| L(s) = 1 | − 4·7-s + 4·13-s − 6·17-s − 19-s + 6·23-s − 5·25-s + 6·29-s + 2·31-s + 4·37-s − 6·41-s + 4·43-s − 6·47-s + 9·49-s + 6·53-s − 12·59-s − 14·61-s − 8·67-s + 14·73-s − 10·79-s − 12·83-s + 6·89-s − 16·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.10·13-s − 1.45·17-s − 0.229·19-s + 1.25·23-s − 25-s + 1.11·29-s + 0.359·31-s + 0.657·37-s − 0.937·41-s + 0.609·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s − 1.56·59-s − 1.79·61-s − 0.977·67-s + 1.63·73-s − 1.12·79-s − 1.31·83-s + 0.635·89-s − 1.67·91-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
| \(L(1)\) |
\(\approx\) |
\(1.259832705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259832705\) |
| \(L(\frac{3}{2})\) |
|
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.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.51762817955516, −15.83144338314339, −15.42417625172976, −15.13462081741881, −13.90094987171804, −13.67202363391693, −13.06005796677550, −12.67843627315619, −11.91741686820867, −11.22181376510869, −10.71886552696123, −10.07011007362676, −9.423649518208472, −8.888943528248260, −8.397356028623453, −7.465595932886789, −6.702949266102787, −6.339719705027767, −5.811344655873985, −4.713277245336622, −4.134255921807142, −3.245347680473402, −2.787946276182461, −1.676293147180452, −0.5143389851485300,
0.5143389851485300, 1.676293147180452, 2.787946276182461, 3.245347680473402, 4.134255921807142, 4.713277245336622, 5.811344655873985, 6.339719705027767, 6.702949266102787, 7.465595932886789, 8.397356028623453, 8.888943528248260, 9.423649518208472, 10.07011007362676, 10.71886552696123, 11.22181376510869, 11.91741686820867, 12.67843627315619, 13.06005796677550, 13.67202363391693, 13.90094987171804, 15.13462081741881, 15.42417625172976, 15.83144338314339, 16.51762817955516
