| L(s) = 1 | + 5-s + 2·7-s + 2·11-s − 2·13-s + 4·23-s + 25-s − 6·29-s + 4·31-s + 2·35-s − 8·37-s − 8·41-s + 43-s − 4·47-s − 3·49-s − 2·53-s + 2·55-s + 2·59-s + 8·61-s − 2·65-s + 12·67-s + 8·71-s − 2·73-s + 4·77-s − 8·79-s − 6·83-s + 6·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.603·11-s − 0.554·13-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.338·35-s − 1.31·37-s − 1.24·41-s + 0.152·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s + 0.269·55-s + 0.260·59-s + 1.02·61-s − 0.248·65-s + 1.46·67-s + 0.949·71-s − 0.234·73-s + 0.455·77-s − 0.900·79-s − 0.658·83-s + 0.635·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−13.83788860567732, −13.26446463115097, −12.89070474307390, −12.26257701803586, −11.87111449029865, −11.26217043206898, −11.04661228263248, −10.33386027317540, −9.837379250309378, −9.471014451889736, −8.867581510761985, −8.360172750775951, −8.003586512126080, −7.194997213167538, −6.863009702550833, −6.422657417032790, −5.540602837717206, −5.261897535790929, −4.754663233961795, −4.104792836301411, −3.485759211275005, −2.883694425166732, −2.092173962910292, −1.659796979108062, −0.9862334735061494, 0,
0.9862334735061494, 1.659796979108062, 2.092173962910292, 2.883694425166732, 3.485759211275005, 4.104792836301411, 4.754663233961795, 5.261897535790929, 5.540602837717206, 6.422657417032790, 6.863009702550833, 7.194997213167538, 8.003586512126080, 8.360172750775951, 8.867581510761985, 9.471014451889736, 9.837379250309378, 10.33386027317540, 11.04661228263248, 11.26217043206898, 11.87111449029865, 12.26257701803586, 12.89070474307390, 13.26446463115097, 13.83788860567732
