| L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s + 2·7-s − 8-s + 9-s − 4·10-s + 12-s − 6·13-s − 2·14-s + 4·15-s + 16-s − 18-s + 4·19-s + 4·20-s + 2·21-s − 6·23-s − 24-s + 11·25-s + 6·26-s + 27-s + 2·28-s + 4·29-s − 4·30-s + 6·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.288·12-s − 1.66·13-s − 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s + 0.894·20-s + 0.436·21-s − 1.25·23-s − 0.204·24-s + 11/5·25-s + 1.17·26-s + 0.192·27-s + 0.377·28-s + 0.742·29-s − 0.730·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.293350815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293350815\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.522141410579076542625431061009, −8.623210362724267034247608731207, −7.84253913019476823710858358599, −7.10043852506497274232713842564, −6.15744346307609821396804060542, −5.34777681077972405320527300661, −4.49371969105134698855653078527, −2.75000607216763176095328952178, −2.25996527018388001879662209200, −1.24659789595230381621816494252,
1.24659789595230381621816494252, 2.25996527018388001879662209200, 2.75000607216763176095328952178, 4.49371969105134698855653078527, 5.34777681077972405320527300661, 6.15744346307609821396804060542, 7.10043852506497274232713842564, 7.84253913019476823710858358599, 8.623210362724267034247608731207, 9.522141410579076542625431061009
