| L(s) = 1 | + 5-s + 4·11-s + 6·13-s + 6·17-s + 4·19-s + 4·23-s + 25-s + 2·29-s − 8·31-s + 6·37-s + 6·41-s + 8·43-s − 6·53-s + 4·55-s − 4·59-s − 10·61-s + 6·65-s + 8·67-s − 12·71-s + 14·73-s − 16·79-s − 12·83-s + 6·85-s + 14·89-s + 4·95-s − 18·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s + 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.824·53-s + 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.744·65-s + 0.977·67-s − 1.42·71-s + 1.63·73-s − 1.80·79-s − 1.31·83-s + 0.650·85-s + 1.48·89-s + 0.410·95-s − 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.711399392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.711399392\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.02193761180334, −13.84673979509869, −13.09344647057665, −12.64164623543400, −12.22596727333570, −11.48180509915474, −11.15784667149881, −10.73532855163740, −10.00639760301007, −9.452963160021651, −9.127731019019438, −8.675254430523961, −7.890491248483117, −7.488441419062770, −6.845652388601899, −6.192811351392173, −5.809365499447307, −5.402105706946502, −4.490019746064206, −3.970772647684121, −3.287303592890146, −2.969207020245445, −1.830339260113409, −1.210532849303166, −0.8639689070590989,
0.8639689070590989, 1.210532849303166, 1.830339260113409, 2.969207020245445, 3.287303592890146, 3.970772647684121, 4.490019746064206, 5.402105706946502, 5.809365499447307, 6.192811351392173, 6.845652388601899, 7.488441419062770, 7.890491248483117, 8.675254430523961, 9.127731019019438, 9.452963160021651, 10.00639760301007, 10.73532855163740, 11.15784667149881, 11.48180509915474, 12.22596727333570, 12.64164623543400, 13.09344647057665, 13.84673979509869, 14.02193761180334
