| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 4·11-s + 4·13-s − 15-s + 4·17-s + 8·19-s + 2·21-s − 8·23-s + 25-s − 27-s + 6·29-s + 31-s − 4·33-s − 2·35-s + 8·37-s − 4·39-s − 2·41-s + 4·43-s + 45-s − 2·47-s − 3·49-s − 4·51-s + 12·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.258·15-s + 0.970·17-s + 1.83·19-s + 0.436·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.179·31-s − 0.696·33-s − 0.338·35-s + 1.31·37-s − 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.291·47-s − 3/7·49-s − 0.560·51-s + 1.64·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.198474781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.198474781\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 31 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−7.81486411756735342179854836982, −7.08537468783503144795199186470, −6.18237263743993004254509366209, −6.08162630759581180074340589914, −5.27406389264556173936922335613, −4.22437732306284281013036341050, −3.60912770364517003693509228352, −2.82603757389062857706354196996, −1.48212619673198409863892561510, −0.863831055934298594474597807553,
0.863831055934298594474597807553, 1.48212619673198409863892561510, 2.82603757389062857706354196996, 3.60912770364517003693509228352, 4.22437732306284281013036341050, 5.27406389264556173936922335613, 6.08162630759581180074340589914, 6.18237263743993004254509366209, 7.08537468783503144795199186470, 7.81486411756735342179854836982
