| L(s) = 1 | + 4·7-s + 4·11-s + 2·17-s − 19-s + 2·23-s − 5·25-s − 6·29-s + 6·31-s + 8·37-s − 10·41-s + 12·43-s − 10·47-s + 9·49-s + 2·53-s + 4·59-s + 10·61-s + 16·71-s − 2·73-s + 16·77-s + 10·79-s − 16·83-s + 2·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.20·11-s + 0.485·17-s − 0.229·19-s + 0.417·23-s − 25-s − 1.11·29-s + 1.07·31-s + 1.31·37-s − 1.56·41-s + 1.82·43-s − 1.45·47-s + 9/7·49-s + 0.274·53-s + 0.520·59-s + 1.28·61-s + 1.89·71-s − 0.234·73-s + 1.82·77-s + 1.12·79-s − 1.75·83-s + 0.211·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.069589008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.069589008\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−16.75157824621791, −15.89900368030127, −15.20475955736730, −14.71229708847080, −14.37778008619741, −13.75802097382361, −13.15972686337792, −12.32394438444584, −11.82536608120132, −11.19792777156100, −11.06292867826007, −9.881209164096606, −9.626338882787104, −8.682145773987507, −8.254768440674015, −7.648329412757399, −6.974312946640946, −6.207821907449775, −5.524999049433093, −4.821069140655545, −4.159300327084081, −3.553915959144130, −2.377616113547066, −1.646357527889799, −0.8787371008903181,
0.8787371008903181, 1.646357527889799, 2.377616113547066, 3.553915959144130, 4.159300327084081, 4.821069140655545, 5.524999049433093, 6.207821907449775, 6.974312946640946, 7.648329412757399, 8.254768440674015, 8.682145773987507, 9.626338882787104, 9.881209164096606, 11.06292867826007, 11.19792777156100, 11.82536608120132, 12.32394438444584, 13.15972686337792, 13.75802097382361, 14.37778008619741, 14.71229708847080, 15.20475955736730, 15.89900368030127, 16.75157824621791
