| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 6·13-s − 15-s + 6·19-s − 2·21-s − 8·23-s + 25-s + 27-s + 4·29-s + 4·31-s + 2·35-s + 6·37-s − 6·39-s − 8·41-s − 10·43-s − 45-s − 3·49-s + 6·53-s + 6·57-s − 4·59-s − 14·61-s − 2·63-s + 6·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s + 1.37·19-s − 0.436·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 0.718·31-s + 0.338·35-s + 0.986·37-s − 0.960·39-s − 1.24·41-s − 1.52·43-s − 0.149·45-s − 3/7·49-s + 0.824·53-s + 0.794·57-s − 0.520·59-s − 1.79·61-s − 0.251·63-s + 0.744·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9192692007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9192692007\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 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.ao |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.71309058127970, −13.18197860485666, −12.52878084718771, −12.10140380067483, −11.87596740872973, −11.34632841550303, −10.43003081510149, −9.982322882849506, −9.816732233437238, −9.314529853946660, −8.641561144609341, −8.012616746131347, −7.768495527731914, −7.191763705979066, −6.670178592209112, −6.188648768042378, −5.428619146028511, −4.845663119353249, −4.413238492169977, −3.700367069431174, −3.113875463757592, −2.759370002587108, −2.051310961494372, −1.276242931012099, −0.2830294727594810,
0.2830294727594810, 1.276242931012099, 2.051310961494372, 2.759370002587108, 3.113875463757592, 3.700367069431174, 4.413238492169977, 4.845663119353249, 5.428619146028511, 6.188648768042378, 6.670178592209112, 7.191763705979066, 7.768495527731914, 8.012616746131347, 8.641561144609341, 9.314529853946660, 9.816732233437238, 9.982322882849506, 10.43003081510149, 11.34632841550303, 11.87596740872973, 12.10140380067483, 12.52878084718771, 13.18197860485666, 13.71309058127970
