| L(s) = 1 | + 5-s + 4·11-s − 2·17-s + 6·19-s + 6·23-s + 25-s − 2·31-s + 2·37-s − 2·41-s + 4·43-s − 8·47-s + 10·53-s + 4·55-s − 4·59-s + 2·61-s + 12·67-s + 8·71-s − 8·73-s − 8·79-s + 4·83-s − 2·85-s − 10·89-s + 6·95-s + 4·97-s + 18·101-s − 4·103-s + 6·107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s − 0.485·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.359·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1.37·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 1.46·67-s + 0.949·71-s − 0.936·73-s − 0.900·79-s + 0.439·83-s − 0.216·85-s − 1.05·89-s + 0.615·95-s + 0.406·97-s + 1.79·101-s − 0.394·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.732845846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.732845846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−7.66796039663428013290022451827, −6.93206142629651974387235581983, −6.53725707498444511149659368832, −5.62877323767792275457758067194, −5.08042818145001392780789886937, −4.21670746443051937372736235715, −3.45438029737175729648305046798, −2.67507489003143511474590574053, −1.63629041099122014857726910967, −0.860083883234595553970752632382,
0.860083883234595553970752632382, 1.63629041099122014857726910967, 2.67507489003143511474590574053, 3.45438029737175729648305046798, 4.21670746443051937372736235715, 5.08042818145001392780789886937, 5.62877323767792275457758067194, 6.53725707498444511149659368832, 6.93206142629651974387235581983, 7.66796039663428013290022451827
