| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 2·11-s − 2·12-s + 2·13-s + 16-s + 17-s + 18-s − 2·22-s + 4·23-s − 2·24-s − 5·25-s + 2·26-s + 4·27-s + 4·29-s + 32-s + 4·33-s + 34-s + 36-s + 8·37-s − 4·39-s + 2·41-s − 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.603·11-s − 0.577·12-s + 0.554·13-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.426·22-s + 0.834·23-s − 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s + 0.742·29-s + 0.176·32-s + 0.696·33-s + 0.171·34-s + 1/6·36-s + 1.31·37-s − 0.640·39-s + 0.312·41-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.697304605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.697304605\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−9.565973118904632348522417349792, −8.418803126683971073229362251105, −7.60714990886078355619416842934, −6.62571407180722531145707218998, −6.00776466444475617807558833728, −5.30864553466560467506407844365, −4.60637393914019774217709833201, −3.53705408459719078408753256035, −2.41873000952486740035473257334, −0.870161894655473165678233970356,
0.870161894655473165678233970356, 2.41873000952486740035473257334, 3.53705408459719078408753256035, 4.60637393914019774217709833201, 5.30864553466560467506407844365, 6.00776466444475617807558833728, 6.62571407180722531145707218998, 7.60714990886078355619416842934, 8.418803126683971073229362251105, 9.565973118904632348522417349792
