| L(s) = 1 | + 5-s − 2·7-s − 3·9-s + 11-s + 8·19-s − 8·23-s + 25-s − 10·29-s + 8·31-s − 2·35-s + 10·37-s − 2·41-s + 6·43-s − 3·45-s − 8·47-s − 3·49-s − 14·53-s + 55-s + 4·59-s − 10·61-s + 6·63-s − 4·67-s − 8·73-s − 2·77-s − 4·79-s + 9·81-s − 10·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 9-s + 0.301·11-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 1.85·29-s + 1.43·31-s − 0.338·35-s + 1.64·37-s − 0.312·41-s + 0.914·43-s − 0.447·45-s − 1.16·47-s − 3/7·49-s − 1.92·53-s + 0.134·55-s + 0.520·59-s − 1.28·61-s + 0.755·63-s − 0.488·67-s − 0.936·73-s − 0.227·77-s − 0.450·79-s + 81-s − 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.073004415109078395166753955639, −7.62383282251326668192601549484, −6.49296184978571056212601605530, −5.98583751713496721947094788282, −5.38495677956491360703357103932, −4.29614249789034997459872502092, −3.30233480408079732148149093321, −2.69378417609535220658682224751, −1.45771330037166308796265906162, 0,
1.45771330037166308796265906162, 2.69378417609535220658682224751, 3.30233480408079732148149093321, 4.29614249789034997459872502092, 5.38495677956491360703357103932, 5.98583751713496721947094788282, 6.49296184978571056212601605530, 7.62383282251326668192601549484, 8.073004415109078395166753955639
