| L(s) = 1 | + 3-s + 4·5-s + 9-s + 2·13-s + 4·15-s − 6·23-s + 11·25-s + 27-s − 4·29-s + 37-s + 2·39-s + 2·41-s − 12·43-s + 4·45-s + 8·47-s − 7·49-s + 2·53-s + 6·59-s − 10·61-s + 8·65-s − 4·67-s − 6·69-s − 8·71-s + 6·73-s + 11·75-s + 4·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 0.554·13-s + 1.03·15-s − 1.25·23-s + 11/5·25-s + 0.192·27-s − 0.742·29-s + 0.164·37-s + 0.320·39-s + 0.312·41-s − 1.82·43-s + 0.596·45-s + 1.16·47-s − 49-s + 0.274·53-s + 0.781·59-s − 1.28·61-s + 0.992·65-s − 0.488·67-s − 0.722·69-s − 0.949·71-s + 0.702·73-s + 1.27·75-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.607301085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.607301085\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−10.03541314056879586505466729776, −9.326276358729378913599212691706, −8.663131876614315022997990473138, −7.65068985974677607576378335214, −6.49112050108937775107194549006, −5.90300642646781966838078609484, −4.92491531080994339658250546487, −3.60918115277375939687709761101, −2.39893627860958541259989683925, −1.55375785719617605512591602936,
1.55375785719617605512591602936, 2.39893627860958541259989683925, 3.60918115277375939687709761101, 4.92491531080994339658250546487, 5.90300642646781966838078609484, 6.49112050108937775107194549006, 7.65068985974677607576378335214, 8.663131876614315022997990473138, 9.326276358729378913599212691706, 10.03541314056879586505466729776
