| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s − 4·11-s + 2·13-s − 4·14-s − 16-s + 6·17-s − 4·19-s + 4·22-s − 2·26-s − 4·28-s − 6·29-s − 5·32-s − 6·34-s + 10·37-s + 4·38-s + 6·41-s − 12·43-s + 4·44-s + 9·49-s − 2·52-s + 2·53-s + 12·56-s + 6·58-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s − 1.20·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.852·22-s − 0.392·26-s − 0.755·28-s − 1.11·29-s − 0.883·32-s − 1.02·34-s + 1.64·37-s + 0.648·38-s + 0.937·41-s − 1.82·43-s + 0.603·44-s + 9/7·49-s − 0.277·52-s + 0.274·53-s + 1.60·56-s + 0.787·58-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 31 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.16795143367172, −12.84344781518091, −12.35831602368008, −11.59292678073140, −11.24204911154939, −10.87071633695502, −10.34001836414714, −10.01597144123018, −9.471256871572980, −8.883996772024943, −8.405847480996443, −8.057731497751326, −7.665487175404254, −7.464644121630918, −6.568860378632595, −5.827808981989517, −5.458016689277529, −4.920250602794521, −4.575924437091968, −3.897907991509357, −3.421639798753338, −2.497198422176868, −1.996818101789398, −1.339046699993333, −0.8427063441357958, 0,
0.8427063441357958, 1.339046699993333, 1.996818101789398, 2.497198422176868, 3.421639798753338, 3.897907991509357, 4.575924437091968, 4.920250602794521, 5.458016689277529, 5.827808981989517, 6.568860378632595, 7.464644121630918, 7.665487175404254, 8.057731497751326, 8.405847480996443, 8.883996772024943, 9.471256871572980, 10.01597144123018, 10.34001836414714, 10.87071633695502, 11.24204911154939, 11.59292678073140, 12.35831602368008, 12.84344781518091, 13.16795143367172
