| L(s) = 1 | + 5-s − 2·11-s − 2·13-s − 7·19-s + 3·23-s − 4·25-s + 8·29-s + 4·31-s − 6·37-s − 12·41-s + 8·43-s + 8·47-s − 4·53-s − 2·55-s − 4·59-s − 13·61-s − 2·65-s + 2·67-s − 5·71-s + 14·73-s + 11·79-s − 12·83-s + 14·89-s − 7·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s − 0.554·13-s − 1.60·19-s + 0.625·23-s − 4/5·25-s + 1.48·29-s + 0.718·31-s − 0.986·37-s − 1.87·41-s + 1.21·43-s + 1.16·47-s − 0.549·53-s − 0.269·55-s − 0.520·59-s − 1.66·61-s − 0.248·65-s + 0.244·67-s − 0.593·71-s + 1.63·73-s + 1.23·79-s − 1.31·83-s + 1.48·89-s − 0.718·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.480052016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480052016\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 5 T + p T^{2} \) | 1.71.f |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−14.07399276070566, −13.77730966891883, −13.35771205343368, −12.63738531802522, −12.29078147683022, −11.91080272828329, −11.03408650906852, −10.64869479314351, −10.22189812792750, −9.771069009065454, −9.054172418539788, −8.618333881097759, −8.104924189694824, −7.505674861172631, −6.916262111861500, −6.280212382799094, −5.995570777805458, −4.994027262406106, −4.872147459714783, −4.092339672174537, −3.353566189124304, −2.606937899136915, −2.190208213309803, −1.407196963344600, −0.4018848851601407,
0.4018848851601407, 1.407196963344600, 2.190208213309803, 2.606937899136915, 3.353566189124304, 4.092339672174537, 4.872147459714783, 4.994027262406106, 5.995570777805458, 6.280212382799094, 6.916262111861500, 7.505674861172631, 8.104924189694824, 8.618333881097759, 9.054172418539788, 9.771069009065454, 10.22189812792750, 10.64869479314351, 11.03408650906852, 11.91080272828329, 12.29078147683022, 12.63738531802522, 13.35771205343368, 13.77730966891883, 14.07399276070566
