| L(s) = 1 | − 3-s − 2·5-s + 9-s − 11-s − 2·13-s + 2·15-s − 6·17-s − 8·19-s − 4·23-s − 25-s − 27-s + 2·29-s + 8·31-s + 33-s + 6·37-s + 2·39-s − 6·41-s − 8·43-s − 2·45-s + 4·47-s + 6·51-s + 10·53-s + 2·55-s + 8·57-s + 4·59-s + 14·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 1.83·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s + 0.174·33-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.298·45-s + 0.583·47-s + 0.840·51-s + 1.37·53-s + 0.269·55-s + 1.05·57-s + 0.520·59-s + 1.79·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−15.50343390471146, −15.17005088693024, −14.81768563701599, −13.81842103911155, −13.46428983958075, −12.77131944497331, −12.36906788659783, −11.67468684402270, −11.44478014255603, −10.75972228653917, −10.21511624338760, −9.792760373496632, −8.858958194995384, −8.282889227124715, −8.024341957723153, −7.117471664123332, −6.568362415127132, −6.254755896238647, −5.260121746896808, −4.688392944164633, −4.147396112269810, −3.674607963825100, −2.399007564360478, −2.154395409184630, −0.7168042214555630, 0,
0.7168042214555630, 2.154395409184630, 2.399007564360478, 3.674607963825100, 4.147396112269810, 4.688392944164633, 5.260121746896808, 6.254755896238647, 6.568362415127132, 7.117471664123332, 8.024341957723153, 8.282889227124715, 8.858958194995384, 9.792760373496632, 10.21511624338760, 10.75972228653917, 11.44478014255603, 11.67468684402270, 12.36906788659783, 12.77131944497331, 13.46428983958075, 13.81842103911155, 14.81768563701599, 15.17005088693024, 15.50343390471146
