| L(s) = 1 | + 2·7-s + 3·11-s + 13-s + 6·17-s − 2·19-s + 3·23-s − 6·29-s + 4·31-s + 7·37-s + 2·43-s − 3·47-s − 3·49-s − 6·53-s + 15·59-s + 5·61-s + 2·67-s + 9·71-s − 2·73-s + 6·77-s + 10·79-s + 12·83-s − 18·89-s + 2·91-s − 17·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.904·11-s + 0.277·13-s + 1.45·17-s − 0.458·19-s + 0.625·23-s − 1.11·29-s + 0.718·31-s + 1.15·37-s + 0.304·43-s − 0.437·47-s − 3/7·49-s − 0.824·53-s + 1.95·59-s + 0.640·61-s + 0.244·67-s + 1.06·71-s − 0.234·73-s + 0.683·77-s + 1.12·79-s + 1.31·83-s − 1.90·89-s + 0.209·91-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.876097508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.876097508\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−16.60795147365727, −16.08324014353100, −15.20322716121941, −14.71237664242957, −14.43036610830876, −13.75472095913605, −13.05184025615647, −12.48741916600150, −11.86626777466587, −11.22725053688923, −10.96637313511488, −9.917679086904427, −9.614663025389597, −8.805143446390608, −8.165122243227800, −7.711209589724688, −6.896397549207876, −6.272004735664753, −5.517640149347436, −4.924278833044458, −4.062298968280993, −3.515091214114256, −2.531887676212457, −1.563640041837919, −0.8639172228251283,
0.8639172228251283, 1.563640041837919, 2.531887676212457, 3.515091214114256, 4.062298968280993, 4.924278833044458, 5.517640149347436, 6.272004735664753, 6.896397549207876, 7.711209589724688, 8.165122243227800, 8.805143446390608, 9.614663025389597, 9.917679086904427, 10.96637313511488, 11.22725053688923, 11.86626777466587, 12.48741916600150, 13.05184025615647, 13.75472095913605, 14.43036610830876, 14.71237664242957, 15.20322716121941, 16.08324014353100, 16.60795147365727
