| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s + 4·11-s − 4·13-s − 15-s − 4·21-s + 2·23-s + 25-s + 27-s − 2·29-s − 6·31-s + 4·33-s + 4·35-s + 2·37-s − 4·39-s + 10·41-s − 2·43-s − 45-s + 9·49-s + 6·53-s − 4·55-s − 10·59-s − 10·61-s − 4·63-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 0.258·15-s − 0.872·21-s + 0.417·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.07·31-s + 0.696·33-s + 0.676·35-s + 0.328·37-s − 0.640·39-s + 1.56·41-s − 0.304·43-s − 0.149·45-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 1.30·59-s − 1.28·61-s − 0.503·63-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 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
−15.16592898418075, −14.75472831704718, −14.21107619532493, −13.69580671901936, −13.03872719700187, −12.55359740544316, −12.27512847929913, −11.63480855045934, −10.92646561872520, −10.40825001198182, −9.552158078738115, −9.394822654657739, −9.053116123507393, −8.194053005911676, −7.492660054359167, −7.131649206297823, −6.537746428198908, −6.023847442868681, −5.213478780625301, −4.389127680750349, −3.866159731465265, −3.307604286232078, −2.731561569111934, −1.967846079270709, −0.9236167939601515, 0,
0.9236167939601515, 1.967846079270709, 2.731561569111934, 3.307604286232078, 3.866159731465265, 4.389127680750349, 5.213478780625301, 6.023847442868681, 6.537746428198908, 7.131649206297823, 7.492660054359167, 8.194053005911676, 9.053116123507393, 9.394822654657739, 9.552158078738115, 10.40825001198182, 10.92646561872520, 11.63480855045934, 12.27512847929913, 12.55359740544316, 13.03872719700187, 13.69580671901936, 14.21107619532493, 14.75472831704718, 15.16592898418075
