| L(s) = 1 | + 3-s − 3·5-s + 4·7-s − 2·9-s + 2·13-s − 3·15-s + 8·17-s − 6·19-s + 4·21-s + 5·23-s + 4·25-s − 5·27-s − 4·29-s + 31-s − 12·35-s + 3·37-s + 2·39-s + 6·41-s + 6·43-s + 6·45-s − 12·47-s + 9·49-s + 8·51-s − 6·53-s − 6·57-s + 3·59-s − 8·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1.51·7-s − 2/3·9-s + 0.554·13-s − 0.774·15-s + 1.94·17-s − 1.37·19-s + 0.872·21-s + 1.04·23-s + 4/5·25-s − 0.962·27-s − 0.742·29-s + 0.179·31-s − 2.02·35-s + 0.493·37-s + 0.320·39-s + 0.937·41-s + 0.914·43-s + 0.894·45-s − 1.75·47-s + 9/7·49-s + 1.12·51-s − 0.824·53-s − 0.794·57-s + 0.390·59-s − 1.00·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.105012857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.105012857\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−8.356091227631432112744147565352, −7.80600810663169791652738346727, −7.50198472674154373278179225694, −6.24541169109682856670095677012, −5.35473284436664335832114757918, −4.60055864279777275442227190873, −3.78378333009399584673787262577, −3.14981024706725465946793634323, −2.00046778372803770255204265095, −0.836282176392189959886657427424,
0.836282176392189959886657427424, 2.00046778372803770255204265095, 3.14981024706725465946793634323, 3.78378333009399584673787262577, 4.60055864279777275442227190873, 5.35473284436664335832114757918, 6.24541169109682856670095677012, 7.50198472674154373278179225694, 7.80600810663169791652738346727, 8.356091227631432112744147565352
