| L(s) = 1 | + 5-s + 7-s − 13-s + 6·17-s − 4·23-s + 25-s − 2·29-s + 4·31-s + 35-s + 10·37-s − 2·41-s − 8·43-s + 49-s − 2·53-s + 2·61-s − 65-s − 4·67-s + 12·71-s − 6·73-s − 8·79-s − 4·83-s + 6·85-s − 2·89-s − 91-s − 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.277·13-s + 1.45·17-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s + 0.169·35-s + 1.64·37-s − 0.312·41-s − 1.21·43-s + 1/7·49-s − 0.274·53-s + 0.256·61-s − 0.124·65-s − 0.488·67-s + 1.42·71-s − 0.702·73-s − 0.900·79-s − 0.439·83-s + 0.650·85-s − 0.211·89-s − 0.104·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 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.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.99607806488588, −12.62051360921243, −12.00482570186643, −11.76722792117928, −11.24960744377032, −10.70286263555995, −10.10280275788888, −9.894809570495536, −9.471831550791035, −8.876729213837648, −8.206123638509486, −7.986573296661607, −7.507809290197172, −6.894476565564966, −6.340474543266306, −5.928613288145185, −5.300213180657667, −5.091139646320701, −4.237569675241896, −3.964176215484576, −3.091165881875833, −2.778695039267752, −2.022645746867776, −1.463568736457600, −0.9069559459831621, 0,
0.9069559459831621, 1.463568736457600, 2.022645746867776, 2.778695039267752, 3.091165881875833, 3.964176215484576, 4.237569675241896, 5.091139646320701, 5.300213180657667, 5.928613288145185, 6.340474543266306, 6.894476565564966, 7.507809290197172, 7.986573296661607, 8.206123638509486, 8.876729213837648, 9.471831550791035, 9.894809570495536, 10.10280275788888, 10.70286263555995, 11.24960744377032, 11.76722792117928, 12.00482570186643, 12.62051360921243, 12.99607806488588
