| L(s) = 1 | + 4·5-s + 4·13-s + 2·17-s + 11·25-s + 10·29-s − 2·37-s − 10·41-s − 7·49-s + 4·53-s + 12·61-s + 16·65-s − 16·73-s + 8·85-s − 16·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.10·13-s + 0.485·17-s + 11/5·25-s + 1.85·29-s − 0.328·37-s − 1.56·41-s − 49-s + 0.549·53-s + 1.53·61-s + 1.98·65-s − 1.87·73-s + 0.867·85-s − 1.69·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.375963859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.375963859\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−14.81761866817988, −14.24039972263513, −13.88468444707116, −13.49096250809614, −12.93976963393383, −12.52541983344083, −11.71070363354975, −11.28460491895558, −10.38467438048999, −10.20158528241625, −9.807138457634126, −8.961041371578039, −8.636467476087455, −8.125307685563215, −7.102387776402051, −6.643481363238015, −6.100452712357377, −5.657374991006647, −5.041199528177119, −4.444711943461854, −3.421851009863840, −2.943974722199484, −2.093177145463434, −1.505872339656363, −0.8260385518378512,
0.8260385518378512, 1.505872339656363, 2.093177145463434, 2.943974722199484, 3.421851009863840, 4.444711943461854, 5.041199528177119, 5.657374991006647, 6.100452712357377, 6.643481363238015, 7.102387776402051, 8.125307685563215, 8.636467476087455, 8.961041371578039, 9.807138457634126, 10.20158528241625, 10.38467438048999, 11.28460491895558, 11.71070363354975, 12.52541983344083, 12.93976963393383, 13.49096250809614, 13.88468444707116, 14.24039972263513, 14.81761866817988
