| L(s) = 1 | + 2-s − 3-s + 4-s − 3·5-s − 6-s + 2·7-s + 8-s + 9-s − 3·10-s + 3·11-s − 12-s + 2·14-s + 3·15-s + 16-s + 18-s + 4·19-s − 3·20-s − 2·21-s + 3·22-s + 9·23-s − 24-s + 4·25-s − 27-s + 2·28-s + 9·29-s + 3·30-s + 2·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.904·11-s − 0.288·12-s + 0.534·14-s + 0.774·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.670·20-s − 0.436·21-s + 0.639·22-s + 1.87·23-s − 0.204·24-s + 4/5·25-s − 0.192·27-s + 0.377·28-s + 1.67·29-s + 0.547·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.749124408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.749124408\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−12.58916158407783, −12.13357985199773, −11.75109935763919, −11.37341542282384, −11.26449631526560, −10.63740718912926, −10.10228967151068, −9.534719979632309, −8.952168227477622, −8.298794707085252, −8.116143254759310, −7.460435549374511, −6.938731568187334, −6.733885398664763, −6.143912972099918, −5.401380369750984, −4.940858527276302, −4.626918585600804, −4.222471356768807, −3.472863062974378, −3.213971551643723, −2.560399080606409, −1.533015650792086, −1.169667145573769, −0.5316165079130357,
0.5316165079130357, 1.169667145573769, 1.533015650792086, 2.560399080606409, 3.213971551643723, 3.472863062974378, 4.222471356768807, 4.626918585600804, 4.940858527276302, 5.401380369750984, 6.143912972099918, 6.733885398664763, 6.938731568187334, 7.460435549374511, 8.116143254759310, 8.298794707085252, 8.952168227477622, 9.534719979632309, 10.10228967151068, 10.63740718912926, 11.26449631526560, 11.37341542282384, 11.75109935763919, 12.13357985199773, 12.58916158407783
