| L(s) = 1 | − 7-s + 2·13-s − 2·19-s − 6·29-s − 4·31-s + 4·37-s + 9·41-s − 43-s − 3·47-s − 6·49-s − 6·53-s + 61-s + 13·67-s + 12·71-s − 16·73-s + 10·79-s − 12·83-s + 3·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.554·13-s − 0.458·19-s − 1.11·29-s − 0.718·31-s + 0.657·37-s + 1.40·41-s − 0.152·43-s − 0.437·47-s − 6/7·49-s − 0.824·53-s + 0.128·61-s + 1.58·67-s + 1.42·71-s − 1.87·73-s + 1.12·79-s − 1.31·83-s + 0.317·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.618227578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.618227578\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.59586102313373, −13.04142507775726, −12.79071563686875, −12.39686382357325, −11.55521913015222, −11.18861623768021, −10.91436187232552, −10.18494305247497, −9.700518865898222, −9.261879459094908, −8.803361378658161, −8.145201956881049, −7.754597140423222, −7.140576794986240, −6.604591699355530, −6.046349193023473, −5.685832967400426, −4.957426479310277, −4.401030799550378, −3.722023378150122, −3.378298553779763, −2.561360422249508, −1.981258360947908, −1.255801234649464, −0.4051064746579640,
0.4051064746579640, 1.255801234649464, 1.981258360947908, 2.561360422249508, 3.378298553779763, 3.722023378150122, 4.401030799550378, 4.957426479310277, 5.685832967400426, 6.046349193023473, 6.604591699355530, 7.140576794986240, 7.754597140423222, 8.145201956881049, 8.803361378658161, 9.261879459094908, 9.700518865898222, 10.18494305247497, 10.91436187232552, 11.18861623768021, 11.55521913015222, 12.39686382357325, 12.79071563686875, 13.04142507775726, 13.59586102313373
