| L(s) = 1 | + 5-s − 3·9-s − 2·13-s − 6·17-s − 4·19-s − 4·23-s + 25-s − 6·29-s + 8·31-s − 2·37-s − 2·41-s + 4·43-s − 3·45-s + 12·47-s − 7·49-s − 2·53-s − 4·59-s + 10·61-s − 2·65-s + 16·67-s − 8·71-s − 14·73-s + 8·79-s + 9·81-s − 4·83-s − 6·85-s + 10·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 9-s − 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s + 0.609·43-s − 0.447·45-s + 1.75·47-s − 49-s − 0.274·53-s − 0.520·59-s + 1.28·61-s − 0.248·65-s + 1.95·67-s − 0.949·71-s − 1.63·73-s + 0.900·79-s + 81-s − 0.439·83-s − 0.650·85-s + 1.05·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.240622525\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240622525\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.69556880125895993906919048601, −6.92114469673781934514976443303, −6.21371933475241794414167140387, −5.79962506110486358982246461760, −4.87966949514469953615819754787, −4.30523069678046427722855688335, −3.37416713721777180685271968198, −2.37289889085752151543981096662, −2.04716871831388657730114879319, −0.49631415514383905112999669566,
0.49631415514383905112999669566, 2.04716871831388657730114879319, 2.37289889085752151543981096662, 3.37416713721777180685271968198, 4.30523069678046427722855688335, 4.87966949514469953615819754787, 5.79962506110486358982246461760, 6.21371933475241794414167140387, 6.92114469673781934514976443303, 7.69556880125895993906919048601
