0.755·7-s − 0.554·13-s + 16-s − 1.69·17-s − 1.37·19-s + 0.755·28-s − 1.29·29-s + 0.493·37-s − 0.762·43-s − 3/7·49-s + 0.554·52-s − 0.961·53-s + 1.82·59-s + 0.128·61-s − 64-s − 0.244·67-s + 1.69·68-s + 0.830·71-s + 0.351·73-s + 1.37·76-s − 1.12·79-s − 1.53·83-s − 0.741·89-s + 0.419·91-s + 1.21·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 41 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−12.82436823582937, −12.55635611751393, −11.74787003172042, −11.32620152899394, −10.95916062971060, −10.19118311495740, −10.09488787878044, −9.468406541593388, −9.034074733815548, −8.792068765290938, −8.200065770026286, −7.815984370260016, −7.088059176283905, −6.706103685889185, −6.270297290339822, −5.739390622617197, −5.146672188946981, −4.703287493852727, −4.153785583188022, −3.881172905158931, −3.231520406607339, −2.555811890168406, −2.092310644711255, −1.409043392901820, −0.4012425844559446, 0,
0.4012425844559446, 1.409043392901820, 2.092310644711255, 2.555811890168406, 3.231520406607339, 3.881172905158931, 4.153785583188022, 4.703287493852727, 5.146672188946981, 5.739390622617197, 6.270297290339822, 6.706103685889185, 7.088059176283905, 7.815984370260016, 8.200065770026286, 8.792068765290938, 9.034074733815548, 9.468406541593388, 10.09488787878044, 10.19118311495740, 10.95916062971060, 11.32620152899394, 11.74787003172042, 12.55635611751393, 12.82436823582937
