| L(s) = 1 | + 3·7-s + 3·11-s + 4·13-s − 17-s + 19-s + 2·23-s − 5·29-s + 4·31-s + 5·37-s + 5·41-s + 2·43-s − 3·47-s + 2·49-s − 9·53-s + 4·59-s − 6·61-s − 14·67-s − 8·71-s − 11·73-s + 9·77-s − 12·79-s + 4·83-s − 12·89-s + 12·91-s − 6·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.13·7-s + 0.904·11-s + 1.10·13-s − 0.242·17-s + 0.229·19-s + 0.417·23-s − 0.928·29-s + 0.718·31-s + 0.821·37-s + 0.780·41-s + 0.304·43-s − 0.437·47-s + 2/7·49-s − 1.23·53-s + 0.520·59-s − 0.768·61-s − 1.71·67-s − 0.949·71-s − 1.28·73-s + 1.02·77-s − 1.35·79-s + 0.439·83-s − 1.27·89-s + 1.25·91-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.17867804979920, −12.71757915666830, −11.97723372685179, −11.67574317844527, −11.27756878267971, −10.88305394505859, −10.51795856138397, −9.697217857447790, −9.414100667423758, −8.764463713451177, −8.534421573882556, −7.966572293732734, −7.433337400145418, −7.081979549233699, −6.278111185190314, −6.010392569179652, −5.533022415086488, −4.687026919374861, −4.472458834645872, −3.947390032060561, −3.263932789912261, −2.768338932730122, −1.925082954929641, −1.367038637931201, −1.088820092426510, 0,
1.088820092426510, 1.367038637931201, 1.925082954929641, 2.768338932730122, 3.263932789912261, 3.947390032060561, 4.472458834645872, 4.687026919374861, 5.533022415086488, 6.010392569179652, 6.278111185190314, 7.081979549233699, 7.433337400145418, 7.966572293732734, 8.534421573882556, 8.764463713451177, 9.414100667423758, 9.697217857447790, 10.51795856138397, 10.88305394505859, 11.27756878267971, 11.67574317844527, 11.97723372685179, 12.71757915666830, 13.17867804979920
