| L(s) = 1 | + 5-s + 11-s − 2·13-s − 2·17-s + 4·19-s + 25-s + 2·29-s − 2·37-s − 6·41-s − 8·47-s + 2·53-s + 55-s − 4·59-s + 10·61-s − 2·65-s − 4·67-s − 14·73-s − 16·79-s − 2·85-s + 10·89-s + 4·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s − 0.328·37-s − 0.937·41-s − 1.16·47-s + 0.274·53-s + 0.134·55-s − 0.520·59-s + 1.28·61-s − 0.248·65-s − 0.488·67-s − 1.63·73-s − 1.80·79-s − 0.216·85-s + 1.05·89-s + 0.410·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.79504010561702, −12.09199780511862, −11.75492710598393, −11.52906971587350, −10.77472560107484, −10.44753450820133, −9.828138746601450, −9.697275401955715, −9.060514215288688, −8.627720985657865, −8.233592930184132, −7.563155817970910, −7.162184680052513, −6.735092745146894, −6.245300913223145, −5.710050982588885, −5.203917802331881, −4.804569235022949, −4.260428820241410, −3.672446214169037, −3.010484135553454, −2.721305775845087, −1.866036103677326, −1.550743275990736, −0.7574483744374510, 0,
0.7574483744374510, 1.550743275990736, 1.866036103677326, 2.721305775845087, 3.010484135553454, 3.672446214169037, 4.260428820241410, 4.804569235022949, 5.203917802331881, 5.710050982588885, 6.245300913223145, 6.735092745146894, 7.162184680052513, 7.563155817970910, 8.233592930184132, 8.627720985657865, 9.060514215288688, 9.697275401955715, 9.828138746601450, 10.44753450820133, 10.77472560107484, 11.52906971587350, 11.75492710598393, 12.09199780511862, 12.79504010561702
